On the Characters and the Plancherel Formula of Nilpotent Groups ...

JOURNAL OF FUNCTIONAL ANALYSIS 1, 255-280 (1967) 

On the Characters and the Plancherel Formula 

of Nilpotent Groups* 

L. PUKANSZKY 

Department of Mathematics, University of Penmyvlvania, 

Philadelphia, Pennsylvania 19104 

Communicated by Irving E. Segal 

Received April 13, 1967 

1. Let G be a connected and simply connected nilpotent group 

with the Lie algebra 2’. We denote by T an irreducible unitary 

representation of G on a Hilbert space. Let da be a left-invariant 

Haar measure on G, and v a C” function of compact support. Then 

the operator defined by 

T, = s da) T(a) da 

G 

is of the trace class (cf. [4], p. 108), and we have Eq. (1) below for 

its trace in terms of v. We denote by 9’ the dual of the underlying 

space of 9, and by (8, d’) the canonical bilinear form on 9 x 9’. 

Let us form the Fourier transform of the C” function of compact 

support defined on 2 by ~(6’) = v (exp k’) (8 E Z), through the formula 

q(8) = I9 y(4) ei(cJ’) d8 (t” E Z’), 

dl denoting a positive translation-invariant measure on 9. Then we 

have 

Tr (T,) = Jo $5(P) dw. (1) 

Here 0 stands for an orbit of the representation, contragredient 

to the adjoint representation, of G on Y, and dv denotes an invariant 

measure on 0. The orbit 0 is well-determined by the equivalence 

class of T and conversely, given any orbit in 9’, there exists a unique 

equivalence class of irreducible representations of G, which corre- 

* Prepared with partial support from Nonr-551 (57). 

@J 1967 by Academic Press Inc. 

580/1/3-I 

255

256 PUKANSZKY 

sponds to it by virtue of the formula (1) (cf. Theo&me, p. 145in[4]). 

The measures da and dr! having been fixed, the measure dv is uniquely 

determined on each orbit; it will be referred to as the canonical 

measure in the sequel. Up to now, no direct characterization of the 

canonical measure has been known; this question is of interest, in 

particular, for the derivation of the Plancherel formula for G (cf. [4], 

Chap. III, Sections 4-6). 

Let us consider now an arbitrary connected Lie group G with the 

Lie algebra 9’. If 0 is an orbit of the kind considered above, and of a 

positive dimension, in 9’, we can assign to it an invariant measure as 

follows. We write 0 for the adjoint representation of G, and p for the 

representation, contragredient to o. Choosing an arbitrary element p 

of 0, let us consider the map CX~ from G onto 0 defined by 

a,(a) = p(a)p (a E G). Its differential vp = da+, is a map of 5’ onto 

the tangent space TP of 0 at p, and its kernel, identifiable to the Lie 

algebra of the stable group of p, coincides with the radical of the skewsymmetric 

bilinear form B,(tr ,8s) = ([/i , es], p) on 9 x 3’. Therefore 

there exist a well-determined nondegenerate skew-symmetric 

bilinear form wp on TP x TP , such that A,(,) = BP . Varyingp on 0 

we obtain in this fashion a 2-form w, which is invariant under the 

action of G. In fact, writing s(a)x = axa-l (a, x E G) and q = p(a)p 

(p E 0) we have mp o s(a) = p(a) o ap whence, by taking differentials, 

it follows that CJ.I* o u(a) = dp(a) Ip o vp . But then, setting 

Ma) lp up = up , we obtain for all &, and tZ in 9: 

4%(6)7 %(a = %M44 4), 

and therefore w; = op , p roving the statement. Let us denote by d 

the dimension of 0. Since this is necessarily an even number, we can 

form the exterior power (w)~/~ of w. We shall call the positive invariant 

measure, corresponding to it on 0, the K-measure of 0. 

The author is indebted to B. Kostant for the conjecture that, in the 

nilpotent case, the canonical measure and the K-measure are essen- 

tially the same. This is strongly suggested by his construction 

(unpublished), making possible, in particular, to set up a one-to-one 

correspondence between certain orbits and the set of all equivalence 

classes of irreducible representations of a compact semisimple group 

(cf. Section 2 for more details), together with the observation, that the 

ratio of the (necessarily finite) total K-volume of an orbit and of the 

dimension of the corresponding irreducible representation is a constant 

depending only on the group.

PLANCHEREL FORMULA OF NILPOTENT GROUPS 257 

The main purpose of the present paper is to verify this conjecture 

for the nilpotent groups (cf. Section 3), though its domain of validity 

seems to include all cases, where a formula of the type (1) can be set 

up at all. We shall return to the discussion of the case of the expo- 

nential groups1 in a further publication. 

The plan of this paper is as follows. Though not directly related to 

the main result, because of its great heuristic value, we thought it 

useful to begin with a self-contained discussion of the situation, 

just quoted, offered by a compact semisimple Lie group. The sole 

purpose of Section 2 is to place the subsequent considerations con- 

cerning the nilpotent case in a more general context. We show, how 

the l-l correspondence, pointed out by Kostant, between the set of 

all equivalence classes of irreducible representations of a connected 

and simply connected compact semisimple Lie group G and certain 

orbits in the corresponding algebra 9 (identified now with its dual 

through the Killing form) can be established, just as in the nilpotent 

case, by aid of a formula like (1). The only difference comes from the 

fact that, as suggested by Kirillov in the case of SU(2) (cf. [3], Section 

5 in $S), prior to forming the Fourier transform of the function v on 

9, we have to multiply it by a certain expression, depending only 

on $4. All this follows easily from the Weyl character formula along 

with a theorem of Harish-Chandra (cf. [2], Theorem 2) implying a 

simple relationship between the character of a given representation 

and the Fourier transform of a positive measure on 9, which is 

invariant with respect to the adjoint representation and is concentrated 

on the corresponding orbit. Let us observe, incidentally, that in the 

nilpotent case, the Fourier transform (formed with 9’), which has 

just been referred to, does not admit any manageable expression on 

9. In Section 3 we prove the main result, according to which for 

niipotent groups the ratio of the canonical measure and of the K- 

measure of an orbit depends only on the dimension of the latter. 

Taking into account the picture found in the compact semisimple case 

in Section 2, one can interpret formula (l), along with the precise 

expression, obtained in Section 3, for the canonical measure, as the 

analog for nilpotent groups of the Weyl character formula. Finally, 

in Section 4 we apply this result to obtain an explicit algorithm for the 

computation of the Plancherel measure. In fact, to obtain this, by 

virtue of Section 3 it suffices to represent a translation-invariant 

measure on 2’ as a continuous sum of the K-measures of the orbits 

1 A connected and simply connected solvable Lie group is called exponential, if the 

exponential map is onto.

258 PUKANSZKY 

of the representation p. We show, that the result implies the theorem 

of J. Dixmier ([4], p. 171), according to which the Plancherel measure 

is representable by a rational differential on the quotient space, 

according to p, of the union of orbits of maximal dimension in 3”. 

There is good reason to believe, that the same algorithm remains in 

force for exponential unimodular groups too. 

As indicated above, the reading of Section 2 is not necessary for the 

understanding of the main results of this paper. 

2. As mentioned above, the purpose of this Section is to discuss, 

following the main idea of the presentation of Kirillov’s theory in [4], 

Part II, Chap. II, Sections 6-8, certain facts concerning irreducible 

representations of compact semisimple groups, the main objective 

being the demonstration of the constancy of the ratio of the dimension 

of an irreducible representation and of the total K-volume of the 

corresponding orbit. 

In what follows throughout this Section, 9 will denote a fixed 

compact semisimple Lie algebra, and b a fixed Cartan subalgebra of 9. 

Let P be the collection of all nonzero roots of 9 with respect to $ 

We assume the elements of P under the form ior (h E $), where i 

is a fixed square root of - 1. Writing for x and y in 

2 : (x, y) = - Tr (a& ady), f or any root (y. there exists a welldetermined 

element ~1’ in b, such that or(h) = (h, a’) (h E a); in the 

sequel OL’, too, will be referred to as a root, and we omit the prime. Let 

(aj;j = 1, 2,..., r} (r = dim) be a system of simple roots and P, 

the collection of the corresponding positive roots. We write 

hj = 201,/(01, 3 aj) (j = 1, 2 ,..., Y). Then, as it is known (cf. [I], 

Exposes 17-21), there is a l-l correspondence between the set of all 

equivalence classes of irreducible representations of the connected and 

simply connected group G determined by 9, and the set II of all 

elements X of 0, for which (h, h,) ( j = 1, 2,..., r) is nonnegative and 

integral. We denote by p the half sum of all positive roots, and by W 

the Weyl group corresponding to IJ; for s in W and h in 8, sh will stand 

for the action of s on h. Then the function xI on Q, which corresponds 

to the restriction to exp t, of the character of an irreducible representation 

of class X (h E /I) is given by 

where 

xr(h) =$#, 

QG) = &44 exp W, P + 41

and 


W) = J (exp [i t (h, 41 - exp [-- i S Vb 41) 

= -$N exp [W, P)l [E(S) = det (s) for s in w]. 

In what follows, &and dh will denote volume elements on 9 and E, 

respectively, corresponding to the Euclidean metric determined by 

the negative Killing form. We denote again by o(a) the adjoint repre- 

sentation of G and by da the element of the Haar measure on G, 

such that J da = 1. We put r(h) = (z]” nNEP al(h) (h E b), where m 

is the number of positive roots. Then ([2], 6. 105) there exists a 

constant Co , such that we have for all functions f, which are con- 

tinuous and of a compact support on 9: 

We shall also use the following formula, due to Harish-Chandra 

(cf. [2], Theorem 2, p. 104)2: 

p(h) T( - h’) 1, exp (a(a) A, h’) da = 4 (& 4~) exp (% W) , 

valid for any two elements h and h’ in 0; d, stands for neEp+ (OL, p). 

Given any function, invariant with respect to the Weyl group, on 0, 

we denote with the same letter the unique u-invariant function on 9 

determined by it. We write u(e) for D(e>/+) (6’ E 9) and recall, that in 

a neighborhood of the neutral element in 9, where the exponential 

mapping is l-l, 1 a(e) I2 &is th e ex p ression for a Haar measure on G. 

Finally, for any A in A we write O(h) for the orbit, with respect to the 

adjoint representation, of X + p. 

With these notations, we have the following 

PROPOSITION 1. For any h in A and f E C” of a compact support 

on 9 we have 

Here dv is an invuriunt measure on O(h); fi(6’) s f (e) FQ and 

f(F) = f9f(t’) exp [+!, G’)] dt! (6” E 9). 

s Substitute -H for the H used in [Zj’, and observe that

260 PUKANSZKY 

Moreover, the orbit O(h) and the measure dv are uniquely determined 

kY (2). 

Proof. We start by observing, that 

xX4 I 44 I2 = QM W) I 44 F2, 

and therefore, we have for the left-hand side of (2) 

where we have put 

On the other hand, writing dA for the volume of O(A) with respect to 

dv, we have for any g(e) continuous on 9, putting x = X + p, 

and therefore the right-hand side of (2) gives 

where 

= dn s afV> W) dl 

2 

G(4) = j, exp [;(~(a) k’, A)] da. 

Since this function is clearly o-invariant, we can conclude that 

Summing up, to establish the equality appearing in the Proposition, 

it suffices to show, that Q,(h) ZE d&h) G(h) (h E b), or that 

dp(h) j, exp [~(u(cz) h, A)] da = C E(S) exp [i(sh, A)]. 

SSW


But for this it suffices to substitute iA in place of h’ in the formula of 

Harish-Chandra, and to choose for dA 

Observe, that this is the dimension of the representation, correspond- 

ing to A in A. 

To prove, that the orbit O(A) in the right-hand side of (2) is uniquely 

determined, since a(O) = 1, it is enough to show, that if 0, and 0, 

are two orbits of u and dv, and dv, are nontrivial invariant measures 

on them, and if we have 

for all C” functions, vanishing outside a given neighborhood of the 

neutral element in 9, then 0, = 0, and dv, = dv, . Let L be a 

translation-invariant differential operator on 9. Replacing f first by 

Lf, then by an approximate identity in the above equation, we conclude, 

that 

for any polynomial function P(e) on $4. If 0, and 0, are different, 

then, since they are compact and without common point, by an 

appropriate choice of P, we can arrange that the left-hand side should 

be arbitrarily small, while the right-hand side remains close to the 

volume of 0, with respect to dv, which, according to assumption, 

is nonzero. Hence 0, = 0,; but then dv, and dv, , too, coincide, and 

this finishes the proof of Proposition 1. 

Summing up once more, Proposition 1 establishes a l-l correspond- 

ence between the family of orbits, determined by elements of the form 

X + p (A E A), in 9 and the set of all equivalence classes of irreducible 

representations of G in the following fashion: The character xA(h) 

equals D(h)/?r(h) t imes the Fourier transform of the positive, o-inva- 

riant measure, concentrated on the orbit of h + p, in 9, the total 

mass of which equals dA = x,,(O). 0 ur next Proposition shows, that 

the K-volume (cf. Section 1) and dA are proportional. 

PROPOSITION 2. For a $xed h in A, Zet us denote by V(K) the

262 PUKANSZKY 

K-volume of the orbit of h + p. Then, with the notations used above, 

we have 

In particular, 

does not depend on A. 

Proof. As it is known, we can choose an orthonormal basis 

(E, , Ei; CY, /3 E P+> in the orthogonal complement l+- of IJ in 2 in such 

a fashion, that adhE, = or(h) EL and adhE: = - al(h) E, for all h in lo. 

Next we observe, that the Lie algebra of the stabilizer of X + p in Osp 

coincides with Q. To this end we remark first, that the subalgebra of 2, 

which has just been referred to, coincides with the centralizer of the 

element x = X + p, of lo, in S?. On the other hand, if 

is some element of -Ee, we have 

But since (01, p) is positive for all (Y in P+ (cf. [I], Lemma 2 in Expose 

19), and since X is in II, a(X) is always positive. Therefore, if 6’ is in the 

centralizer of X, we must have c, = CL = 0 for all CX, and thus 6’belongs 

to TJ. We are going to compute the K-volume of the orbit of X + p, 

to be denoted again by O(X), in two steps. First we determine the 

volume V(E) of O(X) with respect to the metric induced on O(h) by 

the Euclidean metric of 2, and then we compute the ratio V(K)/V(E). 

As regards the first point, we start by observing, that the tangent space 

to O(X) at A, since it is spanned by the vectors {a&A; I E ,Ep), by virtue 

of what we saw above, coincides with bl. Therefore, since c(a) is an 

orthogonal transformation of .P into itself for any a in G, at the 

point o(a) x of O(h) the orthogonal complement of the tangent space 

can be identified with u(a) $. Let us set again I = dim $. For a 

positive E, we denote by S, the r-dimensional solid sphere of 

radius e in 6, and we write v(e) for its volume. Also, we write

PLANCHERRL FORMULA OF NILPOTRNT GROUPS 263 

f&‘) for the characteristic function of the set uafc u(a) (A + S,) 

in 14. Then we have the following relation: 

in other words, we have V(E) = C, naEp+ (a, X + p)“. Let us denote 

by d the dimension of O(A); we have d = 2m. Next we observe, that 

V(K)/V(E) is the same as the ratio of the two d-forms, representing 

the volume element in the K-metric (cf. Section 1) and in the Euclidean 

metric of O(X), at A, say. We denote by {ej; j = 1,2,..., d} the 

vectors {B. , EL; OL E P+} in some order. Then, by virtue of a known 

formula, the coefficient of I-$1 A ej in the (d/2)th power of 

equals 2” * m! times the square root of det (([ei , ej], A); i,j = I,..., d); 

one verifies by an easy computation, that the latter is the same as 

naEp+ (cw, h + p). On the other hand, the factor of I-&, A ei in the d 

form, corresponding to the Euclidean volume element of O(A), at 

x is equal to the square root of det ((adei x, adei 1); i,j = l,..., d); 

one verifies easily, that this is the same as 1 ~(1) 12. Therefore the 

ratio V(K)/V(E) is given by 2” * m! nssp+ (01, h + p)/l r(X) 12, and 

thus finally we obtain that V(K) = C,, 2” * m! nmEP+ (LU, X + p), 

finishing the proof of Proposition 2. 

We close this section with the following observation. Let q(a) be a 

C” function which vanishes outside a neighborhood, where canonical 

coordinates of the first kind can be introduced, of the unity of G. 

We write ~(4) for the corresponding function on G. For a h in A, we 

denote by T(“) an irreducible representation, of the class A, of G. Then, 

by virtue of the previous considerations, we have 

where dw. is the element of the K-measure for O(h), provided the 

Haar measure da on the left, and the volume element d/ (by aid of 

which the Fourier transform in the right-hand side is formed) of

264 PUKANSZKY 

2 are chosen in such a fashion, that the ratio da/d/ at the neutral 

element of 2 should be the same as V(K)/d, = C, 2” * m! nasP+ (a, p). 

We shall see in the next section that, in the nilpotent case the value 

of the analogously defined constant is 4” * m! rrm. 

3. In this Section we consider again a nilpotent Lie algebra P’ of 

dimension n > 0, with the corresponding connected and simply- 

connected group G. We denote by u the adjoint representation of G, 

and by p the representation, which is contragredient to u, on 2’. 

In other words, denoting by (/, el) the canonical bilinear form on 

2 x ZZ”, we have, for all a in G, 

(u(a) rt, P) = (l, p(a-‘) 8’) (t E Y, t’ E 9’). 

Let H be a subalgebra of 2, of a dimension h < n. We shall say, 

that the ordered (n - h)-tuple of elements {t; , /a ,..., &‘n-h} of $P is a 

supplementary basis of H, provided for each j = 1, 2,..., n - h the 

subspace, spanned by Hand (8, ,..., e,>, of 2 is a subalgebra of dimen- 

sion h + j. Let us denote by exp H the subgroup, which is the image 

of H through the exponential map, of G and let us set gj(t) = exp (J’+) 

(j = 1, 2,..., 12 - h). The map of exp H x Rn-h into G, which 

assigns to (h, (ti , t, ,. .., tn-h)) (h E exp H) the element 

of G, is a homeomorphism between exp H x Rn-h and G ([4J p. 96, 

Remarque 2), and dt, dt, *a+ dt,-, defines an invariant measure on the 

homogeneous space of right-classes of G according to exp H ([4J, 

p. 121, Remarque). 

An element c!; of 2’ having been fixed, we shall denote by B(lr ,&J 

the skew-symmetric bilinear form on 9 x 2 defined by ([/r , t.J, 8;) 

(/i , tz E 2). Given a subspace H of JZ’, we shall write (H)i for its 

orthogonal complement, with respect to B, in 2, and H-L for its 

orthogonal complement, with respect to the canonical bilinear form 

in 9’. 

We start with a new proof for a statement, announced first by 

Kirillov ([4], Theo&me on p. 50, and also the Remark below). 

LEMMA 1. Let & be a nonzero element in 9”, and let 0 be its orbit 

with respect to p; we set d = dim 0, and suppose d > 0. Let


be a Jordan-H6lder sequence in 9, and {ej; j = 1, 2,..., n} a basis in Y 

such that ej’ E S?&., - 9’; (j = 1, 2,..., n). Then there exist n polyno- 

mials {Pi; j = 1, 2,..., n} of the d variables {xk; k = 1, 2,..., d} and d 

indices 0

266 PUKANSZKY 

we obtain a system of polynomials {P,}, having the properties claimed 

in the Lemma. 

Since Q,(T) E (fj , p(g( T)) r$), to prove (2) it is clearly enough 

to show, that 

(ai > 0), if, for some k with jk > j, we have ak > 0. But in this case 

the left hand side can be written as B( fd, r$, where 4 is some element 

in sj and j, > j. But this is zero, since fc lies in FtPl C (9&. To 

establish (1) it suffices to prove that, for j = j, , 

((adf$” (adj,&“~-’ *** ((adfipe, , e;) = 0 

if uk > 1 or uk = 1 and uG > 0 for some & < k, and - 1 if uk = 1 

and a, = 0 for G < k. The second statement being implied by 

B(q , -7-J = %j , we consider only the first. In this case, since 

[9, ,Ep] C Pi-, , the left hand side has the form B(fk , GT1), where 

&i lies in Zj-r (j = jk). But as above, we can show, that this is zero, 

finishing thus the proof of the Lemma. 

Remark. Let G as above; we recall, that a continuous representa- 

tion U, of G, on a finite-dimensional vector space V is called unipotent, 

provided the differential dU of this representation contains nilpotent 

operators only. We wish to show, that in the previous Lemma 9 

and p can be replaced by V and U, respectively. In fact, considering 

the dual I” of V as an Abelian Lie algebra, and setting 

$8) = - [transpose of dU(r!)] (I E 5?), let us form the semidirect 

product 8 of dp and V’ by defining the bracket of c! in 9 and v’ in 

V’ as I v’. Then the dual V of the ideal V’ of 9’ can be identified 

with the quotient space of the dual of 9’ according to $P’, and the 

representation induced in it by dp, formed for 8, is identifiable with 

dU. Therefore, to obtain the desired parametrization for the orbit of 

an element V,, of V, with respect to U, it suffices to choose a Jordan- 

Holder sequence of 9, containing V’ and then to consider the param- 

etrization (in the sense of the previous Lemma) of the orbit, with 

respect to p for exp 8, of an element of 9, lying over v, , and finally 

to take the result mod 9’. 

Let again 8; be a nonzero element of the dual of 9’. We recall 

(cf. [q, pp. 153-154), that a subalgebra H, of 9, is said to be sub- 

ordinated to 8;) if this is perpendicular to the first derived algebra of 

H. In this case we can form a character x of K = exp H by setting 

x (exp h) = exp [i(h, Gi)] (h E H). Let us form the representation T,


induced by x, of G. We denote again by d the dimension of the orbit 

0 of 8; with respect to p. Then, as it is known ([4], p. 154), T is 

irreducible if and only if we have dim H = dim $P - + d. 

The content of the following lemma is a special case of known 

results quoted before. The purpose of the new proof given for it 

here is to obtain a suitable algorithm for the computation of the cano- 

nical measure (cf. the Remark directly following the proof). 

LEMMA 2. Let 8; a$xed element of 3”; we suppose that the dimension 

d of the corresponding orbit is positive. Assume that the subalgebra H, 

of dimension dim 9 - 4j d, is subordinated to & , and form the unitary 

representation T as above. Let z,h be a C” function of compact support on G. 

Then the operator T$ is of class Hilbert-Schmidt, and we have 

where 0 is the orbit of t$ , dv an invariant measure on 0, and $(el) 

the Fourier transform of the function v,(l)>, which corresponds on 9, through the 

exponential map, to #” x $ (#“(a) = [#(a-‘)]). The integral on the right- 

hand side converges absolutely. 

Proof. Let {8r , 4, ,..., 4,) be a supplementary basis of H in 3’; 

we write gj(t) = exp (t’&) (j = 1, 2,..., m; 2m = d). We set 

K = exp H, and using the observations made before Lemma 1, we 

identify the homogeneous space I’ of the right classes of G according 

to K with the closed subset {gl(tl) *a. gm(t,); (tI , t, ,..., tJ E Rm} of 

G. Then every element of G can uniquely be written in the form kr 

(K E K, y E r). In particular, for any y in r and a in G, we set 

ya = k(ya) yd (k(ra) E K, yd E r). Let dy be any positive invariant 

measure on r (dt, dt, *** dt,, say). Then, if x (exp h) = exp [i(h, &)I 

(h E H), the representation T can be realized in the Hilbert space 

L2(r, dr), such that T(a) f (y) = x(k(ya)) f (yd). Let us write 

v(a) = (#- x #) (a) (a E G). We divide the proof in several steps. 

(a) First, we show, that T, is implemented in L2(P, dr) by a 

kernel, thus leading to an expression of Tr (T,) in terms of CJJ. If 

f(y) is continuous, and of compact support on r, we have

268 PUKANSZKY 

We denote by dk the invariant measure on K, satisfying 

for any function g, which is continuous and of compact support, on G. 

Then the previous relation gives 

T,f(Y) = j,,, dr-lb'> xVWb4 dk 4 = j, K&Y, r')fW 4' 

where we have 

&h r’> = j, dPh4 x(4 dk. 

(a = PW), 

Since, on account of our choice of y, the operator T, is positive, 

we conclude, that the continuous function K, on r x r is positive- 

definite and therefore 

Tr V,> = j, K&J, r> 

4 (3) 

irrespective of whether the expressions on both sides of this equation 

are finite or not (cf. [4], pp. 116-l 19). 

(b) Let us put q(e) = q~ (exp k) (k’ E 2); the left-hand side is C” 

and has a compact support on 5?. We write again 

W) = j9 ~(4 exp [@,~‘)I dt 

(e’ E 9’) 

and express &,(y, y) by aid of @(e’) as follows. We observe first, that 

choosing a linear measure dh on H in a suitable fashion, we have 

where (T is the adjoint representation of G. Next, by fixing the volume 

element de’ on Y in an appropriate manner, we obtain 

q(4) = j,, $3(P) exp [ - i(/,P)] dP. 

In this case we shall call de’ the measure, dual to dtf. Choosing elements


y and h in I’ and H, respectively, and replacing G by a&-l) h in the 

above equation, we conclude, that 

cp(u(y-l) h) = I‘,, +(I’) exp [- i(o(y-l) h, t’)] dk’ 

= s ~, +(p(y-l) e’) exp [ - i(h, d’)] de’. 

Let us take the measure A’, dual to dh, on H’ and let us form the 

measure dhl, uniquely determined by de’ and dh’, by aid of which we 

have 

Then the previous equation gives 

d4P) h) = IH, (/HL$%(~-l) (f’ + AL)) dhl) exp [- @, 41 dh’. 

In the exponential on the right-hand side, (h, h’) stands for the 

canonical bilinear form on H x H’, and it is understood that the 

class, containing el, on 2’ modulo HI is h’. Substituting this in (4), 

and applying the Fourier inversion formula on H, we obtain 

(c) Next we observe, that the orbit U of r$, with respect to the 

restriction of p to K = exp H, coincides with/i + HL (cf. [4], p. 158). 

In fact, one verifies easily, using dim H = dim 2’ - 4 d, that U 

is open in 8; + H-L, but since it is the orbit of a unipotent representa- 

tion of the nilpotent group K = exp H (by virtue of the Remark 

after Lemma l), U necessarily coincides with 6’; + HJ-. Let us denote 

by S the stabilizer, with respect to p, of ei in G. Its Lie algebra R 

is the radical of B(fl , /J = ([/1 , &,I, 8;) (tr , z$ E 9). By virtue of our 

choice, H is maximal self-orthogonal with respect to B, hence it 

contains R (cf. [4], p. 157). S ince, by Lemma 1, S is connected,3 

we have S = exp R C K. Let us denote by A the homogeneous space 

of right classes of K according-to S. The map CO, of A into 8; + H-L, 

defined for the class K of K in K mod S by w(R) = p(kl) 8; , is a 

bijection between A and 6’; + Hl. We denote by dA the inverse 

3 In fact, the orbit 0 in Lemma 1 is simply connected.

270 PUKANSZKY 

image, through w, of dbl. Then dA is an invariant measure on A. To 

see this, it suffices to observe, that, for any k in K, we have 

p(k) 4; = 8; + &, where hi E Hl, and therefore, if g is continuous of 

compact support on Y, we obtain 

Using dA, (5) can be rewritten as 

We recall, that K,(y, y) is continuous and nonnegative on I’. Using (3), 

we obtain finally 

(d) Observe, that de, = dA dy is an invariant measure on G/S; 

we denote by the same symbol the corresponding measure on 0. 

Using the notations of Lemma 1 we set dz = dz, dz, *+* dz, . Taking 

into account the remarks made before Lemma 1 we see that dz, too, 

is an invariant measure on 0, and therefore it differs only by a positive 

factor from dv. From this we conclude that, in order to prove Lemma 

2, it suffices to show that the function +(z, , za ,..., zd), obtained from 

$(e’) by replacing d’ through the parametrization of 0, is summable 

with respect to the Lebesque measure on Rd. Since ~(4’) is C” and 

has a compact support, +(e’) is rapidly decreasing, and therefore there 

exists a constant C, such that 

for all z in R”. But by virtue of the properties of the polynomials {Pj}, 

described in Lemma 1, the right-hand side is certainly summable. 

We recall finally, that ~(a) = ($- x 4) (a) (a E G), and therefore 

T, = T,*T,. 

Remark. Observe, that given a Haar measure da on G, and a 

translation-invariant measure d/ on 9, Lemma 2 yields the following


algorithm to determine the canonical measure dw on 0. Take first 

invariant measures dk and dr on K and r, respectively, such that 

da = dk dr holds. Next, take the measure dh, which is the inverse 

image of dk through the exponential map from H onto K, on H, 

and forming the dual measure de’ of uY, determine the measure dhl 

on HJ-, such that aY” = dh’ dhl holds; here dh’ is dual to dh. Denoting 

finally by M the measure on A = K/S, which is the inverse image of 

dhl through the map k -+ p(k-l) 4’; of A onto 8; + Hl, the canonical 

measure on 0 is obtained by transferring to it the measure dh dy of 

G/S. 

It is easy to verify directly, that the result is independent of the 

factorization da = dk dy with which we started. 

THEOREM. Let G be a connected and simply connected nilpotent Lie 

group with the Lie algebra 9. Let da be a Haar measure on G, and dr? 

a translation-invariant volume element on 3’. We choose an element 

e$ in the dual 9’ of 9, such that the corresponding orbit 0 has a positive 

dimension. Then the ratio of the K-measure and of the canonical measure 

belonging to 0 depends on the dimension d of 0 only, and is given by 

(d/2)! #I2 2d divided by the ratio of da and & at the neutral element 

of 3. 

Proof. We choose a subalgebra H, subordinated to /i , and having 

a dimension h = dim 2 - Q dim 0, of 2 (cf. [4], p. 159, Remarque 

2). We form again B(4’i , f2) = ([/i , t2], Ci) (4, , e2 E g), denote its 

radical by R, and set r = dim R. Then, if d = 2m, we have 

m = n - h = h - r. Let us choose a decreasing sequence of subalgebras 

2% = 3’ 3 A?,+, 3 2%-Z 3 *** 3 .A?i 3 Z,, = (0), such that 

dim g* = j, g,-, = H and gnMd = R. We select for each 

j = 1, 2,..., n a nonzero element 8j in 3’j - gi-i , and we write 

gj(t) = exp (tt,). Since the system {/r ,4, ,..., &} is a supplementary 

basis to the trivial subalgebra of _Lp, the canonical coordinates of the 

second kind {tl , t, ,..., tn} in g = g,( tl) gz(t2) *** gJt,J are valid 

globally on G, and dt, dt, *** dt, defines a Haar measure da for G. 

Let us put 4’ = cj”=i y& and de = dy, dy, se* dy, . The measure de, 

when transferred to G by means of the exponential map, gives rise to a 

Haar measure ([4], p. 90), which, in the present case, coincides with 

da. In fact, in order to see this it suffices to observe that the Jacobian 

of the variables {tk}, connected with the variables {yj} through 

580/1/3-z

272 PUKANSZKY 

according to the latter has the value 1 at the origin of R”. Observe, that 

it is enough to prove the above theorem for this choice of da and dt 

respectively; in fact, one sees at once from the character formula (1) 

(cf. Section 1) that, if we change the ratio of da and d/from 1 to c, 

the canonical measure on each orbit gets multiplied with the same 

constant. By virtue of our choice, the system {8r+r ,..., tn,) is a sup- 

plementary basis to R. Therefore, putting T = (t,,, , tr+2 ,..., tn) E Rd 

and g(T) = gr+lPr+l) - g&J, we can conclude, that the map 

sending T into p([g( T)]-l) tfi , of Ra into 0 is a diffeomorphism, and 

that dt = dt,,, dt,,, e.0 dt, defines an invariant measure on 0. To 

prove the theorem it suffices to establish, that if at /i , say, cr dt and 

ca dt correspond to the d-forms representing the K-measure 

and the canonical measure, respectively, on 0, then we have 

cl/cz = (d/2)! # 2d. We do this in two steps. 

(a) First we compute ce using the Remark following Lemma 2. 

Using the canonical coordinates of the second kind {ti;j = 1, 2,..., n} 

introduced above, we observe, that dk = dt, dt, *** dt, and 

dr = d&+1 *a* dt, are invariant measures on K = exp H and on 

r = G/K respectively, and, in addition, da = dk dy. The measure 

dh on H, corresponding to dk, is dh = dy, dy, *** dyh . Let us choose 

a basis {ei} satisfying (e, , Cj) = aij (;,j = 1, 2,..., a) in 9’. We write 

d’ = ~Yryjr!~ and observe, that the measures dt!” and dh’, which are 

dual to dt and dh, respectively, on 9’ and H’ are given by 

a?!‘= (237)~” ay; fly; *-* ay; and dh’ = (23~)-~ dy; dy; .** dy; . 

Therefore, the measure dhl on HJ- satisfying de’ = dh’ dhl is the same 

as 

Let us write now 

If 

dhL = (24” dy;+, dy;+z ..* dy: . 

T = (b+l , t,+z ,..., t,J E Rm and W”) = g,+&+d a** g&d 

then c2 is the same as (2n)-m times the value D of the Jacobian of the 

variables (y;; j = h + l,..., n) according to the variables


(tj;j = r + l,..., h} at T = 0. Let us denote by a*, the partial deriva- 

tive of y; according to tk at T = 0. Then we have 

and thus 


D = det (B(t,.+i , /h+k); i, k = 1,2 ,..., m). 

(b) To determine c,, b y virtue of the definition of the K-measure 

(cf. l), we have to proceed as follows. We consider the 2-form 

w = C B(lr+d 9 tT+j) dtr+i A &+j 

i.j>O 

at 8; , and form its mth exterior power; then this is the same as ci dt. 

Thus, for the value of ci , we obtain 

m! 2m[det (B(87+i , /,.+j); i, , = 1, 2 ,..., d]l12. 

Taking into account, that c!,.+$ lies in H for i = 1,2,..., m, and that H, 

being subordinated to 8;) is self-orthogonal with respect to B, we 

conclude, that B(c!,+{ , l+.+$) = 0 for i, j = 1, 2 ,..., m. Therefore, the 

above determinant has the value D2, and thus we obtain finally, that 

cl/c2 = (2~)~ m! 2” = (d/2) ! #I2 2d, finishing the proof of the theorem. 

4, The purpose of this concluding section is to obtain, by aid of the 

expression of the canonical measure found in the preceding section, 

an algorithm for the Plancherel measure. 

In what follows we continue to denote by $4 a non-Abelian nilpotent 

Lie algebra and by G the corresponding connected and simplyconnected 

group. For an element x in the dual 9 of 3, we write 

B& , /,) = ([/r , &a], x) (8, , l2 E S), and denote by R(x) the radical 

of B,. 

Let 9 = -E”, 3 Yn-, 3 a** 3 dpO = (0) be a Jordan-Holder sequence 

for 9, such that dim ~j = j (j = 0, l,..., n). We write 

6pi(x) = ..Yj + R(X) and o b serve (cf. the proof of Lemma 1) that

274 PUKANSZKY 

there exists a well-determined system of d = dim O(x) inte- 

gers 1 < j, < j, < ..* < j, < n (we assume d > 0), such that 

dim Yj(x) = dim Z&x) + 1 if and only if j = j, for some 

k = 1, 2,..., d; we write f(x) = &}. The purpose of the following 

lemma is to prepare the proof of Lemma 4, which brings a proof, 

adopted to the needs of the present context, of Theorem, Section 5, 

Chap. 1, Part II of [4]. 

LEMMA. 3. There exists a uniquely determined system f of integers 

between 1 and n = dim 9, such that the set 0 = {x; f (x) = f > in 

2’ is open in the Zariski topology. 

Proof. For each j = 1, 2 ,..., n let t; be a nonzero element in 

e!Zj - LFj-1 a We denote by Mi(x) the matrix 

{B5(ti , &.); 1 

Then we have dim R(x) = n - rank A&(x), and therefore the set 

0, - {x; dim R(x) = r} (I = min dim R(x), x E 9’) is open in the 

Zariski topology. Next we observe that, since 

writing 

the set 

dim sj(x) = dim R(x) + rank i&(x), 

di = sup dim 5$(x) (x E Q 

0, = 0, n {x; dim Z&X) = dJ 

is open in the Zariski topology. Obviously r = d,, < dl < *-- < d, = n; 

we denote by f the set of integers 1 < j1 < j, < .** < j, < n, such 

that dj = d,-, + 1 if and only if j E f. Let US write 8 = (J~-i Oj; 

we claim, that putting E = {x; f (x) = f} we have E = 0. In fact, 

we observe first, that evidently 0 C E 2 0, . Let p be an element in 

E - 0. Then there exists an index j such that dim Z??&J) < dj . 

But this is impossible, since [by virtue off Cp) =f] we have 

dimS!(p)= 1 1 +r=dj 

I&i 

Thus 0 = hf(4 =fl is o P en in the Zariski topology and evidently, 

f is uniquely determined by this property. 

Remark. 1. We observe that in general 8 is strictly contained in 

0, . In fact, let us consider the four dimensional nilpotent algebra 9


defined by a basis {ei; j = 1, 2, 3, 4) with the commutation relations 

Eee 9 e,l = e2 , [e, , e23 = el , all other brackets being zero. Then 

L$ = {ej ,..., el> (j = 1, 2, 3, 4) is a Jordan-Holder sequence. One 

can verify through a simple computation, that in coordinates with 

respect to a basis, dual to {e,}, in 9” we have o0 = (x; xi2 + x22 # 0} 

but 0 = {x; x1 # O}. 

Remark. 2. For an x in 9’ with dim O(X) > 0 let us denote by 

TP the tangent space, to the orbit O(x), at p, and let us consider again 

the non-degenerate 2-form wP defined by 

%k%m~ rp,V2)) = &(4 , e2) (P E w4; 4 > 42 E =w 

(cf. Section 1). For each element t in TP there exists a unique element 

#t in its dual, such that oP(s, t) = (s, #t) for all s in TP . We identify 

the exterior algebra /l( TP) over TP with that over its dual through #. 

Then wP gives rise to a scalar product on II( TP) such that, if 

7 s 

U=~AhUk and v = n AV~ (ok 9 vj E Tp), 

k=l j-1 

the scalar product wP(u, v) is 0 if 7 # s and det{wP(Ui, 0,); 1 < i, k < Y} 

otherwise. We consider again the system f determined in Lemma 3, 

and for each k = 1, 2,..., d we select a nonzero element ek in 

5?j - 9,-i (j = j,). Let us denote by EP the d-vector 

h 'dek) =%(fi h ek). 

k=l k=l 

Then the function Q(p) defined as c$,!$, , E,) if dim O(p) > 0 and 

zero otherwise is the square of a p-invariant polynomial function,4 

homogeneous of degree d/2, on Y, and Q(p) = 0 if and only if p 

lies in 0, where 0 has the same meaning as in Lemma 3. In fact, we 

observe first, that we have Q(p) = det{([ei , e,], p); 1 < i, j < d) for 

all p in 9’ implying that Q(p) is the square of a homogeneous poly- 

nomial function of degree d/2. As far as the invariance of Q(p) is 

concerned, by virtue of the invariance of wP , it suffices to establish 

the p-invariance of EP . We observe first that, if 4 = p(a) p, we have 

R(q) = u(a) R(p) and thus 

44 -%I4 = 4-4 (=% + XP)) = 3 + w = -%I)1 

’ The p invariance of Q will not be used in the sequel.

276 PUKANSZKY 

which implies at once that f(p(a) p) =f(p). Hence, in particular, 

0 is invariant under p. Therefore it is enough to show, that 

E4 = dp(a) IP Ep if q = p(a) p on p E 0. But since 

for all G in 9 (cf. Section l), it suffices to verify, that vq maps 

n$=, A ek and n$=, A ~(a) ek onto the same element in Ad(Tq). We 

have o(a) ek - ek E q C q(q) ( j = j, - l), and therefore to obtain 

the desired conclusion it suffices to note, that by virtue of the choice 

of the system (e3, e, E q(q) - q-,(q) (j = ji , i = 1, 2 ,..., d), and 

thus q-I is spanned by R(q) and {e,; 1 the 

vectors {ek) are independent mod R(p) if p E 0, we have in this case 

Q(p) # 0. On the other hand, if p lies outside the set 

8, = {p; dim O(p) = d}, 

then clearly Q(p) = 0. H ence, to finish the proof of the statement 

made above, it is enough to show that we have Q(p) = 0 for all p 

in 0, - 0. With the notations of the previous lemma, let j be the 

smallest integer, for which dim .A@) < di . Then evidently j = j, 

for some k = 1, 2,..., d and -Y&(q) = q(q) (j = jk), and thus the 

system {ei; 1 

Q.E.D. 

LEMMA. 4. Let 9 be a non-Abeliun nilpotent Lie algebra, 

9n = 3’ 3 ,EpW1 3 **a 3 .ZYO = (0) a Jordan-Holder sequence in 9 

(dim 4 = jfor j = 0, 1, 2 ,... n) and {ej’; j = 1, 2 ,..., n} a basis in 3” 

such that ej, is in q*, - 9j1. Then there exists a nonconstant p-invariant 

polynomial function Q(x) on Y, a positive integer d, d indices 

0


B,(ej , h(x)) = 8, . Let us write again, as in Remark 2 above, 

Q(x) = det (R& , e.)}; 3 then for any el in .A?‘, Q(X) (f(x), el) is the 

restriction of a polynomial function in x to 0. To obtain Lemma 4, 

it suffices to repeat the reasoning of Lemma 1 for each x in 0, replacing 

{fi} in the lemma with {fi(x)} as determined above. Then 

Q(x) = det VW+ , e.)} 3 will satisfy the requirements since, by virtue 

of Remark 2, we have x E 0 if and only if Q(x) f 0. Q.E.D. 

Remark. 3. Let us denote by K the complement off in the set of 

all integers between 1 and n. Then for any fixed z and k in K, Pk(z, x) 

is of the form xk + Qk(xl, x2 ,..., x& (Qi = const.), where 

xi = (li , x) (i = 1, 2 )..., n). In fact, let us denote by n the canonical 

projection of 9 onto 9i = -E”‘/5?$. Writing 0(+x)) for the orbit 

of n(x) under rr o p, we have O(n(x)) = rr(O(x)). In order to be able 

to conclude, that Pk(z, x) depends only on the class of x modulo 

.9’;, it suffices to take into account the following easily verifiable 

fact. Let T be a unipotent representation of G on a real vector space V 

of dimension n [T is said to be unipotent if &(e) is nilpotent for all e 

in 91. Suppose, that V = V, 1 V,-, 3 a** 1 V, = (0) is a Jordan- 

Holder sequence with respect to T, and let us choose a basis {Us> in V 

such that vj E Vj - V,+, (j = 1, 2,..., n). Then a parametrization, 

of the type as described in Lemma I with {vi} instead of {&}, of any 

orbit of 7 is uniquely determined. Finally we observe that, since 

r(&) is invariant under 7r o p, we have, for any real a, 

40(x + 4)) = 0(4x)) + a&>, 

and therefore also Pk(z, x + a&) = Pk(z, X) + a, finishing the proof 

of our statement. 

Remark. 4. Let us write X,(x) = Pk(O, X) (k E K); then the 

system {A,(x)} so obtained is functionally independent by virtue of 

Remark 3. Furthermore, it generates the field of all rational invariants 

of p. To see this, we observe first, that any rational invariant is the 

quotient of two invariant polynomial functions (cf. [4], p. 61). Let 

P(x) be such a polynomial function, and let Q(+) (x, = (8, , x), k E K) 

be its restriction to the hyperplane (x; (4 , x) = 0,j ef }. Then we 

have evidently P(x) E Q(hk(x)) for all X. 

Before passing to the description of the Plancherel measure, we 

make several preliminary observations. Let p be an element of 0; 

we continue to identify the tangent space T, , to O(p) at p, with its

278 PUKANSZKY 

dual through wz, , as this was done in Remark 2. We denote by L the 

identity map from O(p) into 0. Identifying the tangent space to the 

open submanifold 0 of $P’, at p with Y, the differential dL (at p) of L 

maps Tp into 9’ and the transposed map, &, coincides with - ?D . 

In fact, we have, for all tin 9, dcy#) = - (a&)‘p, and therefore 

4v,(k), W)) = (4 dv,,(k)) = - (4 Wk)’ P) = - ([k 4, P) 

= - 4dk)s vk9> (k E 3). 

Let us consider again the system {e,; K = 1, 2,..., d} and the function 

Q,(p) determined in Remark 2. We denote by (Q(p))1/2 a polynomial 

function on S’, of which Q(p) is the square, and with which, putting 

vp = m! 2”(Q&~))-l/~ nd- k-l A ek (2~2 = d), we have Sb(qJ = up” 

(p E 0). For each p, qp is uniquely determined by this property 

modulo the ideal generated in A(Y) by R(p). 

Next we consider the quotient space A = 8/p. We identify it 

by means of the system {AL(x); k E K} of Remark 4 with the (Zariski) 

open set, which is the image of 0 in Rd. Thus, in particular, A gets 

equipped with the structure of a differentiable manifold of dimension 

r = n - d. Let 71 be the canonical map from 0 onto A. If p is a point 

of@, t(p) the tangent space of A at n(p), and Tztpj its dual, we have 

&r(T&,) = R(p) and 

Let e’ be a nonzero element in A”($“); we denote by tithe correspond- 

ing volume element on 9. We choose a Haar measure da on G, such 

that the ratio of da and dJ at 0 should be 1. For h in A we denote by 

OA , T’“) and dvA the corresponding orbit, irreducible representation 

and canonical measure, respectively. Concerning dvA , it is understood, 

that it corresponds to the choice, just made, of d/and da (cf. Section 1). 

Thus we have, according to (1) in Section 1, for all function F, which 

is C” and of compact support, on G: 

Tr (T$ = ~,,$5(&“) dv, 

To determine the Plancherel measure means to find a positive measure 

dp on A, such that 

Let e be such in Am(Y), that (e, e’) = (277)-n according to the duality

PLANCHERRL FORMULA OF NILPOTENT GROUPS 279 

between A”(9) and A”(Y). Then the volume element de’, corre- 

sponding to e on Y, will be dual to &in the sense, that 

holds. We recall, that according to the Theorem of Section 3, the 

ratio on any OA (A E A) of the K-measure and of the canonical measure, 

for the normalization of the latter as above, equals to (d/2)! ~4~ * 2d. 

Let us put E = n$=i A ek and observe, that for each p in 0 there 

exists a well-determined element cp in Ar(R(p)), such that 

(Q(p))ll” (2~‘)~/~ e = g A E. With respect to any basis in /I’(9), E?, , 

multiplied with (Q(p)) r - lj2, will have polynomial coefficients. Further- 

more, cp is invariant (that is l q = u(u) l r, if q = p(a) p), since for any a 

in G u(a) R(p) = R@(Q) p), u(a) E = E modulo the ideal generated 

by R(p(a)p) in A(9) (cf. R emark 2), and det ~(a) = 1. By virtue of 

Lemma 4 and Remark 4 0 is diffeomorphic to A x Rd. Therefore to 

obtain dp it sufices to take the positive measure which corresponds to the 

r-form 6 satisfying &~(a) = 4, on A. 

Remark. 5. In terms of the parameters {Ak; k E K) (cf. Remark 4) 

8 has the form R(h) (nksK A dh,), where R is a rational function, as it 

can easily be seen by what preceeds. It is in this form that the Planche- 

rel theorem for nilpotent groups is stated in [q, Part II, Chap. III, 

Section 6. 

We sum up the result in the following 

PROPOSITION 3. Let G be a connected and simply connected nilpotent 

Lie group of dimension n, with the Lie algebra 9. We denote by d the 

7,ximum of the dimension of the orbits of p(a) = (Ad(a-l))‘, (a E G) in 

and assume, that d > 0. Then we can select d elements 

{e,; k = 1, 2,..., d} in 

Q(x) = det {([e, , e] 5 , x);l 

9 in 

280 PUKANSZKY 

ACKNOWLEDGMENT 

The author is indebted to B. Kostant for stimulating conversations on the subject. 

REFERENCES 

1. CARTIER, P., Seminaire Sophus Lie, Paris, 1955. 

2. HARM-I-CHANDRA, Differential operators on a semisimple Lie algebra. Am. 1. 

Math. 79 (1957), 87-120. 

3 KIRILLOV, A. A., Unitary representations of nilpotent groups. usp. Mat. Nauk 17 

(1962), 57-110. 

4 PUKANSZKY, L., “Le9ons sur les representations des groupes.” Dunod, Paris, 1967.


Sur les Vecteurs SCparateurs des Alg5bres 

de von Neumann 

MICHEL BROISE 

C.N.R.S., Paris 

Communicated by J. Dixmier 


Le but de cet article est la demonstration du theoreme 1 et des 

diverses propositions qui s’y rattachent. 

THBORBME 1. Soient J%’ et %” deux algbbres de Von Neumann dans 

un espace de Hilbert H. On suppose que 27 est commutative de genre 

dhombrable et que tout vecteur skparateur de .Z est vecteur skparateur 

de sk’ et inversement. Alors sl = 3. 

Ce probleme m’a CtC pose, pour &’ et 3’ commutatives et maximales, 

par J. M. Jauch et B. Misra. J’ai d’abord etabli le thCor&me 1 dans 

ce cas. B. Misra m’a fait remarquer que le theoreme restait vrai 

pour & et B commutatives quelconques (son argument est signal6 

dans la demonstration du theoreme 1, il est aussi utilise dans la 

demonstration de la proposition 1). J’ai enfin Ctabli le theoreme 1 en 

general. Un resume est paru dans les Comptes Rendus de 1’AcadCmie 

des Sciences de Paris. 

NOTATIONS ET RAPPELS DIVERS 

Soient d une algebre de Van Neumann dans un espace de Hilbert 

H, p et q deux projecteurs de &, x un Clement de ~2, .$ et r] des 

elements de H et tn et n deux sous-ensembles de H. 

On note x(H) le sous espace vectoriel form6 par les x(t), lorsque .$ 

parcourt H et x(H) l’adherence dans H de x(H). 

Si m et tt sont des sous-espaces vectoriels, on note m v n le sous- 

espace vectoriel engendre par m et n. De m&me, on note p v q le 

projecteur (hermitien) sur le sous-espace p(H) v q(H); c’est le plus 

petit projecteur de & majorant p et q. On note E(&‘, 5) le projecteur 

sur le sous-espace vectoriel ferme de H engendre par les vecteurs 

281

282 BROISB 

x’(t), lorsque x’ parcourt J$. Alors E(H, 5) est le plus petit des 

projecteurs p de JF/ tel que p(f) = e. De meme, on note E(&‘, m) 

le plus petit des projecteurs p de .&’ tel que p(v) = r) pour tout 77 E m. 

Remarquons que E(&‘, x(H)) est le support de x*; en particulier on 

a E(&‘, p(H)) = p. 11 est clair que E(&‘, m U n) = E(d’, m) v E(&, n). 

Pour que m soit separateur pour &, il faut et il suffit que E(&‘, m) = 1 

(Cf. [l], p. 6). P our qu’une algebre de Van Neumann commutative ait 

un Clement separateur il faut et il suffit qu’elle soit de genre denom- 

brable (Cf. [l], p. 20). 

LEMME 1. Soient JZ?’ une a&&e de Von Neumann, p et q deux 

projecteurs de &. Les conditions suivantes sont kquivalentes: 

(i) 4 GP. 

(ii) PVC1 -(PV!l)) = 1. 

Dtmonstration. (i) G- (ii) est evident 

(ii) * (i). Les projecteurs p et 1 - (p v q) sont orthogonaux, done 

la condition p v (1 - (p v q)) = 1 entraine 1 - (p v q) = 1 - p. 11 

en resulte p v q = p; d’ou q < p. 

PROPOSITION 1. Soient &I et JX?~ deux alg&res de Von Neumann duns 

un espace de Hilbert H. On suppose que tout sous-ensemble de H skparateur 

pour 541 est skparateur pour dz et inversement. Alors &I = J&‘II , 

De’monstration. Soient m un sous ensemble de H et v un Clement 

de H tel que E(&i, r)) < E(&‘;, m). Posons p, = E(d;, m), 

q2 = E(di , 7) et r2 = p, v (1 - (pa v qJ). 11 est bien clair que 

q2vr2 = 1. Done 

E(d; ,T u r,(H)) = E(di, 7) v EC&L, r,(W) = 1. 

Ainsi 7 u r,(H) est un sous-ensemble separateur pour 5;42 .I1 en resulte 

Comme on a 

il vient 

E(dol; , r] u r,JH)) = 1. 

1 = E(&l, r]) v E(d; , r,(H)) < E(d;, m) v E(d;, r,(H)). 

E&d;, m u r,(H)) = 1 done E(d; , m u r,(H)) = 1. 

On en deduit r2 = p, v r, = 1. Compte tenu du lemme 1, il vient 

q2 < p2 0~ Q4 ,rl) d Q4 p 4.

VECTEURS SePARATEURs DES ALGhBRES DE VON NEUMANN 283 

La fin de la demonstration est due a B. Misra. Comme les conditions 

7 E E(&“, m) ou E(&‘, 7) < E(&‘, m) sont Cquivalentes (pour 

toute algebre de Von Neumann A#’ C 5?(H)) il en resulte E(J.T?‘~ , m) < 

E(cc4; , m). On montrerait de meme que E(.zZ; , m) ,< E(&; , m). 

Done E(&i, m) = E(&; , m). Comme tout projecteurp d’une algebre 

de Von Neumann & C 9(H) 

&I et de &a coincident. 

est Cgal a E(&‘, p(H)) les projecteurs de 

LEMME 2. Soient J@’ une algBbre de Von Neumann dans un espace 

de Hilbert H, g une sous-algdbre de Von Neumann de &‘, x un &me& 

de LX?, 5 un vecteur skparateur pour LZ?‘. Alors 

(i) Jw’, x(5)) = E(a’, x(H)). 

(ii) E&f’, x(5)) = E(==f’, x(q). 

Df?monstration. (i). On a E(B”, x(H)) x(5) = x( [), done 

E(g’, x(H)) 2 E(g’, 45)). C omme E(9’, x( 5)) x(5) = x(5) et que 5 

est un vecteur separateur pour d on a E(#, x(5)) x = x. 11 en 

resulte E(4?‘, x(C)) > E(W, x(H)). Done E(9, x(c)) = E(9’, x(H)). 

(ii). On se ram&e au cas precedent, en prenant g = ~2. 

PROPOSITION 2. Soient ~9 une a@bre de Von Neumann dans un 

espace de Hilbert H, ~3~ et ~3~ deux sous-algdbres de Von Neumann de ~2. 

On suppose que tout vecteur skpayateur de k& est vecteur skparateur de L%‘;z 

et inversement, et que ST? possLde un vecteur skparateur. Alors 3f1 = az . 

Dkmonstration. Soit p, un projecteur de 3Yi . Posons qI = 1 - p, , 

pz = E(% , p,(H)), q2 = (% , q,(H)) et y = p, + (1 - p2). 11 est 

clair que r est un projecteur de J&’ et que E(ai , r(H)) = 1. Comme { 

est un vecteur separateur pour ~2, le lemme 2(z) nous donne 

E(g; , r(5)) = 1. d one E(99’; , r(c) = 1.11 en resulte 

1 = -JW; , (P, + 1 -P,)(O) G EW; 9 P,(O) v E@; j(1 - P&N 

= P, ” -fw; 9 (1 - P,KN. 

Comme 1 -p, < 1 -pi et que 1 -p, ~37~) on a E(58’; , (1 -p2)([) < 

1 -P1. 11 en r&ulte E(&?i , (1 - p,)(c) = 1 - p, et de m&me 

E(a’; , (1 - q&C) = 1 - q1 = p, . Les projecteurs 1 - p, at 1 - q2 

sont orthogonaux. 11 est done clair que 

EW; 3 (1 - P,)(W) < EW; , (1 - p2 + 1 - #W,

284 BROISE 

et par suite que W’; , (1 - PdH)) v WC , (1 - MU < 

E(9’; , (1 - p, + 1 - q,)(H)). En appliquant 3 fois le lemme 2(i), 

il en resulte 

1 = E(g; , (1 -p, )(t;)) v E(9; , (1 - 4&J) = -w; > (1 -P, + 1 - Q(0) 

Done 

1 = E@q, (1 -P, + 1 - Q&N- 

Le lemme 2(i) entraine alors 1 = E(9’;: , (1 - p, + 1 - Q&H)) = 

I-pp,+l--q,. Commeona l-pp,

VECTEURS SfiPARATEURS DES ALGkBRES DE VON NEUMANN 285 

famille de projecteur orthogonaux. L’algebre de Von Neumann 3’ 

&ant suppoke de genre denombrable, il existe un nombre reel A, 

tel que p,, = 0 c’est a dire tel que E(%“‘, 5 + h,~) = 1. 

LEMME 5. Soient &’ et 9’ deux algkbres de Von Neumann duns un 

espace de Hilbert H, 5 un vecteur skparateur de 3, 5 et 7 des e’lt+ments de 

H tels que 

W”, 4 < E(z”‘, I). 

On suppose que 9’ est commutative et que tout vecteur skparateur de &’ 

est vecteur skparateur de 3 et inversement. De plus on suppose que pour 

tout projecteur r de ,clz, il existe un scalaire X tel que 

Alors 

EC@“, rl + WI) = Et&‘, 4 v EW’, r(5)). 

El&‘, rl) < E&f’, 5). 

De’monstration. Posons p = E(&‘, t) et q = E(&‘, r]). Con- 

siderons le projecteur r = p v (1 - (p v q), d’aprb le lemme 2(ii), 

on a r = I?(&‘, r(c)). C omme on a q v r = 1, d’aprb I’hypothese 

il existe un scalaire h tel que E(&“, n + h(c)) = 1, done 

Et%“‘, rl + W’)) = 1. 

Si I’on suppose E(A?‘, 7) < E(%O’, 0, il en resulte 

1 = E(T’, 7) v E(LY, r(5)) = E(%“‘, t) v I??(%“‘, r(5)). 

Le lemme 4 entraine alors l’existence d’un nombre reel A’ tel que 

E(2’, 5 + A’@)) = 1, d one tel que E(&‘, 4 + h’r(.

286 BROISE 

Le lemme 5 entraine alors l’implication 

EW’, p(t)) < -qb’, 5) * E&f’, P(O) < Et&‘, I). 

Soit r] un vecteur de H. D’apr&s le lemme 4 il existe un scalaire I\, 

tel que 

Jw’, 7) v EW’, r(5)) = JYJ’, 7 + WN, 

done tel que 

W’s +3) d EW’, 7 + W5)). 

D’aprb l’implication qui prC&de, il en rbsulte 

Le lemme 3 entraine alors 

EC&‘, 45)) < EW’, 7 + b(5)). 

EC&‘, f(5)) v EW’, 7) = -J-q&‘, 7 + W)). 

Du lemme 5 rkulte l’implication 

-W’, 7) < E(z’, (5) z-e- EC@“, q) < Et&‘, 8). 

La fin de la dkmonstration est due a B. Misra. Les conditions 

7) E I!?(&‘, f)(H) et E(.#, 7) < E(&‘, 4) &ant Cquivalentes, on en 

dCduit E(ZZ”, 5) < E(&‘, 6) p our tout 5 E H. En particulier, compte 

tenu du lemme 2(ii) pour tout projecteur p de nous dCduisons 

et 

EW’, 2%)) < Et&‘, PCJ) = P 

-qb’, (1 -P)(5)> < E(@,‘l’, (1 - p)(O) = 1 -P. 

Comme 5 est sbparateur pour 9, on a 

E(%“‘, P(5)) v -qb’, (1 -P)(5)) = 1 

et par suite E(9”, p(c)) = p. A insi les projecteurs de & appartiennent 

B 9’. Done &’ C 9’. JZ? &ant commutative, en permutant les r8les de 

&’ et de 9’ on dCduirait de m&me 9’ C J&‘. Done %” = &’ (notons 

d’ailleurs que la proposition 2 entraine JJZ = d d&s que l’on sait que 

dcq. 

COROLLAIRE. Soient 2 un espace localement compact ci base &nombrable, 

v une mesure positive SW 2, L2(Z, v) l’ensemble des classes de 

fonctions SW Z de car& v-sommable et L”(Z, v) l’ensemble des classes de 

fonctions SW 2, mesurables et essentiellement bornkes pour v. De m&e 

soient 2, , v1 , L2(ZI , vl), et L”(Z, , v 1 ). Soit 4 une application Zinkaire 

bijective et isomdtrique de L2(Z, v) sur L2(2, , vl). On suppose que 

pour 5 E&z, v).

VECTEURS SltPARATEURS DES ALGhBRBS DE VON NEUMANN 287 

Alors il existe un ensemble v-nkgligeable N dans Z, un ensemble vlnkgligeable 

NI dans Z, et une bijection 0 de Z - N SW Z, - NI , telle 

que les mesures v1 et ecv) ( res p v et &l(vJ) soient &uivalentes. I1 existe 

une fonction v,-mesurable p, sur Z, telle que 

%x1> = P,(L) 5uw-lN 

pour toute function 6 de L2(Z, v) et pour presque tout Kd = fP’Kd 

pour [r E Z, - N1 et telle que les mesures e(v) et v,(resp. v et F(v,)) 

soient Cquivalentes. Done il existe une fonction rr positive v,-mesurable 

sur Z, telle que 0 < r,(&) < co pour & E Z, et telle que ecv) = rlvl . 

Soient f un Clement de L”(Z, v), .$ et 7 deux elements de L2(Z, v). 

D’une part on a 

@“At), 4 = jzf 671 dv = j, -N ( f 0 e-W 0 @-Wj 0 s-l) d ‘W 

1 1 

= q f )(d 0 e-l)(;i 0 e-1) de(v). 

4-N I 

D’autre part comme % est une application isometrique, on a 

= 

I @( f ) @‘(5) WI) dvl. 

=1 

La fonction G(f) &ant arbritraire dans L”(Z, , vr), il en resulte 

(8 0 e-l)(ij 0 e-1) rl = 4?(f) a(~) dans Ll(Z, , vI). 

5W1/3-3

288 BROISE 

Choisissons v tel que G?(q) soit un vecteur separateur pour 3F’i . 

11 est clair que @(‘I) peut se mettre sous la forme 

4(~) = (7 0 ~-1)(~1)1/2 eisl 

ou s1 est une fonction reelle v,-mesurable sur 2, . On a done 

d’oh 

(6 0 F)(ij 0 0-l) rl = U(Q(ij 0 &l)(~l)l/z e-Q,. 

(6 0 &1)(r1)1/2 eisl. = a([). 

Ainsi il existe une fonction vi-mesurable p, sur 2, telle que 

wEN(51) = Plbl) WYL)) 

pour tout 6 EL2(Z, v) et pour presque tout {i E 2, - N1 . 

Remarque 1. On pourrait penser que si LT’i et si 3a sont deux 

algebres de Von Neumann commutatives de genre denombrable et 

si tout vecteur separateur de .3i est vecteur separateur de 8,) alors 

q2; , ‘5) < -WC 9 4 P our tout 5.11 n’en est rien. Prenons pour %oi 

l’algebre commutative maximale engendree par les projecteurs 

pour d, l’algebre engendree par les projecteurs 

11 est clair que tout sous-ensemble separateur pour Z1 est un ensemble 

separateur pour ~%‘a . Mais on a 

Remarque 2. Soit .z&’ une algebre de Von Neumann finie standard 

et de genre dtnombrable. D’apres [I] (chap. III, 5 1, 5, Cor. du

VECTEURS SiPARATEURS DES ALGBBRES DE VON NEUMANN 289 

thtor. 5) & et JZ” ont memes vecteurs separateurs. Le theoreme 1 ne 

s’ttend done pas en gCnCral au cas non commutatif. 

PROPOSITION 3. Soit zz? une ai@bre de Van Neumann dans un espace 

de Hilbert H, ayant un vecteur skparateur. Les conditions suivantes sont 

kquivalentes. 

(i) ~4 est commutative. 

(ii) pour tous vecteurs f et 7 de H, il existe un scalaire A, tel que 

E(d’, 5) v &d’, rl) = El&‘, 6 + w. 

(iii) pour tous x et y E &, il existe un scalaire A, tel que 

- - 

x(H) v y(H) C (x + AY) H. 

Dtmonstration. (i) z= (ii) est la consequence du lemme 4. 

(ii) * (iii). Supposons la condition (ii) remplie. Soient x, y E &. 11 

existe un scalaire h tel que 

E(d’, x(5)) v -qd’, Y(5)) = E(d’, (x + hY@)) 

Le lemme 2(ii) entraine alors 

EC@“, x(H)) v EW’, y(W) = JW”, (x + ~YW)), 

c’est a dire (iii). 

(iii) * (i). Si Lc4 n’est pas commutative, il existe une sous *-algebre 

66’ de CQI qui est un facteur de type I, (Cf. par exemple la fin de la 

demonstration du Cor. 3, Chap. III, $ 1, p. 230 de [I]). Si G? satisfait 

a (iii), B satisfait a (iii). Mais (iii) ne depend que du type algebrique de 

l’algebre de Von Neumann envisagee. Done l’algebre Ys(C) des 

matrices complexes a 2 lignes et 2 colonnes satisfait a (iii) ce qui est 

absurde, comme on le voit en considerant dans -P&C) deux elements 

de rang 1 qui ont mCme noyau mais pas mCme image. 

BIBLIOGRAPHIE 

1. DIXMIFX, J., “Les Algkbres d’Optrateurs dans 1’Espace Hilbertien.” Gauthier- 

Villars, Paris, 1957.

JOURNAL OFFUNCTIONAL ANALYSIS 1, 290-330 (1967) 

The Sizes of Compact Subsets of Hilbert Space 

and Continuity of Gaussian Processes 

R. M. DUDLEY* 

Department of Mathematics, Massachusetts Institute of Technology, 

Cambridge, Massachusetts 02139 

Communicated by Irving E. Segal 


1. THE SIZES OF COMPACT SETS 

The first two sections of this paper are introductory and correspond 

to the two halves of the title. 

As is well known, there is no complete analog of Lebesgue or Haar 

measure in an infinite-dimensional Hilbert space H, but there is a need 

for some measure of the sizes of subsets of H. In this paper we shall 

study subsets C of H which are closed, bounded, convex and symme- 

tric (- x E C if x E C). Such a set C will be called a Banach ball, 

since it is the unit ball of a complete Banach norm on its linear span. 

In most cases in this paper C will be compact. 

We use three main measures of the size of C. One is as follows. 

Let V, = V,(C) be the supremum of (n-dimensional Lebesgue) 

volumes of projections P,(C) where P, is any orthogonal projection 

with n-dimensional range. Then we define the exponent of volume of C, 

J-V), by 

1% vn 

EV(C) = lim sup ~. 

n+cc nlogn 

Another numerical measure of the size of C involves the notion 

of E-entropy [12]. Let (S, d) b e a metric space. The diameter of a set 

T C S is defined as 

sup (4% r) : x, y E T). 

Given E > 0, one defines N(S, E) as the minimal number of sets of 

diameter at most 2~ which cover S. Then the r-entropy of S, H(S, E), 

* Fellow of the Alfred P. Sloan Foundation. 

290

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 291 

is defined as log N(S, E). (The logarithm is taken to the base e. 

The ideas of information theory and thermodynamics play no explicit 

role in this paper.) Finally, we define the exponent of entropy r by 

(In case the lim sup is equal to a limit, r has been called the metric 

or&v of S-see [12], p. 22.) 

We prove below (Proposition 5.8) that if EV(C) > - l/2, then 

r(C) = + co, while if H’(C) < - l/2, then 

r(C) >, - 

2 

1 + 2EV(C) * 

If the above inequality becomes an equality C will be called volumetric. 

In Section 6 we prove that ellipsoids, rectangular solids, certain 

“full approximation sets”, and, if Ev(C) < - 1, octahedra, are 

volumetric. The question is left open for - 1 < Ev(C) < - l/2, 

but I conjecture (5.9) only that a Banach ball C with Ev(C) < - 1 

is volumetric. 

Our third general measure of the size of a Banach ball C involves 

the canonical “normal distribution” L on H ([I8], pp. 116-119; 

[9]). L is a linear mapping of H into a set of Gaussian random variables 

with mean 0, which preserves inner products. Let A be a countable 

dense subset of C and 

L(C) = sup {] L(X) 1 : X E A}. 

Then E(C) is a well-defined functionoid; i.e., a different choice of 

A will affect E(C) only on a set of zero probability. 

For any K > 0, r(K) = r(C) and EV(KC) = EV(C), but the ran- 

dom variable L(C) d oes not have this homothetic invariance. We call C 

a G&set if E(C) is finite with probability one. This property is 

homothetically invariant, and for other reasons which will become 

clearer in the next section, we study mainly the GB-property rather 

than the entire random variable z(C). To relate this property to T 

and Ev we have the following main results: if r(C) < 2 then C is a 

GB-set (V. Strassen (unpublished) and Corollary 3.2 below). If 

r(C) = 2, C need not be a GB-set (Section 6) and I conjecture (3.3) 

that if Y(C) > 2 it never is. If Ev(C) > - 1, C is not a GB-set 

(Theorem 5.3); I conjecture (5.4) that C is a GB-set if EV(C) < - 1, 

and prove this for Ev(C) < - 3/2 (Proposition 5.5). The conjectures 

are proved in all four classes of special cases considered in Section 6.

292 DUDLEY 

However, at r = 2 and EV = - 1 there is some “overlap” and 

the GB-property is not a monotone function of the H(S, c) as E JO 

nor of V, as n -+ co (Proposition 6.10). 

2. CONTINUITY OF GAUSSIAN PROCESSES 

We shall study sample function continuity and boundedness of 

Gaussian processes from a general viewpoint. Let (S, d) be a metric 

space and let (X t , t E S) be a real-valued Gaussian stochastic process 

over S (for definitions see, e.g., IS], p. 72). Then the xt are all elements 

of a Hilbert space H = La(D, 9, Pr) over some probability space 

(Q, a, Pr). (Q is a set, 99 a u-algebra of subsets, and Pr a probability 

measure on a). If two Gaussian processes over the same S have the 

same mean and covariance functions Ex, and Ex,xt , then they have 

the same joint probability distributions for (xt , t EF} for any finite 

or countable subset F of S ([.5], p. 72, (3.3)). Such processes will be 

called “versions” of each other. We say that a process is samplecontinuous 

if it has a version (xt , t E S> such that for almost all w in J2, 

t -+ .X,(W) is continuous on S. (In case S is e.g. the real line it is well 

known that not all versions of a process will be continuous.) 

Sequential convergence of functions on 12 almost everywhere 

implies convergence in measure and then, for the Gaussian case, 

convergence in H. Thus since S is metric, sample continuity implies 

that t + x1 is continuous from S into H and we can and will restrict 

ourselves to this case. Then, Ex, is continuous on S, and xt is samplecontinuous 

if and only if xt - Ex t is, so we may and do assume 

Ex, = 0. 

A subset C of an abstract Hilbert space Hi is realized as a Gaussian 

process {x f , t E C} with Ex t = 0 and Ex$x, = (s, t) by letting 

x1 = L(t) where L is the “normal” random li ear functional or weak 

distribution mentioned in Section 1. We callnC a G&et (Gaussian 

continuity set) if L is sample-continuous on C. Thus if (xt , t E S) 

is a (Gaussian) process with Ex, E 0, the function t -+ xb is continuous 

from S into H, and its range is a GC-set, then the process is 

sample-continuous. 

For any set A C H there is a sample-continuous process (xt , t E S} 

whose range is A, letting S be A with discrete topology, but such 

examples are rather artificial and much of the study of samplecontinuous 

Gaussian processes reduces to the study of GC-sets. 

(See e.g. the end of Section 4.) 

A process (x1 , t E S> will be called sample-bounded if it has a version


such that the sample functions t --+ x1(o) are bounded uniformly on S, 

for each w. Here we have a perfect correspondence: a Gaussian 

process is sample-bounded if and only if its range is a GB-set. The 

convex, closed, symmetric hull of a GB-set is a GB-set and is compact 

(Proposition 3.4 below). We shall on the whole restrict ourselves to 

compact sets, and a compact GC-set is a GB-set. Conversely, most, 

but not all, GB-sets are GC-sets. Sample continuity and boundedness 

are equivalent for ellipsoids and rectangular blocks (Propositions 6.3 

and 6.6 below) and stationary processes on a finite interval ([2], 

Theorem 1). A narrow class of GB-sets which are not GC-sets appears 

among octahedra (Propositions 6.7 and 6.9 below), and other examples 

can be constructed by the law of the iterated logarithm. We shall 

prove severe narrowness of the class of GB-sets which are not GC-sets 

in general (Theorem 4.7). 

V. Strassen proved (unpublished) in 1963 or 1964 that, if S is a set of 

Gaussian random variables with r(S) < 2, then (in our terminology) 

it is a GC-set. Strassen’s result is sharpened somewhat (Theorem 3.1 

below) to include some sets with r(S) = 2 and to yield a result of 

Fernique [7], [7a] for processes over the unit cube as a corollary 

(Theorem 7.1 below). 

Conjecture 3.3 (if r(S) > 2 S is not a GB-set) is verified for certain 

random Fourier series with independent Gaussian coefficients, 

both those covered by a result of Kahane [IO] and some others 

(Propositions 7.2 and 7.3 below). 

In Section 4 we give some general results about E(C), convergence 

of series defining L, etc. Among other things, we establish an exact 

natural correspondence between GC-sets and the “measurable 

pseudo-norms” of L. Gross [9] (see Theorem 4.6 below). 

Section 8 gives some brief comments on possible methods of 

attack in proving the conjectures. 

3. SAMPLE CONTINUITY AND E-ENTROPY 

Here is a sufficient condition for sample continuity in terms of 

e-entropy: 

THEOREM 3.1. Suppose S is a subset of a Hilbert space and 

Then S is a CC-set. 

m H(S, 1/291'S< co 

c n-1 2” 

(1)

294 DUDLEY 

Proof. Given a positive integer n, we decompose S into N(1/2n+“) 

sets of diameter at most l/2”+“, and choose one point from each set, 

forming a set A, . Let G, be the set of all random variables L(x - y) 

for x and y in A,-, u A, and jl x - y (1 < l/2%. Then the cardinality 

of G, is at most 4N(2-n-4)2. 

We shall use below the well-known estimate, for a > 0, 

i 

m 00 xe-“=I= dx e-a2/2 

e-z212 dx & =-* 

a I a a a 

Let (b,) be any sequence of positive real numbers. Let 

P, = Pr (max {l(z) 1 : z E G,} $ b,) < 4N (2-n-4)2 (exp [- 4%,2/2])2”/b,. 

Thus P, < b, if n > 2 and - 4”bm2/2 + 2H(2-n-4) < 2 log b, , 

or 

[H(2-“-4) - log !&J/4”-” < &a. 

Let an2 = H(2-“-4)/4n-1. Then 2 a, < co by (l), and a, is inde- 

pendent of b, . But now we specify b, , letting b, = max (2a, , l/n”). 

Then an2 < bm2/2 and log b, > - 2 log n, so for n large enough 

(- 4 log &J/4” < 1/2n4 < bn2/2, 

and then P, < b, . Since C b, < co, we have 2 P, < co. 

Thus for almost all w there is an no(w) such that 1 z 1 < b, for all 

n > no(o) and all x in G, . 

Now let T be any countable dense subset of 5’. We shall show that 

on T, L is uniformly continuous with probability one. Its extension 

to S is then a version of L with continuous sample functions, as 

desired. 

Given 6 > 0, we choose n, so that 

where 

sa(fkJ = +J : no(w) > no) 

For any s in T, we choose points A,(s) in A, such that 

II S - An(s) II < l/2 n+3. Now if n > n, , s, t E A, and (1 s - 

then 11 A,(s) - A,(t) (1 f l/2%. Thus L(A,(s) - A,(t)) 

t 11 < 1/2n+3, 

E G, . Also, 

WW - An+,(s)) E Gn,, . Thus for w $ Q(n,,), L(A,(s)) (w) +L(s) (w)


for all s in T, and for any t in T such that d(s, t) < 1/2no+a, we have 

I L(s) (w) -L(t) (w> I < 8. 

Letting 6 4 0, we see that L is uniformly continuous on T with 

probability 1. Q.E.D. 

COROLLARY 3.2. If S is a subset of a Hilbert space and r(S) < 2, 

then S is a GC-set. 

There are numerous examples of sets S with 7(s) = 2 which are 

neither GC- nor GB-sets; see, e.g., Section 6 below. Moreover, 

Theorem 7.1 below and its partial converse, due to Fernique [7], 

indicate that, even when specialized to stochastic processes on the real 

line, Theorem 3.1 is essentially the best possible result of its kind. 

However, we prove in Proposition 6.10(a) below that no sufficient 

condition for the GC-property of a Banach ball in terms of H(S, E) 

is necessary, i.e., the GC-property is not a “monotone function” of 

the function E + H(S, l ) as E J 0. Yet I make 

Conjecture 3.3. If S is a GB-set (and hence if S is a compact 

GC-set), then r(S) < 2. 

In Sections 6 and 7 below, Conjecture 3.3 is proved in a number 

of special cases. In the general case, I shall prove at present only the 

following: 

PROPOSITION 3.4. If S is a GB-set then S is totally bounded 

(i.e., its closure is compact). 

Proof. If S is a GB-set, it is certainly bounded. Suppose it is not 

totally bounded. Then for some E > 0 there is an infinite sequence 

{ fj}& in S such that the distance of fi+l from the linear span Fi of 

fi ,..., fi is at least E for all j. Let 

where ]I g, I/ >, E and g, 1. F, . Given M > 0, let 

Then 

A,={w:max(lL(f,)I:l~j~n} WI 

= Pr W5J > M) Pr &J/2.

296 DUDLEY 

Now for some S > 0, we have, for all n, 

Pr (L&) > M) > 28; so Pr (A,) < (1 - S)+l 

by induction. This contradicts the fact that S is a GB-set and com- 

pletes the proof. 

The method of proof just used will yield a stronger result. Using 

also (5.2) and Lemma 5.6 (cf. also Proposition 6.9), it can be shown 

that, if S is a GB-set, then for any 6 > 0 

N(S, c) < exp (exp ( l/~z+s)) 

for E sufficiently small. Since the examples in Section 6 indicate that 

this inequality has an unnecessary extra exponentiation, no further 

details will be given. 

4. PSEUDO-NORMS 

Let V be a real linear space and let W be a linear space of linear 

functionals on V. Then for any set C C V, the polar Cl is defined by 

When C is symmetric, 

Cl = {w E W : W(X) < 1 for all x in C}. 

C1 = {w E W : 1 w(x) / < 1 for all x in C}. 

If A is a linear transformation of V into itself and W is closed under 

the adjoint A* (i.e., composition with A), then for any CC V, 

A(C)1 = (A*)-1 (Cl). 

(Here (A*)-l is a set mapping and A* need not be invertible.) In 

particular V may be a Hilbert space and IV its dual space, possibly 

identified with V. 

On K-dimensional Euclidean space Rk, let h or A, be Lebesgue 

measure and let G be the standard Gaussian probability measure; 

dG = (27r)ekj2 exp (- r2/2) dX, 

where r is the distance from the origin. 

PROPOSITION 4.0 (Gross [9]). Let A be a linear transformation


from Rk into itself with norm 11 A (1 < 1 and let C be a convex symmetric 

set in Rk. Then 

G(A(C)l) > G(C). 

Proof. This follows directly from [9], Theorem 5, stated in dif- 

ferent language. For A symmetric and invertible it is Lemma 5.2 of 

[9], and arguments to reduce to this case are given in the proof of 

Theorem 5. Q.E.D. 

Now as usual, let H be a separable, infinite-dimensional Hilbert 

space. Every GB-set in H is included in some Banach ball which is 

still a GB-set. 

For any subset C of H, let s(C) be its linear span. Then if C is 

convex and symmetric, it is the unit ball of a norm 11 * ]Ic on s(C). If 

C is a Banach ball, then (s(C), II l Ilo) is a Banach space and its natural 

injection into H is continuous. 

Let H* be the dual space of H. (For clarity, we do not identify 

the two.) For each 93 in H*, let 

II 9J II;: = SUP {I d$) I : $J E 0 

Then if C is a Banach ball in H, 11 * 11; is the dual norm to II * lIc 

(composed with the natural map of H* into the dual space 

MC)‘> II * IILL) of (4Ch II l IICN. 

L is an assignment of random variables to elements of H, or equivalently 

to continuous linear functionals on H*. The assignment can 

be extended to some nonlinear functionals in various ways. For 

example, if q~ is a Bore1 measurable function on R” and fi ,..., f, E H, 

then ol(fi ,-.., f,) defines by composition a function on H*. (Such a 

function is called “tame.“) The assignment 

L(dfl 9.*-Y fn>) = dL(fl)Y.,L(fnN 

is well-defined, as is well known [9], [18]. Thus, e.g., we let 

L(lf I> = IL(f) I ,f EH- 

Now in general, an assignment such as 

L (SUP &) = SUP b%Ad 

will not be well-defined, but if g, = If, I , f, E H = (H*)*, then 

sup g, = II * 11; where C is the closed symmetric convex hull of the 

f, . Also 

SUP L(&) = SUD (1 L( fn) I) = acj,

298 DUDLEY 

and the assignment 

is well-defined. 

WI * Ilc> = W) (9 

By f.d.p. (finite-dimensional projection) we shall mean an ortho- 

gonal projection of H onto a finite-dimensional subspace. For pro- 

jections P and Q, one says P < Q if the range of P is included in that 

of Q, and P, 7 I if PI < P2 < *** and P,(f) +f in (Hilbert) norm 

for each f in H. Also P 1 Q means the ranges of P and Q are ortho- 

gonal. If {fn} is an orthonormal basis of H, g, are independent, 

normalized Gaussian random variables, and L,(f) = (f, f,) g, , then 

the series 

Ii1 Ln(9 (1’) 

is a version of L. If P, 7 I (and the P, are f.d.p.‘s) then there is an 

orthonormal basis { fi} of H such that for each n, { fi ,..., fk.} is a basis 

of the range of P, for some k, , k, t co. The convergence of L o P, 

to L is equivalent to convergence of a certain sequence of partial 

sums of (1’). 

We shall need an infinite-dimensional form of Proposition 4.0. 

PROPOSITION 4.1. Let A be a linear operator from H into itself with 

([AI/, Pr (L(C) < t). 

Proof. If C is finite, the result follows immediately from Proposi- 

tion 4.0. In general, let C, be finite sets which increase up to a dense 

set in C. Then 

Pr (L(C) < t) = $i Pr (L(C,) < t) < $i Pr (L(AC,) < t) 

= Pr (L(AC) < t), Q.E.D. 

PROPOSITION 4.2. If P,, are f.d.p.‘s, P, t I, C C H, and t >, 0, 

then 

Pr (L(C) < t) = $rr Pr (E(P,C) < t). 

Proof. Let A be countable and dense in C. Then L(P,f) -L(f) 

as n -+ co for all f in A, with probability 1. Hence 

Pr (X(C) < t) < li?AdPr (yP,C) < t).


On the other hand Proposition 4.1 yields 

liy+yp Pr (L(P,C) < t) < Pr (YC) < t), 

completing the proof. 

The following definition is essentially that of Gross [9]. 

DEFINITION. A pseudo-norm 11 l I] on H* is measurable (for L) if 

for every E > 0 there is a f.d.p. P, such that, for every f.d.p. P 1 PO , 

Pr (L(ll * II 0 P) > c) < E. 

Note that ]I l 11 o P is a tame function on H* so that L of it is defined. 

If C C H then I] * 11: is * measurable if and only if for every E > 0 there 

is a f.d.p. P,, such that, for every f.d.p. P 1 PO , 

Pr @(PC) > c) < E. 

It then follows by Propositions 4.1 and 4.2 that 

Pr (EP,IC) > 6) < E. 

(For any projection P, P 1 = I - P where I is the identity operator.) 

THEOREM 4.3. If C is a Banach ball in H, the following are equiv- 

alent: 

(a) C is a GB-set, i.e., Pr (EC) < co) = 1. 

(b) Pr (E(C) < co) > 0 

(4 [resP* WI WnC) converges in law for some (resp. every) 

sequence of f.d.p.‘s P, 1 I. 

(e) L restricted to s(C) has a version linear and continuous with 

probability I for /I * II=. 

Proof. Let {f,J b e an orthonormal basis of H. For each f in H, 

the series (1’) converges almost everywhere on 52 and in L2(Q). For 

any finite N, 

@(9 b) 

is bounded on C for each w in Sz, and finiteness of L(C) (0) thus 

depends on the g, for n > N. Thus by the zero-one law ([13], B, 

p. 229), Pr (E(C) < co) = 0 or 1, and (a) is equivalent to (b). 

(a) is equivalent to (c) and (d) by Proposition 4.2.

300 DUDLEY 

(a) is equivalent to (e) since a linear functional on a normed space is 

continuous if and only if it is bounded on the unit ball. The proof 

is complete. 

Before treating GC-sets, we introduce some facts we need about 

function-valued random variables. Let S be a metric space with a 

countable dense subset A = {+‘ZZ1 . Let %(S) be the Banach space 

of bounded continuous real-valued functions on S, with supremum 

norm II * IL . We say Xi, X, ,... are given as a set of V(S)-valued 

random variables if probabilities 

Pr (Xi(ti) E Aij , i, j = 1, 2 ,...) 

are defined for any points t, , t, ,..., in S and Bore1 sets A, in the 

real line. Then the norms 

II Xi I/m = SUP {I xi(t) I : t E A) 

are measurable. Note, however, that W(S) will not be separable if S 

is not compact. Then, the distributions of the Xi will not be expected 

to be defined on all open sets in V(S) for the supremum norm topo- 

lo!3 (cf. PI). 

A random variable X in V(S) will be called symmetric if - X has 

the same distribution as X. Independence of random variables Xi 

in q(S) is defined also, naturally, to mean that the sets of real random 

variables 

Ai = {Xi(t) : t E S} 

are independent for different values of i. 

Let X, be independent and symmetric in %‘(S) and 

The following generalization of a Lemma of P. Levy is proved much 

like the classical version (Loeve [23], p. 247). 

LEMMA 4.4. For any LY > 0, 

Pr (max {\I S, 11 : k = l,..., 4 > 4 < 2 Pr (II S, II > 4. 

Proof. For each k = l,..., m, j = 1, 2 ,..., and s = f 1, let 

Afk, j, s) = {w : )I Si 11 < a, i = l,..., k - 1, ) &(x0) 1 < OL, 

q = l,..., j - 1, sS,&) > a}.


Then {w : 11 S, (1 > 01 for some k, 1 < k < m} is the disjoint union 

of the A(R, j, s). We also have for each k, j and s, 

Pr (A@, j, s) and II S,,, I/ > a) b Pr (A(k,j, s> and s(S, - Sk> (q) 2 0) 

Hence 

2 Pr (A@, .A s))P. 

2Pr (II S, II > a> t C Pr (A&j, s)) = Pr (max{I\ S, jl : K = I,..., m} > a); 

k,f,s 

Q.E.D. 

PROPOSITION 4.5. The series X:=1 X, of independent symmetric 

%?(S)-valued random variable converges in S(S) (i.e., uniformly on S) 

with probability 1 if and only if it converges (uniformly) in probability. 

Proof. “Only if” is obvious. “If” is proved from Lemma 4.4 

just as in the classical case where S has only one point: see [13], 

p. 249. 

THEOREM 4.6. For any compact Banach ball C in H, the following 

are equivalent : 

(a) for any E > 0, Pr (XC) < c) > 0; 

(b) C is a GC-set; 

(4 [rev. WJ L 0 Pm converges uniformly on C in probability for 

some (resp. all> sequences of f.d.p.‘s P, t I; 

(c’) [resp. (d’)] replace “in probability” by “with probability I” 

in (c) [resp. (d)]; 

(e) 11 l IIc is a measurable pseudo-norm on H*. 

Proof. Throughout let A b e a countable dense subset of C. 

(a) * (b): Let P, be f.d .p.‘s and P, t I. Given E > 0, let 

C,(c) = {w : q&&T) < E/3}, 

K(e) = lim sup CJE) 

= {w : C,(e) holds for arbitrarily large n}. 

Then K(E) is a tail event, having a probability 0 or 1. 

By (a) and Proposition 4.1, 

0 < Pr (L(C) < c/3) < Pr (ZP,% < c/3) 

for all n, where P,’ = I - P 7&* Thus K(E) has positive probability,

302 DUDLEY 

hence probability 1. Hence almost every w belongs to Cm(e) for some n. 

Then since L o P, is continuous, there is an 01 > 0 such that if 

x, y E A and (1 x - y 11 < (II, then 

I Q) -L(Y) I b) 

Since E was an arbitrary positive number, (b) is proved. 

(b) z- (c): given E > 0 we use uniform continuity on C with 

probability 1 to infer that for some 6 > 0, 

Pr (sup {I L(x - y) I : x, y E A, /I x - y II < 63 > e) < E. 

We choose a finite-dimensional subspace F such that F n C is within 

6 of every point of C. Let P be the projection onto F. Then by Pro- 

position 4.1 

since, for any x in C, there is a y in F n C with 11 x - y 11 < 8 and 

Ply = 0. Thus 

Pr(j(L--LOP)-(C)I>,E) 0 there 

is an n such that 

Pr (E(Q,“C) > c/2) < c/2. 

Now the operator norm 11 PmlQn I/ -+ 0 as VI -+ co since Qn has finitedimensional 

range and Pml-+ 0 pointwise. Hence E(P,-LQ,,C) -+ 0 

in probability as m -+ co. Also 

Pr (~(P,IQmLC) < 42) 2 Pr (yQfl’C) < e/2) >, 1 - c/2, 

so, for m large enough, 

and (d) holds. 

~(P,IC) < ~(P,IQnC) + ~(P,IQdC); 

Pr (L(P,T) > c) < E


(d) * (d’) by Proposition 4.5. 

(d’) + (e): clearly (d’) => (c), and (c) =- (e) by Proposition 4.1. 

(e) * (a): given E > 0, we choose a f.d.p. P such that 

Then also 

Pr (L(P%) < e/2) > 0. 

Pr (L(K) < l /2) > 0 

and since L(PX) and E(PC) are independent, we have 

Pr (E(C) < c) >, Pr @(PC) < c/2 and J?(Z’%) < c/2) > 0. 

Q.E.D. 

Not every GB-set is a GC-set, as we shall see below (Propositions 

6.7 and 6.9). Thus all possible implications among the conditions 

listed in Theorems 4.3 and 4.6 are settled. However, these conditions 

suggest others, e.g., replacing “in law” in (c) and (d) of Theorem 4.3 

by “in probability” or “with probability one”. If P,C C C for all 71, 

then L(P,C) is nondecreasing, so the different forms of (c) are equiv- 

alent in this case. In Section 6 we present a GB-set (octahedron with 

axes (log n)-+), which is not a GC-set, and for which P,C -+ C for 

certain natural projections P, t I. Thus the stronger forms of (c) 

do not imply that C is a GC-set, but other possible implications are 

not settled. 

We shall conclude this section with a result showing that the class 

of GB-sets which are not GC-sets is quite narrow. 

If B and C are Banach balls in H, we shall say B is C-compact 

if B C s(C) and B is compact for ]I l IIc . If B is a GB-set, we call it 

maximal if whenever B is C-compact, C is not a GB-set. (No GB-set 

A is maximal in a strict set-theoretic sense since 2A includes A 

strictly and 2A is a GB-set.) 

THEOREM 4.7. Every GB-set is either maximal or a GC-set. 

Proof. Suppose B is C-compact where C is a Banach ball and a 

GB-set. Then L restricted to s(C) has a version which is linear and 

continuous for I] . /le. The ]I l IIc topology is stronger than the original 

Hilbert topology on s(C) since C is bounded, hence these two topolo- 

gies are equal on the compact set B ([II], Theorem 8, p. 141). Thus 

B is a GB-set. Q.E.D.

304 DUDLEY 

If 11 * I/ is a measurable pseudo-norm on H*, then L is defined by a 

countably additive probability measure on the completion of H* 

for I/ * 11.l At the moment the converse seems to be an open question. 

Suppose (xt , t E S) is a sample-continuous Gaussian process over a 

compact metric space (S, d). Then t -+ xt is continuous from S into 

H, and 

e(s, t) = (E(xs - xJ2)l12 

defines a pseudo-metric e on S which is continuous for d and hence 

defines a weaker topology. If (S, e) is Hausdorff, i.e., if x, # xt 

for s # t, then the d and e topologies on S are equal, and hence the 

range of the process in H is a GC-set; its closed convex symmetric 

hull is a GB-set, which, by Theorem 4.7, is not much worse. 

5. SEQUENCES OF VOLUMES 

We shall need the volumes of certain simple sets in Rk. First, 

suppose A is a simplex, i.e., a convex hull of (h + 1) points x,, ,..., xk , 

having an interior. Let F be a face of A, i.e., a convex hull of K of its 

vertices. Let h or h, be Lebesgue measure on Rk and S or Sk-r the 

(K - 1)-dimensional Lebesgue “surface” or “area” measure. Then 

h(A) = S(F) d/k, 

where d is the distance from the vertex not in F to the hyperplane 

through F. Now suppose x0 = 0 and let di be the distance from xj 

to the linear span of x0 ,..., xi-r , j = l,..., k. Then 

h(A) = (fJ dj)/k!. 

i=l 

Now, recalling the definitions of V,(C) and EV(C) given is Section 

1, we have the following fact (a stronger statement is given as Propo- 

sition 5.10 below): 

LEMMA 5.0. If C is a convex set in H and EV(C) < - 1, then C 

is totally bounded. 

1 For the proof, see L. Gross, Abstract Wiener spaces, in Proceedings of the Fifth 

Berkeley Symposium cm Mathematical Statistics and Probability (1964). University of 

California Press, Berkeley, 1967.


Proof. If C is not totally bounded we make the same construction 

as in the proof of Proposition 2.4. Then for some E > 0, V,(C) is 

greater than or equal to the volume of the convex hull of 0, fi ,..., f, , 

so 

VJC) > E”/tz! for all n. 

By Stirling’s formula, this contradicts the hypothesis. Q.E.D. 

Next, let ck = X,(B) w h ere B is a ball of radius 1 in Rk. Then it 

can be shown by induction that, for any positive integer k, 

c&+1 = 22”+wk!/(2k + l)!, 

c2k = m”jk!. 

Thus by Stirling’s formula we have the following estimate: 

y+i Cj(Trj)“” (j/274’2 = 1. (5-l) 

We shall also need the following fact. Let {a,} be a sequence of 

positive real numbers such that a, JO as n + co. For such a sequence 

and E > 0 we define 

n(c) = n({%} , c) = max (?z : a, >, E), 

h = h({u,}) = infict:f (5p 0 such that 

Pr(L(C)y>O. 

C may be replaced in the above inequality by any orthogonal pro- 

jection P(C), according to Proposition 4.1. Multiplying C by a

306 DUDLEY 

positive number leaves the relevant properties unchanged, so we may 

assume M = 1. Suppose the first conclusion is false. Then for any 

K > 0 there is an n such that V, > (K/n)lz. 

Let P, be a projection with n-dimensional range F. Then 

Y < Pr (-@‘,C) < 1) = G(P,CY), 

where the polar is taken in the dual of F and G is normalized Gaussian 

probability measure. We use the general inequality 

where B is any convex symmetric set in Rn (due to Santa16 [17]). 

(Later work by Bambah [I] on a lower bound for h,(B) h,(P) may 

also be noted.) For any /3 > 0 there is a P, such that 

U~nC) 2 tqy, so h&w7m G ca2(nP)n- 

Using (5.1) we obtain for any 01 > 0 

M~nq) < 4cw’2 

for certain arbitrarily large n. Now, given X,(A) for a set A, G(A) 

is clearly maximized when A is a ball E(r) centered at 0, say of radius T. 

Hence 

where r, < (~ln)l/~. Then 

where 

G(PnW < ‘Wt~,d), 

G(E(r,)) < jr"" ~+le-‘~/2 dr/I, , 

I,= I m 0 

~--l~=l~ dr. 

The integrand increases for 0 < r < (n - 1)li2. But (an/(n - 1))li2 --t 0 

as n -+ co and 0110, so G(E(r,)) -+ 0 as n + co through a suitable 

sequence, contradicting the fact that G((P,C)l) >, y > 0. 

Thus for some M > 0, 

log vv2 

M 

-


Conjecture. 5.4. If C is a Banach ball and W(C) < - 1, then 

C is a GC-set. 

The above conjecture may be made plausible by a supporting 

conjecture (5.9 below) and proofs of both conjectures in four classes 

of special cases (Section 6). In the general case, I can prove the follow- 

ing. 

PROPOSITION 5.5. If C is a Banach ball and EV(C) < - $, then 

C is a GC-set. 

Before proving Proposition 5.5 we introduce another construction 

and some other facts. Given a compact Banach ball C in H and an 

orthonormal basis {F~}& of H, let F, be the linear span of v1 ,..., ?n , 


C, = CnF, (Co = F. = {O}). 

Given two sets A and B in H we define their distance as usual, 

e(-%B)=s~p$X-YIl’ 

d(A, B) = e(A, B) + e(B, A). 

We shall say the basis {vi> is adapted to C if 

4G-, , C) = d(G-, , G) 

for 7t = 1,2,... . Since C is compact, a basis adapted to C always 

exists. Then the sequence {F,} of subspaces will also be called adapted 

to c. 

If there is a sequence G,, C G, C +a* of subspaces of H with each 

G, n-dimensional and d(C n G, , C) < a, for all n, a, JO, then the 

sequence {an} will be called adapted to C (whether or not the G, are). 

In order to fmd an upper bound for E-entropies of sets with a given 

adapted sequence {a,} we use the following result. 

LEMMA 5.6. Let B({ai}&) b e a rectangular n-dimensional block 

ofsides2aS,0

308 DUDLEY 

Proof. We consider the cubes of side 2eln112 whose vertices are 

of the form 

t 2?Tljrl?P, 1 mj 1 < 1 + ?PUj/2r, 

j-1 

and the mj are integers. B is included in the union of these cubes, 

their diameters are 2~, and the number of them is bounded as indi- 

cated. Q.E.D. 

The latter, cruder estimate in the above Lemma is sufficient for its 

applications below except for one rather delicate one (Proposition 

6.10). 

PROPOSITION 5.7. Let C be a compact Banach ball in H and 

{an> adapted to C. Then 

Proof. Let s = X({a,)) and let F,, CF, C F, **a be subspaces of H, 

F, n-dimensional, such that for all n, d(C, , C) < a, where 

C,= CnF,. 

If /3 > 01 > s then for small enough E > 0, 

by (5.2). For such an E < 1 and n = n(e/2), 

NC, 4 < WC, ,4). 

Since r and s are homothetically invariant we can assume a, < 1. 

Clearly C, is included in the block B({+}&) of Lemma 5.6, so for 

E small enough 

N(C, G) < exp (n (log 3 + 4 log n + log (l/e))) < exp (E-B). 

Thus Y(C) 9 /3 for all B > s and r(C) < s. Q.E.D. 

Proof of Proposition 5.5. By Lemma 5.0, C is compact. There is a 

c > 8 such that V,(C) < n-w for n large enough. We choose a basis 

(~~1 adapted to C and v, in C,,, such that e(v, , C,) = a, = d(C, C,), 

n = 0, I,... . Then C includes the symmetric convex hull of the v, , so


Then by Stirling’s formula there is a p > 4 such that 

for n large enough, and a, < n-b. Thus 5.7 and 3.2 imply that C is a 

GC-set. Q.E.D. 

Suppose given a Banach ball (= convex symmetric bounded 

closed set) C in H. Suppose also that {FJ is a sequence of subspaces 

adapted to C. Given Fl ,..., F,-, , we assume F, can be and is chosen 

among its possible values so as to minimize h,(F, n C). Then we 

define 

Wn = UFn n C>, 

EW(C) = liT+iup (log Wn)/(n log n). 

For a sufficiently “smooth” set C, e.g., an ellipsoid, we shall have 

W(C) = EW(C) and even V, = W, (see Proposition 6.1 below). 

At the end of Section 6 we show that EW(C) < EV(C) is possible. 

Next we obtain a lower bound for r(C) in terms of EV(C). In each 

of the four classes of examples treated in Section 6, it becomes an 

equality at least for EV( C) < - 1. 

PROPOSITION 5.8. For any convex symmetric set C in H, 

(a) r(C) > - 2/(2EV(C) + 1) if EV(C) < - 4 

(b) r(C) = + co &W(C) > - &. 

Proof. If C is covered by m sets, each of diameter at most E, then 

any n-dimensional projection P,C is covered by m balls of radius E, 


mc,P 2 V, , so w, 4) >, Vn/w” 

for all n. Let EV(C) = - b > - c and c > +. Then for n large 

enough 

V,/c# > nn’z(~)1/2/[(27fe)“/2 &znc] = k, , 

say. The following paragraph gives motivation only. 

To maximize k, , we note that 

kn+$tn = ((n + l)/n)n[(l’+cl (n + 1)(1-c)/42m)l’2, 

which is asymptotic as n + co to 

e-cn(1/2)-c/E(2,)1/2.

310 DUDI.EY 

At any rate, as E JO we choose m = m(c) so that m(1/2)-c is asymptotic 

to e%(2?~)l/~, as is clearly possible. Then for any 6 > 0, and E small 

enough, 

h, = (m(1/2)-C/e(2ne)1’2)n (mn)l12 

2 exp (4 - ?$ - 6)) 

> exp ((c - 4 - 6) [(I - ~)/e”~(2rr)1’2]2’(2e-1). 

Hence for some constant y > 0, 

N(C, E) >/ exp (y~~‘(l-~~)) 

for E small enough. If - b < - + we let c approach b and obtain (a). 

If - b > - 8, we let c approach $ and obtain (b). Q.E.D. 

DEFINITION. A Banach ball C is volumetric if EV(C) < - + and 

Y(C) = - 2/(2EV(C) + 1). 

Conjecture. 5.9. If C is a Banach ball and EV(C) < - 1, then C 

is volumetric (hence Y(C) < 2 and C is a GC-set). 

A weaker inequality in the direction converse to 5.8 (a) is easily 

proved. Let {F,) b e adapted to C and T, = )c,(F, n C). If 

T, < .--no+*) for n large enough, 6 > 0, then since a, *a* a&! < T, , 

we have a, < n-8 for n large enough; hence, by Proposition 5.7, 

r(C) < l/S. Thus: 

PROPOSITION 5.10. I’ 


/3 = W(C) OY /3 = &V(C), /? < - 1, 

r(C) < - l/(8 + 1). 

Suppose given a compact Banach ball C in H for which El+‘(C) 

is defined and equals 

$ (log W,)/(n log n) 

(not just lim sup). Let (F,} and {an} be adapted sequences of subspaces 

and numbers, respectively, and (yn} an adapted orthonormal basis. 

Let A be the linear transformation such that 

44 = 4+?% for all tt, b, J- 0. 

Then the F, and p’n are adapted to A(C), F, now being uniquely 

determined, and 

44CN = b7I 3 

W&l(C)) = b, *** b,W,(C).

Thus 

where 


Thus the following is useful. 

PROPOSITION 5.12. If b, JO, 

-WV(C)) = EW(C) + 4%)) (5.11) 

ew((bJ) = li%tup (i log bj)/n log n. 

j=l 

eea(fbj,)) = - l/x({bjl)* 

Proof. Given 8 > 0, we have by (5.2): 

n(c) = n({b,}, e) < l/CA+8 

for E small enough, and n(e) > l/&-B for arbitrarily small E > 0. Now 

if 71 = n(e), 

(~lw,)/~logn >(hsM%4. 

When n >, l/~~+ and 0 < E < 1, 

log n 2 (A - 8) log (1 /C) and (log c)/log n > - l/(h - 6). 

Thus letting 6 J 0 we have 

e@d) >, - l/X. 

For the converse inequality, we can assume b, < 1. For any positive 

integer m let E = e(m) satisfy m = E-X-~~. Then as m -F CO, E JO. 

Since 

n(c) < l/E”+8 < l/&+26 

for E small enough, 

(fJl log b,)/m log m < (m - n(e(m))) (log 6) th+28/(h + 28) log (l/c) 

< (1 - 4 (log 4/(X + 26) log u/4 

= (- 1 + @)/(A + 26) + - l/(A + 26), 

where E = e(m), m -P co. Thus, letting 6 JO, we have 

~(VJ,)) < - l/h. 

Q.E.D.

312 DUDLEY 

6. SIMPLE SUBSETS OF HILBERT SPACE 

In this section we study symmetric rectangular solids, ellipsoids, 

and “octahedra” and determine when they are GC- and GB-sets. We 

also study certain “full approximation sets” (see [14]), which are 

maximal sets with a given adapted sequence {un}, while octahedra are 

(among the) minimal sets. 

For each class we shall have sequences {b,} of real numbers, b, JO, 

related to an orthonormal set {vn> in H, usually complete. Let F, 

be the subspace spanned by vi ,..., qn. For any orthonormal set 

{qn} and any b, 2 0 we define the ellipsoid 

Clearly, E is compact if and only if the b, for b, > 0 can be arranged 

into a sequence b, JO. Then the {F~}, {F,) and {bn} are adapted to E. 

(The F, are uniquely determined unless some positive b, are equal, 

and the b, are unique.) 

More abstractly, we can define a compact ellipsoid as an image 

A(&) of the unit ball B, = {X : 11 x 11 < l> in H under a compact 

operator A2 

It follows that if E is a compact ellipsoid and S is a bounded linear 

transformation from H into itself, then S(E) is a compact ellipsoid. 

LEMMA 6.0. If E = E({b,), {c&) is a compact ellipsoid and P is 

a f.d.p., 

P(E) = WQ, GM), bn 10, fin 109 

then /3, < b, for all n. 

Proof. We may assume the {q,} are complete. Given n let G, be 

the linear span of #r ,..., #, . G, has at least one-dimensional intersection 

with the set of vectors u orthogonal to P(yJ, j = l,..., n - 1. 

If also u E P(E) then 11 u 11 < 11 Pv I/ for some v E E({b,}, {q~~}~&, so 

11 u II < b, and hence p, < b, . Q.E.D. 

Now we find the exponents of volume of ellipsoids. 

PROPOSITION 6.1. 

EV(E) = EW(E) = - ; - & . 

2 For the equivalence of the definitions, see R. T. Prosser. The r-entropy and e-capa- 

city of certain time-varying channels. J. Math. Anal. Appl. 16 (1966), 553-573.


Proof. Lemma 6.0 implies that 

v, = w, = cnb,b2 * ** b, for all n. 

Let B be the unit ball E({l)) in H. Then 

V,(B) = W,(B) = c, . 

Using (5.11) and (5.12) the proof is complete. 

PROPOSITION 6.2. For any compact ellipsoid E = E({b,}), 

Thus if EV(E) ( - +, E is volumetric. 

Proof.3 We have Y > A by Propositions 5.8 and 6.1, and r < h by 

Proposition 5.7. The second conclusion follows then from 6.1 and 

the definition of “volumetric” (just before Conjecture 5.9). 

PROPOSITION 6.3. The following are equivalent : 

(a) E = E({b,)) is a GC-set 

(b) E is a GB-set 

(c) C& bn2 < co (E is a “Schmidt ellipsoid”). 

Proof. (a) implies (b) clearly if E is compact; if not, both fail. 

If (b) holds, and A is the linear operator such that A(y,) = b,v,, , 

L o A has a version continuous on H (Theorem 4.3(e) above). It 

is known that this is true if and only if A is a Hilbert-Schmidt oper- 

ator (see [8], Lemma 4, p. 344). Thus (b) and (c) are equivalent. 

Next, assume (c). Then for some k, 7 GO, C kn2bn2 < CO. Let 

El = E {k,b,)). Th en E is El-compact and not maximal, so by 

Theorem 4.7, E is a GC-set. Q.E.D. 

It follows immediately from the above results that Conjectures 

3.3, 5.4, and 5.9 all hold for ellipsoids. 

Now we turn to our second class of examples. Let {Fm} be an 

increasing sequence of subspaces of H with F, n-dimensional, 

n = 0, 1, 2 ,... . Let b, 1 0. Specializing [14], we define the full 

approximation set A = A({b,}) as 

{X : for all 71, 11 x - yn I/ < b, for some y, inFn}. 

s r = I\ is also proved by Prosser; see op. cit. in previous footnote.

314 DUDLEY 

It is easy to see that A 1 E({b,}). Also we can choose ylt in 

A n F, 3 A, . Hence {b,} is adapted to A and A is simply a maximal 

set having (b,) as an adapted sequence. 

PROPOSITION 6.4. r(A) = X({b,}). If EV(A) < - Q then A is 

volumetric. 

Proof. Since A 3 E({b,}), we have r(A) > r(E) = h({b,}) by 

Proposition 6.2. r < h by Proposition 5.7, so r = A. 

If EV(A) = - + - 6, 6 > 0, then EV(E({b,})) < - 3 - 6 so 

Y(A) = h({b,}) = r(E) = - 2/(1 + 2EV(zz)) < l/6 

= l/(- 8 - EV(L4)) = - 2/(1 + 2EV(A)). 

The converse inequality holds by Proposition 5.8 (a), so A is 

volumetric. Q.E.D. 

Note that the ellipsoid E with same parameters {b,}, included in A, 

also has the same exponent of entropy and the same exponent of 

volume if that of either is less than - &. We have proved Conjectures 

5.9 and (hence) 5.4 for A. Conjecture 3.3 also holds since if r(A) > 2 

then r(E) > 2 and 3.3 holds for ellipsoids. 

The condition 2 bn2 < CQ is clearly necessary for A({b,)) to be a 

GB-set but I don’t know whether it is sufficient for A to be a GC-set 

or GB-set. 

Next we consider the rectangular solid or “block” 

We assume as usual b, .lO and, to assure B C H, C bm2 < co. 

(Since Wd) 1 E(W), no GB-sets are lost here.) For blocks we 

shall not find adapted subspaces, but we shall characterize GC- 

blocks and GB-blocks and verify the three conjectures. 

PROPOSITION 6.5. If h = h({b,}), t = EV(B({b,})), and I = r(B), 

then t = - l/h = - & - l/r if any of these terms is less than - 4 

( i.e., if t < - 4, h < 2 or I < a~). Thus under these conditions B 

is volumetric. 

Proof. Given 6 > 0, we have for n large enough 

b,, < VA” < nt+‘, 

so h < - l/t if t < 0. Thus by 5.8 (b), any of our hypotheses implies 

x < 2.


Then by 5.2 we have for n large enough 

b, < n-llLw) 

so for 6 < 2 - X we have for k large 

so letting S JO we have by Proposition 5.7 

r < l/C- Q + l/N < a 

so by 5.8 t < - & and r > - 2/(1 + 2t), so 

Thus Conjectures 5.4 and 5.9 hold for blocks. 

PROPOSITION 6.6. The following are equivalent: 

(4 c bn I L(%J I converges with probability I; 

(b) C b, < ~0; 

(cl B = WJ) is included in some GC-ellipsoid; 

(d) B is a GC-set; 

(e) B is a GB-set. 

Q.E.D. 

Proof. (a) implies (b) by an application of the three-series theorem 

([131, P. 237). 

If C b, < co, we let 

an = (6” glbj)li2. 

Then E({a,}) is a GC-ellipsoid by 6.3, and B C E, so (b) implies (c). 

Clearly (c) implies (d) which implies (e). 

If B is a GB-set, then for almost every w, there is an M < co 

such that 

2 w4w) b> G M 

j-1 

for all possible choices of So = f 1. Hence (a) holds, and the proof 

is complete.

316 DUDLEY 

Now if a block B is a GB-set, then r(B) < 2 by (c) so Conjecture 3.3 

holds for blocks. 

If r(B) < 2 and E is the ellipsoid of (c), then it is easily shown that 

r(B) < r(E) < 2. 

Next we discuss some other classes of subsets of H: orthogonal 

sets S({b,)) and their closed symmetric convex hulls, octahedra 

Oc ({b,}). These sets refute a number of conjectures which up to now 

might have seemed plausible (cf. Propositions 6.7, 6.9, 6.10, and the 

remarks between and after them) while satisfying Conjectures 3.3, 

5.4, and 5.9. 

Given b, JO and {p),} an orthonormal basis let 

s = ww = K9 ” @n%x~1 3 

Oc = Oc ({b,}) = symmetric closed convex hull of S 

In this case, as for ellipsoids but not blocks, the {vn} and {b,} are 

adapted to Oc ({b,)). It is easy to see that 

iv (Oc, 4 >, N(S, 4 = 4Y%J, 424 f 4 + B 

for all E > 0 such that b, = E for at most one value of n. 

PROPOSITION 6.7. The following are equivalent : 

(a) Oc ({b,}) is a GC-set; 

(b) S({b,)) is a GC-set; 

(c) b, = 0 (log n)-112. 

Proof. Clearly (a) +- (b). T o p rove the converse, note that (b) is 

equivalent to b,L(y,,) -+ 0 as n -+ co with probability one. Given 

E > 0, for almost all w there is an N such that for all n > N, 

I b%?%J (0) I < E/4* 

and there is a 6 > 0 such that whenever X, y E Oc ({b,}) and 

II x -Y II -=c 69 

and we infer (a).


Now (b) is equivalent by the zero-one law ([13] A, p. 228) to the 

following: for any E > 0, 

or 

i L 

n-1 I 

eox2J2 dx < co. 

00 

e-x2/2 dx is asymptotic to e-M2/2/M. 

As is well known, an integration by parts shows that as M --+ co, 

s M 

Thus (b) is equivalent to 

5 b, exp (- c2/2bn2) < co. 

n-1 

Letting b, = olyr (log n)-l12, n > 2, we obtain the series 

f c$(log 4-l/2 d’~,a. (63) 

n=2 

If (c) holds, i.e., ollt --t 0, then the terms of (6.8) become less than 

W-~ for n large, so (b) holds. 

Conversely suppose (c) is false, so that for some 6 > 0, ol, > 6 

for arbitrarily large values of 7t. For such an n and nil2 42. 

>, & - ?.w - 1)/2?21’2 log n + co 

as n + co (recall that 6 is independent of n). Thus (6.8) diverges and 

(b) fails, so (b) 3 (c). Q.E.D.

318 DUDLEY 

PROPOSITION 6.9. The following are equivalent: 

(a) Oc ({b,}) is a GB-set; 

(b) S({b,}) is a GB-set; 

(c) b, = O((log $-l/2). 

Proof. We use some notation and results of the previous proof. 

(By the way, note that Theorem 4.7 and either of 6.7 and 6.9 make 

the other at least very plausible.) Here the equivalence of (a) and (b) 

is obvious. (b) is equivalent to the statement that for some M > 0, 

4t I GJTJ I < M f or n sufficiently large, with probability 1, or that 

(6.8) converges for E = M. If (c) holds, i.e., if for some N > 0, 

1 ollz 1 < N for all n, we can let M = 2N and infer (b). If {OIJ is 

unbounded, then given M we choose n so that 01, > 2M. Then 

01~ > M for n1i2 < j < n, 

C 

d’= 0. Thus (b) implies (c). Q.E.D. 

We infer from Propositions 6.3 and 6.7 that a GC-set, Oc ({l/log n)), 

is not included in any GB-ellipsoid, since 

F2 u(logn)2 = + cc 

(see [1.5], Lemma 2). 

We next show that the GC- and GB-properties are not monotone 

functions of the “size” of a set as measured by volumes V, or by 

e-entropy. 

PROPOSITION 6.10. There exist a GC-set Oc = Oc ({a,}) and u 

non-GB-ellipsoid E = E({b,J) such that 

(a) H(E, e)/H(Oc, c) --+ 0 as c: JO, 

(b) V,(E)/&(Oc) -+ 0 as n + 00. 

Proof. We let a, = an(log n)-l12, n > 2, where 01~ J 0 sufficiently 

slowly; for definiteness we can let 0~~ = (log log n)-l14, n > 3. Let 

b, = (n log n log log ?z)-1’2, n > 3.


Then Oc is a GC-set and E is not a GB-set. T/,(E) is asymptotic to a 

constant times 

and for 12 large, 

($yiZ n-112 (y2 n-114 5 (logj log logj)-112, 

Vn(Oc) > ($” (3774-1’2 fJ q(logj)-r’~, 

j=2 

Thus there is a K > 0 such that for n large, 

Vn(E)/Vm(Oc) < K fi (4n2/log log j)l14, 

j=3 

which implies (b). 

To prove (a) it suffices to show that 

as E 1 0. Let S = S({a,)). 

Given E > 0, let 

fq-q(n 1% w2>>, 4/~(~(~~~~), 4 + 0 

N(S, c) = n = n((uj}, 6) + g rt * . 

Because of the slow growth of the logarithms, this implies that, for c 

small enough, 

l/98 < (log n)2 log log n < l/8, 

log It < l/8, log log ?z < 2 log (l/E), 

H(S, l ) = log ?z 2 1/5q1og (l/E))““. 

To estimate iV(E, E) from above we take the smallest integer 7t such 

that 

(n log .)-l12 < 42, i.e., n log n > 4/c2. 

For E small enough this implies n log n < 5/e2. Now 

where 

N(E, 4 < w% , 42) d WL 9 4) 

Em = E(W), Bn = B(GW, 

pi = ( j log j)-l12, j = 2 ,..., n, j!15 = 0, j > n.

320 DUDLEY 

By Lemma 5.6, for E small and hence for n large enough, 

N(& ) E/2) < fi (2 + nl’2( j log$l’“/~) 

j=2 

(Note: the logarithms have served to make n smaller, but they 

are no longer needed.) 

For n large we have n! > (n/e)“, so 

N(E, l ) < (3e/0z112)” = exp {n[log 3 + 1 + log (l/c) - 3 log n]}. 

Since n < 5/e2, we have, for E small enough, 

log n < log 5 + 2 log (l/c) < 3 log (l/c), 

n > 4/e2 log n > 4/3c2 log (l/c), 

log 12 > log (4/3) + 2 log (l/E) - log log (l/E), 

H(E, 6) < 5[3 + log log (1 /c)]/e” log (1 /e). 

Thus H(E, e)/H(S, e) -+ 0 as E J 0. Q.E.D. 

Suppose given a sufficient condition that a set C be a GC-set, 

asserting that H(C, 6) is sufficiently small (e.g., Theorem 2.1) or 

that the V,(C) are sufficiently small (e.g., Proposition 5.5, Conjecture 

5.4). Then the GC-octahedron of Proposition 6.10 will never satisfy 

such a condition since the ellipsoid does not. Hence no such 

sufficient conditon can be necessary. 

In the converse direction, likewise, a sufficient condition for a Banach 

ball not to be a GB-set such as Theorem 5.3 or Conjecture 3.3 cannot 

be necessary. 

One may, however, seek “best possible” conditions of the given 

kinds. In the four cases, Theorem 3.1 has a fairly strong claim to be 

best (see the next section). Theorem 5.3 has a weaker claim. Conjec- 

tures 3.3 and 5.4, if they are true, could probably be improved upon. 

The volume of the n-dimensional octahedron 

is 2”/n!, which is asymptotic to (2e)“/n”(2m)1/2 by Stirling’s formula. 

Thus by 5.11 and 5.12


A sequence b, J- 0 such that A({&}) < co is o((log n)-li2) (cf. end 

of the proof of Proposition 6.7). Thus conjecture 5.4 holds for octa- 

hedra. The next proposition implies that conjecture 5.9 also holds for 

octahedra. 

PROPOSITION 6.12. Let X = A({&}), s = EV(Oc ({b,))), I = t(Oc). 

Then r = - 2/(2s + 1) = 2h/(2 + X) ;f any of these terms is less than 

2 (i.e., if r < 2, s < - 1, or X < co). Thus under these conditions 

Oc is volumetric. 

Proof. r > - 2/p + 1) in general by 5.8 (a). If s < 1, then 

X < co and s > - 1 - l/h by 6.11. Thus any of the hypotheses 

implies X < co, and then l/X >, - 1 - s, 

W(2 + 4 = 2/((2/4 + 1) < - 2/p + 1) 

ifs < - 4. It will now suffice to show that if h < co, 

r < 2/((2/4 + 1) 

(since then Y < 2 and s < - 1 < - +). 

Let 0 < y < l/X. Then for n large enough, b, < l/n7 by 5.2. 

Thus for some K > 0, Oc ({b,}) C KC, where C, = Oc ((l@)), 

and r(Oc) < r(KC,) = r(C,). Thus it is enough to prove that 

+q < 2/u + &). 

For x in C, , we have 

x = 1 xjP)jl!P, Cbjl 0, let A(x) be the set of all j such that 

1 x, 1 > j2Q2/4. 

Then the number m of integers in A(x) satisfies 

m1+2y/(l + 2~) = Sr x2Y dx < ,gi jay < 41~~. 

Let (II < /3 < y. Then for some c(r), 

m < c(.y)/&U+2v) < &/U+W) 

for E small enough. (Of course m depends on y and E).

322 DUDLEY 

The largest integer N in A(x) is at most (2/e)lly. Thus the number 

of possible choices of A(x) is at most 

N 

llz < N” < exp (c(y) log (~/E)/~E~/(~+~Y)) 

( 1 

< exp (@/(l+W)) 

for E small enough. For any x in C, , 

Thus 

WC, ,4 G C WyW, 42) 

A 

where the sum is over the possible sets A = A(x) and C,(A) is the 

set of all sums 

Here we use a crude estimate from Lemma 5.6 to obtain for E small 

enough 

N(C, , E) < exp (~-~/(l+~fl)) (3&“/c)” 

< exp (~-~/(l+~)). 

Thus r(C,,) < 2/(1 + 201). Letting 01 t fl t y we infer 

r(q) < 2/u + 2r). Q.E.D. 

By Proposition 6.9, to prove Conjecture 3.3 for octahedra its uffices 

to prove the following, where a, = (log n)-lj2, n > 2. 

PROPOSITION 6.13. r(Oc ({am})) = 2. 

Proof. Y > 2 since this Oc is not a GC-set (Corollary 3.2, Propo- 

sition 6.7), or by volumes (5.8 (a) and 6.11). 

To prove Y < 2 we shall use the method of the previous proof with 

some additional complications. Let E > 0 and 6 > 0. Given x in Oc 

let A(x) be the set of all j such that 

1 Xj 1 > l 2/4Uj2 = (C” lOgj)/4* j 2 2. 

Then (for E small enough) A(x) has at most 4/e2 elements. The


largest possible integer 12 in A(x) satisfies n < exp (4/e2). For any x 

in Oc 

Let Q(c) be the number of possible sets A(x) for a given E > 0. Then 

by Lemma 5.6, 

N(Oc, E) < Q(c) (~/E~)~/s’ < Q(c) exp (e-“-*) 

for E small enough. 

(The estimate Q(e) < n41cB < exp (161~~) is clearly inadequate.) 

Let s be a positive integer such that l/s < 8. For I = 0, l,..., s - 1, 

let 

Z,, = {j : 4.5-2r’s < log j < 4e-2(r+1)/r}. 

If j E A(x) n Z,, , then 

so the number of elements of A(x) n Z,, is at most e2(r-8)/8. Thus the 

number of ways of choosing A(x) n Z,, is at most 

[exp (46-2(r+1)/s)lr*(r-r)‘I = exp [4e-2w+1)/s E2w-8)/s] < exp (E-2u+8))s 

Thus for E small enough 


Q(E) Q 2e4 exp (s~-~(l+a)) < exp (E-S-~*), 

N(Oc, c) < exp (e-2-58). 

Letting 6 1 0 we get r(Oc) < 2. Q.E.D. 

Next we show that EW(C) may be strictly smaller than EV(C). Let 

c = oc ({2/(2~ + 1))) x qw4), 

a Banach ball in H x H which of course is a separable Hilbert space. 

Then subspaces adapted to C are uniquely determined, with 

%, = 2p + 11, %+I = ll(n + 1). 

It follows easily that EW(C) = - 7/4. Taking projections of the 

ellipsoid only we get EV(C) > - 3/2. By 5.8 (a), 6.2, and 6.12 we 

obtain Y(C) = 1, EV(C) = - 3/2. Thus in measuring volumes it 

seems better to use EV primarily, as we have done, rather than EW,

324 DUDLEY 

since, e.g., Conjecture 5.9 is false if EV is replaced by EW, and Y and 

EW are no functions of each other over a reasonable range. 

We have not evaluated EV(Oc {b,)) if A({b,}) = + co, although 

then for (b,} bounded we have EW(Oc) = - 1. Thus it is conceivable 

that Conjecture 5.9 could hold even for EV < - 4, but it seems 

unlikely. 

7. PROCESSES ON EUCLIDEAN SPACES 

In this section we apply Theorem 3.1 to Gaussian processes over 

a finite-dimensional Euclidean parameter set, e.g., the usual one 

dimensional “time”. Conjecture 3.3 is also verified in certain cases. 

Since any compact Banach ball is a continuous image of the unit 

interval,4 our hypotheses in general do not restrict the geometry of 

the Banach balls in H which arise, and we do not try to evaluate their 

volumes. 

THEOREM 7.1 (Fernique [7], [7a] for T = cube). Suppose (x1, t E T} 

is a Gaussian process where T is a bounded subset of Rk. Suppose q.~ 

is a nonnegative real-valued function such that 

(a) Ej~~--x~1~ 0 such that 

N(T, 6) < A/Sk for all s>o 

(see [I2], Section 3, I, p. 20; cf. also Lemma 5.6 above). (b) and (c) 

imply ~(8) J 0 as 6 J, 0. 

For any E > 0 let 

s = F(E) = sup {t : q(t) <


Let 6, = Y(1/2”), defined and positive for n large enough (unless 

y 3 0, in which case the conclusion is trivial). Then 6, 4 0 as n --+ co. 

Now 

so 

NC, 1/w < &$znk, 

H(C, l/2”) < log A + k log (l/S,). 

Let .v~ = (log (1/S,J)li2. By Th eorem 3.1 it suffices to prove that 

(Note how the dimension becomes irrelevant.) 

Now 

so 

I 

p)(e-2’) > l/2” for %a-1 < x < XT& , 

IN cp(e-““) a% >, g (xn+l - x,)/2”+’ = 

n-N 

*sg+l %P - f %n/2*+1 

m--N 

= 8 n ;+l %P - x,PN+‘, 

so the required series converges. Q.E.D. 

Fernique [7j shows that Theorem 7.1 is optimal of its kind in a 

sense, even for k = 1, since if 

i 

m 

fp(e-““) dx = + cc 

and CJZ 

satisfies some additional mild monotonicity assumptions, then 

counterexamples to sample continuity exist. However, note that we 

may take a process xf on T = [0, l] satisfying the hypotheses of 

Theorem 7.1 and transform it by a “steep” homeomorphism f of T, 

e.g.f(t) = l/log (l/t), into a process x!(t) which may no longer satisfy 

7.1. (c) but of course is still sample-continuous. The E-entropy of the 

range is unchanged, so Theorem 3.1 applies to xtfl) and has a broader 

range of applications. Note however that such a transformation 

destroys stationarity of the process, and for stationary processes 

Theorem 7.1 may be essentially the best possible.

326 DUDLEY 

It has been shown [A that for T an interval, hypothesis (c) of 

Theorem 7.1 can be replaced by any of several conditions, of which 

the best ([4J p. 186, 3”) seems to be 

f 2k’2[,(1/29’/” < co. 

k=l 

But this condition is easily shown to imply Fernique’s. 

Next we discuss random Fourier series and the work of Kahane [IO]. 

Let {xt , t E R) be a Gaussian process, stationary and periodic of 

period 27r. [Note: Fernique’s counterexamples showing that Theo- 

rem 7.1 (c) cannot be improved are all of this type, so the additional 

hypotheses do not change that situation.) We assume xt is continuous 

in probability and that Ex, z 0. It is then well known and not hard 

to prove that a version of xt is given by 

q(u) = 9 + f f3,(& sin nt + 7n cos nt), 

n-1 

(1”) 

where the &(o) and Q( w ) are all independent, normalized Gaussian 

random variables and the /3, are nonnegative constants, C &a < 00. 

(Conversely, any such series (1”) defines a process of the given type.) 

Kahane [JO] assumes /3,, = 0, which does not affect the sample 

continuity. 

Let 

11.+,‘+1 

ti2 = c &2. 

s--e'+1 

(Note: ti are not values of t!) Kahane ([IO], p. 2, Theoremes 3, 4) 

proves the 

THEOREM. The condition C& t, < co is necessary for sample 

continuity or boundedness of xt and, if the ta are decreasing, also s@icient 

(even for almost sure uniform convergence of ( 1”)). 

Neither half of the above theorem will be proved here, and I 

doubt that the methods of this paper would give such a complete 

result. However, it will be shown that Conjecture 3.3 holds to the 

extent that Kahane’s rather sharp result applies. Also we shall treat 

some additional cases where Kahane’s theorem does not apply but 

the conjecture still holds. 

PROPOSITION 7.2. Suppose t, > t, 2 - -- and C t, < 00. Let S 

be the set of all xI in H. Then r(S) < 2.


Proof. We can restrict ourselves to 0 < t < 2~. For any s and t 

in (0, 271), 

Let 

Jws - 4”) = E 1 rnfl /%A( cos 11s - cos nt) + fimTn(sin tls - sin &)I21 

= 2 f pm2(1 - cos (n(s - t))). 

n-1 

B” = ;l Pm2 = gl ti2, b = gl ti . 

Given E > 0, we choose a minimal M(E) such that 

For all x, 1 - cos x < x2, so if 


Hence 

1 s - t ) < 42@M, 

2 5 /3%2(1 - cos (n(s - t))) < q4 

n-1 

N(S, E) < 2 %h,BM/e + 1. 

Now M(C) < 2i for the least i such that 

For any 6 > 0, 

for l small enough by (5.2). Thus 

n({ti}, l 2/8b) < ( 1/e2)l+* 

M(r) < 2.~-~‘~+5’ 

for E small enough. Hence r(S) < 2. Q.E.D. 

PROPOSITION 7.3. If C /Ia < co, then series (1”) conwerges uni-

328 DUDLEY 

formly in t with probability 1, so xt is sample-continuous. Then if X 

is the set of all x1 in H, r(X) < 2. 

Pyoof. CMIt,I + hlbndh ence (1”) converge uniformly in t 

by the three-series theorem. We represent X in H as follows: let 

{qn} be an orthonormal basis, and 

X = I f &J{cos nt) y2% + (sin nt) y+i] : 0 4 t < 27f 

t . 

n=1 

Then X C B({b,)) where /I, = bznmI = b,, , and C b, < a~. As 

remarked after Proposition 6.6, r(B) < 2, SO r(X) < 2. Q.E.D. 

For “lacunary” random Fourier series of the form 

%(a) = fj Igk!s&J) cos bk% 

k=l 

where nkfl/nk > y > 1 for all k, & > 0 and the & are independent 

normalized Gaussian random variables, it is easy to see that C t, < 00 

implies C & < co. Thus Kahane’s theorem and Proposition 7.3 

together imply that Conjecture 3.3 holds for lacunary series (without 

any further monotonicity assumptions). 

8. COMMENTS ON THE CONJECTURES 

Of the three Conjectures 3.3,5.4, and 5.9, Conjecture 5.4 is supported 

by 5.9 which has nothing a priori to do with Gaussian processes. 

One might seek similar support for 3.3. But Y(C) > 2 does not imply 

(for octahedra) that the W,(C) are too large to satisfy Theorem 5.3. 

However, Theorem 5.3 may not be the best possible, and the largest 

V,(C) and W,(C) I know for a GB-set C are those of Oc({K(log n)-‘j2}), 

K > 0, namely 

VJC) 2 W,(C) = $ fi (logj)-1’2. 

* 3=2 

The following approach to some of our problems might seem 

natural, Given E > 0 and C, C Rn, let C,c be the set of points within 

E of C, . Then 

N(G ,24 B L.(C,“)lW. 

For C, convex, X,(C,S) can be expressed in terms of “mixed volumes”


of C, (Bonnesen and Fenchel [.?I, Paragraph 29, p. 38; Paragraph 32, 

p. 49; Paragraph 38, p. 61). I believe some estimates of Santalb [16] 

are of this sort. 

However, such estimates do not seem adequate for our purposes. 

Consider for example Oc ({log n)-‘l”)). To estimate N(Oc, E) is more 

or less equivalent to estimating N(Oc, , c/2) where Oc, is the intersection 

of Oc with the span of v1 ,..., vn , and n is approximately 

e4/Ea. Then every point of the boundary of Oc, is at least (n log n)-li2 

from the origin, so 

AJoc,q > c&y 1 + 1/2n19n, 

Wk%~~ B exp (Y exp W2)) 

if y < Q and 12 is large enough. Thus this method seems quite inferior 

to that used to prove Proposition 6.13, in this case, since it produces 

an extra exponentiation. 

ACKNOWLEDGMENTS 

I am greatly indebted to Volker Strassen for the idea of introducing a-entropy into 

the study of sample continuity of Gaussian processes, and for the statement of the 

result which now appears as Corollary 3.2. 

Another main result, Theorem 5.3, is proved using L.A. Santalb’s theorem [17] on 

volumes of convex symmetric sets and their polars. I thank G. D. Chakerian for 

telling me of Santalb’s result via a network of mutual friends. 

REFERENCES 

1. BAMBAH, R. P., Polar reciprocal convex bodies. Proc. Cambridge Phil. Sot. 51 

(1955), 377-378. 

2. BELYAEV, Yu. K., Continuity and Hijlder’s conditions for sample functions of 

stationary Gaussian processes. Proc. Fourth Berkeley Symp. Math. Stat. Prob. 

2 (1961), 23-34. 

3. BONNESEN, T. AND FENCHEL, W., “Theorie der Konvexen Kiirper.” Springer, 

Berlin, 1934. 

4. DELPORTE, J., Fonctions alCatoires presque sQrement continues sur un intervalle 

fern& Amt. Inst. Henri PoincarL B.1 (1964), 111-215. 

5. DOOB, J. L., “Stochastic Processes.” Wiley, New York, 1953. 

6. DUDLEY, R. M., Weak convergence of probabilities on non-separable metric 

spaces and empirical measures on Euclidean spaces. IZZ. I. Math. 10 (1966), 

109-126. 

7. FJSRNIQIJE, Xavier, Continuitb des processes Gaussiens, Compt. Rend. Acad. Sci 

Paris 258 (1964), 6058-60. 

7a. hRNIQUB, Xavier, Continuitl de certains processus Gaussiens. Sbm. R. Fortet, 

Inst. Henri Poincarb, Paris, 1965.

DUDLEY 

8. GELFAND, I. M. AND VILENKIN, N. YA., “Generalized Functions, Vol. 4: Applica- 

tions of Harmonic Analysis (translated by Amiel Feinstein). Academic Press, 

New York, 1964. 

9. GROSS, L., Measurable functions on Hilbert space. Trans. Am. Math. Sot. 105 

(1962), 372-390. 

10. KAHANE, J.-P., Proprietis locales des fonctions a series de Fourier aleatoires, 

Studiu Math. 19 (1960), l-25. 

11. KELLEY, J. L., “General Topology.” Van Nostrand, Princeton, New Jersey, 1955. 

12. KOLMOGOROV, A. N. AND TIKHOMIROV, V. M., c-entropy and e-capacity of sets in 

function spaces (in Russian), Usp. Mat. Nuuk 14 (1959), l-86. [English transl.: 

Am. Math. Sot. Trunsl. 17 (1961), 277-364.1 

13. Lo&, M., “Probability Theory” (2nd ed.). Van Nostrand, Princeton, New Jersey, 

1960. 

14. LORENTZ, G. G., Metric entropy and approximation. Bull. Am. Math. Sot. 72 

(1966) 903-937. 

15. MINLOS, R. A., Generalized random processes and their extension to measures, 

Trudy Moskomk. Mat. Obsc. 8 (1959), 497-518. [English transl.: Selected Trunsl. 

Math. Stat. Prob. 3 (1963), 291-314. 

16. SANTAL~, L. A., Acotaciones para la longitud de una curva o para el numero de 

puntos necesarios para cubrir approximadente un dominio. An. Acud. Brusil. 

Ciencius 16 (1944), 111-121. 

17. SANTAL~, L. A., Un invariante aiin para 10s cuerpos convexos de1 espacio de 

n dimensiones. Portugal. Math. 8 (1950), 155-161. 

18. SEGAL, I. E., Tensor algebras over Hilbert spaces, I. Truns. Am. Math. SOC. 

81 (1956), 106-134.


Algebras with the Same Multiplicative Measures 

JOHN GARNETT* AND IRVING GLICKSBERG+ 

Massachusetts Institute of Technology, 

Cambridge, Massachusetts and University of Washington, Seattle, Washington 

Communicated by John Wermer 


This note is an addendum to [3] and our primary purpose is to 

improve some results given there. For example, Corollary 2.5 of [3] 

is refined to assert (Theorem 1.7) that given algebras A C B C C(X) 

with (ball A-L)” having no completely singular elements, A = B pro- 

vided only that every multiplicative measure for A be multiplicative 

for B. We also show that when X is metric, M,(A) always contains an 

element A* for which f E C(X) n H2(A, A*) implies f E H2(A, A) 

for all X in M,(A), thus giving a general analog of the strongly domi- 

nant representing measures of [2]. Finally a second section gives an 

application to rational approximation. 

1. Notations and definitions are as in [3]. By a “multiplicative 

measure” for A we mean as usual a probability measure on the under- 

lying space X which is multiplicative on A(hence in M,(A) for some 

9 E Vtm,); by a “complex multiplicative measure” we mean a measure 

which is multiplicative but not necessarily nonnegative or real. 

THEOREM 1 .l. Suppose A C B are subalgebras of C(X) and 

vE);mg* If 

%(4 = Jf,(W, U-1) 


EP(A, A) = W(B, A) fm all h in M,(A). U-2) 

Thus by [3, Corollary 2.31, (1.1) implies that for b E B there is a 

bounded sequence {a,} in A with a, -+ b a.e. A, for all h in M,(A). 

* Research sponsored by the Air Force Office of Scientific Research, Office of 

Aerospace Research, United States Air Force, under AFOSR Grant No. 335-63. 

+ Work supported in part by the National Science Foundation. 

331

332 GARNETT AND GLICKSBERG 

Note that conversely (1.2) + (1.1) is trivial since X E M,(A) is 

multiplicative on H2(A, A), hence on B, while M,(B) C M,(A). To 

prove 1.1 it suffices to show f E B lies in H2(A, A) for a fixed h in 

M,(A), and we can assume v(f) = 0. Let f = u + iv with U, v real- 

valued. Since M,(A) = M,(B), J u dh’ = 0 for all A’ in M,(A), and 

( 1.2) of [3] yields 

sup {Re p)(a) : a E A, Re a < U} = inf {h’(u) : A’ E M,(A)} = 0. 

Hence there are f, = u, + iv, in A with u, and 

u - u,, dh = 0 - Re I --f 0. s 

We can of course assume J v, dh = 0, and so we have elements 

of B with u - u, > 0, 

I 

f - fn = (u - 21,) + i(v - %I) 

v-wer,dh=O=lim u-uu,dA. 

I 

Because of the first two of these conditions, a classical inequality 

[6 P. 2541 appl ies, as was observed by Lumer [S]: 

(1 1 w - v, p/2 dq < 2 * j 24. - 24, dh. (1.3) 

Indeed, U = u - u, > 0 insures that the element 

u + iv = (u - un) + i(TJ - t+J 

of B has a root (U + iT/‘)l12 with positive real part in B, and since A is 

multiplicative on B, 

(1 my2 = (1 u + ivdy = J (U + ivy dA 

= Re I (U + iV)li2 dh > cos & 7r I ( U2 + V2)1/2’1/z dA 

2 -& J I v 11’2 dh. 

Without loss of generality we can assume CE==, J u - u, dh < CO, 

so Cl, (u - u,) < 00 a.e. A, and u, + u a.e. A; again we can assume

ALGEBRAS WITH THE SAME MULTIPLICATIVE MEASURES 333 

Cz==, ( J’u - u, dh)1/2 < co as well, so that by (1.3) v, -+ v a.e. h. But 

now et/n --+ en boundedly a.e. X for t > 0, so eIf E H2(A, h) by domi- 

nated convergence. Because f is continuous, t-l(e”f - 1) + f uni- 

formly as t L 0; so f E H2(A, A). 

We should remark that Theorem 1.1 is an easy consequence of [4, 

4.21. Indeed M,(A) = M+,(B) im pl ies, by that result, that for h E M,+,(A) 

and f = u + iv E B with y(f) = 0, u is the real part of 

F = u + iV E H,2(A, X). So v - V = i(F -f) is in H,2(B, h), real- 

valued, hence zero since h is multiplicative on H2(B, A): 

J-(v- V)vh=(jv- q2=0. 

So f = F E H2(A, h). 

Before proceeding to the consequences of 1.1 we note that (1.1) 

infects entire parts. 

COROLLARY 1.2. Suppose A C B C C(X), q~ E %JIB , and 

J%(A) = %W (1-l) 

Then the part PA in 1131, containing q~ coincides with the part PB in %I&, 

containing y, and (1.1) holds throughout the part. 

Proof, Since H2(A, X) = H2(B, X) for all A in M,(A) by 1.1, we 

know from [3, Corollary 2.31 that for b E B there is a bounded sequence 

{a,} in A that converges to b except on an M,(A)-null set. Since 

M,(A)-null sets and M,(A)-null sets coincide if $ lies in PA [3], 

a, + b except on an M&A)-null set, so that by dominated convergence 

each X in M,(A) is multiplicative on B, and all yield the same I,J in 

‘9JlB extending #, with, evidently, 

~$P) = ~,W. (I-4) 

Now (1.4) im pl ies 6 E PB since h in i&(B) = M,(A) and h’ in 

(1.4) are not always mutually singular. So each z+5 E PA is the restriction 

of a 4 in PB , while each such restriction lies in PA; since (1.4) implies 

$ -+ $ j A is l-l, we can identify both parts, and (1.4) becomes (1.1). 

Our next application of 1.1 reflects the fact that (1.1) has some 

continuity properties not at all apparent in (1.2). For the sake of a 

later application we shall consider complex multiplicative measures; 

if h is one we shall take H2(A, h) as the closure of A in L2( / A 1) and 

H”(A, h) as L”(j h 1) n H2(A, h). Note that as usual H”(A, A) is an 

algebra on which A is multiplicative. (We could equally well replace

334 GERNETT AND GLICKSBERG 

L2 and H2 by L* and HP, 1 = 4 (# + ?u (g”> = (8 (ICI + 4’) (d>” = i - 

So M,(A) = M,(A) = M,(B), and f E B C H2(B, h) = H2(A, /\) 

for all h in M,(A) by 1.1. As a consequence of the proof it will be 

worthwhile to set down 

COROLLARY 1.4. Given algebras A C B C C(X) and v E m* , if 

each element of M,(A) is multiplicative on B then 9) has a unique exten- 

sion $Y in %R, and 

M,(A) = M,(A) = M,(B). 

COROLLARY 1.5. If X is metric and q E nA there exist X* in M, for 

which 

c(x)nH2(A,h*) = c(x)n n H2(A,h). 

AOM~ 

Thus [3, 2.31 for f in that set there is a bounded sequence (an} in A with 

a, --+ f a.e. h, all h in M, . 

Proof. Since M, is separable, we have a w* dense sequence {h,} 

in M,, and we need only set X* = CT 2-12hn; now f E H2(A, h*) 

implies f E H2(A, h,), and 1.3 applies.


We shall call an element A* of M, with the property in 1.5 central. 

Somewhat more generally, we can produce central representing 

measures via 

COROLLARY 1.6. If h E M, and M, n L2(h) * h is w* dense in 

M, , or more generally, if a set E of complex measures in L2(h) * A which 

represent q~ has M, in its w* closure, then h is central. 

Proof. Suppose f E C(X) n H2(A, A). Then if g is any polynomial 

in f with coefficients in A, g lies in the same set, so we have a sequence 

{un} in A with a, -+ g in L2(X). Thus for hh in E, 

and hh - h annihilates the subalgebra B of C(X) that A and f generate. 

Since B C H”(A, h), h is multiplicative on B, so all hX represent the 

same functional rj? on B, as must the elements of M,(A) in the w* 

closure of E. So we have M,(A) = M,(A) = M,(B), and our con- 

clusion follows by 1 .l. 

In case there is a single part P for which E = uWp M, is w* dense 

in the set of all multiplicative measures, a central representing 

measure A* for v has a stronger property: f E C(X) n H2(A, A*) 

implies f E H2(A, A) f or all multiplicative A. For if B is again the 

subalgebra A and f generate we have, exactly as in 1.3, an extension 

4 of y to B with 

M,(A) = M,(A) = M,(B); (1.5) 

by 1.2, P is a part for B and all the elements of our w* dense set E 

are multiplicative on B, so the same is true for every h multiplicative 

on A. 

But now by 1.4 any q in m, has a unique extension rjj in ‘9XB satisfying 

(1.5), so our conclusion holds by 1.1. 

The last part of this argument is precisely a proof of the fact that, 

given algebras A C B C C(x), if all multiplicative measures for A 

are multiplicative on B then %RA and !LRB are homeomorphic under 

the natural map and can be identified, with (1.1) holding everywhere 

on %RA = !LRB. This lforms the basis of our extension of [3, Corollary 

2.51: 

THEOREM 1.7. Suppose A C B C C(X) are a&ebras, (ball AI)e has 

no completely singular elements, and every multiplicative measure for A is 

multiplicative on B. Then A = B.


As we have noted, nm, = mB and M,(A) = M,(B) for v therein, so 

W(A, /\) = H2(B, X) f or every multiplicative measure h for A by 1.1. 

Thus B C A follows from [3, 2.41. (Of course we could just as well 

allow (ball A-L)e to have completely singular elements C Bl.) More- 

over [3, 2.41 could itself be improved (replacing “all h in” by “a w* 

dense set of X in,” as is clear from 1.3). We shall state formally only 

one more improvement of a result in [3], that of the analogue of 1.7 

avoiding completely singular measures. The proof is a trivial modifica- 

tion of that of [3, Corollary 2.71, using 1.1. 

THEOREM 1.8. Suppose1 5 C A is a nonvoid set of invertible elements 

of C(X), non-invertible in A, and 5-l u A generates C(X). Then if 

B 1 A is a subalgebra of C(X) f or which, for eachf E 3, Mf(A) = Mf(B) 

(so the probability measures orthogonal to fA are orthogonal to fB), then 

B = A. 

We conclude this section by noting a reformulation of (1.1). 

Recall [3, (1.2)] that for u E CR(X), v E m, , 

sup {Re v(a) : Re a < u, a E A} = inf {h(u) : h E M,(A)}. (1.6) 

If M,(A) = M,(B) th en in particular for u = Re b the right side is 

precisely X(Re b) = Re v(b), so 

Re q(b) = sup {Re p(a) : Re a < Re b, a E A}. (1.7) 

The converse also holds, and (1.7) holds for every b E B 8 

M,(A) = M,(B). For if h* E M,(A)\M,(B) we have u E CR(X) with 

X*(u) + 1 < inf {h(u) : A E M,(B)} = sup {Rev(b) : Re b < u, b E B} 

which coincides with the left side of (1.6) because of (1.7). So 

h*(u) + 1 < inf {h(u) : h E M,(A)}, our contradiction. 

Thus (l.l), which holds throughout a part if at all by 1.2, amounts 

to a condition which could be expressed by saying the A-super- 

harmonic minorant of each element Re b of Re B coincides with Re b 

on the part containing F (or just at p)). 

1 When such an 5 exists and X is metric there is a single probability measure p 

for which A = C(X) n W(A, p). For 3 can be taken countable, $j = {fn}; with h, 

central for &if-, set p = C2-“h,, . Thenf E C(X) A EP(A, cl) implies ffm E W(C + fnA, h,) 

and so in the corresponding set for all h in M fn . Since fn , f;’ E C(X), f E W(A, X) for 

all such h, so fe A by [3, Corollary 2.61.


Now as a variant of 1.7 we have 

THEOREM 1.9. Suppose A C B C C(X) are algebras, (ball Al)” 

has no completely singular elements and one q~ in each part of m, lies in 

Then A = B isf” 

m B’ 

Re q(b) = sup {Re y(a) : Re a < Re b, a E A), all bEB, 

for all such q~. 

Both 1.7 and 1.9 apply to R(K) C A(K) for any compact 

K C C : R(K) = A(K) zfl, f or some z in each R(K)-part of K = !J&(,) , 

Ref(z) = sup {Re r(z) : Re r < Ref, r E R(K)} 

for each f in A(K), or ifl every multiplicative measure on R(K) is multi- 

plicative on A(K). 

2, We conclude with an application of the results of [3] to 

rational approximation, specifically to a case which is not accessible 

via the results of [l, 21. Let K C C be compact, with connected3 

interior K”. Suppose the components Ho (unbounded), Hr , H, ,... of 

C\K have pairwise disjoint closures which accumulate in a set E 

disjoint from Uy=, aHi (so 8K = Uj”=, aHj U E). 

In order to conclude that R(K) = A(K) we must make several 

hypotheses concerned, in a way, with both the size of E\aH, and the 

density with which the Hj approach E\aH,: we suppose 

(a) the harmonic measure X,(E\aH,) = 0, x E K”, 

(b) on E\aH, there is a w* measurable map z -+ vz E M&A(K)) 

with v,(E\BHo) = 0, 

(c) there is a 6 > 0 and for each j > 1 there is a smooth simple 

closed curve y3’ having only Hj in its interior and disjoint from aK, with 

X,(aH,) > 36 for z in (J& yj . 

Under these hypotheses R(K) = A(K). (When E\aH, is in fact a 

singleton, (b) says simply that it is not a peak point for A(K).) 

* In the corresponding variant of Theorem 1.8, Mf(A) = M’(B) is replaced by 

0 = sup inf Re( fb - fa)(x) 

0 z 

(for all b E B,fe 3) which is a simplified version of Eq. (1.7) for ‘p : z + fb + z and 

the algebras C + fA, C + fB. 

3 For convenience. We could also allow the HJ- to meet in finite (or certain infinite) 

clusters (taking H, then as the union of the corresponding components); the basic 

requirement is simply that we can obtain Eq. (2.5) below (cf. [2]).

338 GERNETT AND GLICKSBERG 

On P(8K) the functional 

f-,,$& 

{where f* is the harmonic extension off to K” and the normal derivative 

is multiplied by the element of arc length) represents the period of 

the conjugate harmonic function about Hj and so annihilates. A(K) 

[I], [Z]. It is represented by a real measure Q on aK which we take 

normalized so that Ij Q 11 = 1 and Q < 0 on aHj , >, 0 on X\aHi . 

(Q has opposite signs on these sets, as is easily verified.) We shall 

prove our result by showing (vr , Q ,...} forms a basis for 

{p E (Re R(K))J- : 1 p 1 (E\BH,) = O}. Since Q is also the result of 

sweeping a measure 7; on the boundary of an “annular” region, 

containing yi and bounded by aHj and a curve y(i close to yi , 

bY (a)* 

Our hypothesis (c) is required only to show 

7@Ho> 3 6, j> 1; (2.1) 

evidently, since r; carries half the mass of V,J; , and 117; )I > 11 Q 11 = 1 

by (c). (Actually (2.1) could serve equally well as (c) as our hypothesis.) 

Now if Q” represents the measure we obtain by sweeping qi to the 

boundary of the compact set K,, = C\u,“,, Hi 3 K then for 1 < j < n, 

since ~j > 0 on aK\uiEo aH,, we have 

7lwo) >, 7iWo) b 8. (2.2) 

Moreover, qj” = 0 for j > n, and rlj”, j < n, still gives a multiple of 

the conjugate harmonic function’s period about Hi for f E CR(aK,J. 

Now given constants cj , 1 < j < 1z we set g = s@ [ciqn(aHj)] on 

i?H,, 1 \


if 0 is chosen appropriately, so 

But 

k#i 

(2.3) 

which carries half the mass of vj R. The other half is carried by aH, 

so from (2.3) 

Now let p E (Re R(K))l. Since 

= s i I cj I = 6 i II 9% II * 

j-1 3-l 

in order to show p J- A(K) we can show this for 

(2.4) 

an element of (Re R(K))1 which vanishes on all subsets of E\aH,, by 

(b) again. So it suffices to consider TV E (Re R(K))* carried by 

u&, W,. Let sn denote the operation of sweeping a measure to the 

boundary of K, = C\u,“,, Hj . We have 

(2-5)


since [I], [2] the Q”, 1 < j < n, span (Re R(K,))l. Since s”s”p = P/.L 

for m < n, and smqj = 0 = smvjn for j > m, we have 

with the same cj; thus p uniquely determines a sequence {cj} satisfying 

(2.6) for all m. 

We have 

by (2.4), and thus Cj”=, Cjrlj is an element of A(K)‘- (since each qj is). 

But now 

Sn /J - f Cjr)j = 

( 1 

j=l j=l 

Sn/Jd - C Iz Cjvj" = 0 for all 71, 

while p - cj”=r cj?7j is carried by uj”=, aHj, SO we conclude that the 

measure itself vanishes, and ,U = Cy=, Ciqj E A(K By [3, 2.5 infra], 

R(K) = A(K). 

When E\aH,, lies in a peak interpolation set for R(K) (and {ri} 

is any homology basis of smooth curves in K”) we can obtain 

R(K) = A(K) once we know that for each p in (Re li(K))l, 

{& cjqj} has a bounded subsequence, where {cj> satisfies (2.6): for 

if v is a w* cluster point of such a subsequence (hence in A(K then 

Pv = Cy=, Cj7)j" = S"p since S" is w* continuous, so p - v, which 

necessarily is carried by u,” aHj , vanishes as before. 

For example, it suffices to know that for infinitely many n at most a 

quarter of the mass of zy-, cjqj is carried by uz+, aHj since then 

so 

Thus it is easy to conclude that if we form K by removing from the 

closed unit disc a sequence of disjoint discs clustering on the unit 

circle then R(K) = A(K) p rovided the radii tend to zero rapidly 

enough.


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tions. Am. 1. Math. (to be published). 

2. GLICKSBERG, I., Dominant representing measures and rational approximation 

Trans. Am. Math. Sot. (to be published). 

3. GLICKSBERG, I., The abstract F. and M. Riesz theorem. /. Functiod Anal. 1 

(1967), 109-122. 

4. HOFFMAN, K. AND ROW, H., On the extension of positive weak* continuous 

functionals. Duke Moth. J (to be published). 

5. LUMER, G., Herglotz transformation and HP theory. Bull. Am. Math. Sot. 71 

(1965), 725-730. 

6. ZYCMUND, A., “Trigonometrical Series I” (2nd ed.) Cambridge University Press, 

London.


Scattering Theory with Two Hilbert Spaces 

TOSIO KATO* 

Department of Mathematics, University of California, Berkeley, California 94720 

Communicated by Ralph Phillips 

Received May 9, 1967 

1. INTRODUCTION 

The purpose of this paper is to give a general scattering theory 

which can be applied, among others, to the scattering for wave 

equations. 

In previous publications (see e.g. [I], Chapter X) we have developed 

scattering theory applicable to Schrcdinger equations. In this theory 

one is concerned with two unitary groups Uj(t) = e-itHj, 

- CO < t < co, i = 1,2 in a Hilbert space 5 and tries to construct 

the wave operators 

where Pi denotes the projection of 5 on the subspace of absolute 

continuity for Hi. If W+ exists, it is a partial isometry in 8 with 

initial projection Pi and final projection < Pz . W, is said to be 

complete if the final projection is equal to Pz . Similar results hold for 

W- . If both W, exist and are complete, the scattering operator 

S = W$W- is unitary on P&. Sufficient conditions for the existence 

and completeness of the wave operators have been studied extensively 

(see, e.g., [II-131). 

Scattering theories of different types have appeared more recently. 

These are concerned with wave equations, in the ordinary as well as 

generalized sense. Here it has been observed that one has to do with 

unitary groups Uj(t) acting in d@erent HiZbert spaces $$ , j = 1,2. 

It is true that the two spaces are often the same vector space L! equipped 

with different inner products, so that the wave operators can still 

* Part of this work was done while the author held a Miller Professorship. It was 

partly supported by Air Force Office of Scientific Research, grant 553-64. 

342

SCATTERING THEORY WITH TWO HILBERT SPACES 343 

be defined by (1.1). In more general cases, however, (1.1) does not 

make sense and must be replaced by a different definition like 

where J E B(ljl,-tF2). (F or any two Banach spaces X, 9, we denote 

by B(x, 9,) the set of all bounded linear operators on X to 9 and we 

write B(X) for B(x, w).) We shall call J the identification operator, 

though we do not always assume that J is bijective nor even injective. 

We shall call (1.1) with the single space $j wave operators of Schriidinger 

type to distinguish them from the more general ones (1.2) with two 

spaces. 

In what follows we propose to study general properties of the wave 

operators (1.2) an d su ffi cient conditions for their existence and completeness. 

In general IV, are no longer partially isometric, unlike the 

wave operators of Schrodinger type, but it will be seen that in many 

cases of practical importance they are. Later we shall consider more 

specific problems, with applications to the scattering for the wave 

equations and Maxwell equations. In particular we shall deduce some 

of the results found in [5]-[S]. 

It seems to the author that one thing has not been properly 

recognized, or at least not mentioned explicitly, regarding the identification 

operator J. Namely, J is not necessarily uniquely determined 

on the physical basis in individual problems. This fact has been 

obscured by the definition (l.l), which is usually used under the tacit 

assumption that the identification by the identity in the common 

space 2 is natural and unique. 

But a little reflection reveals that this is not so. To illustrate the 

point, let us take the Maxwell equations, which are a special case of 

the systems considered in [S]. Here it suffices to note that the unperturbed 

and perturbed electromagnetic fields are described by unitary 

groups Ui(t) acting in the Hilbert spaces Bj , which are the [L2(R3)lS 

with certain positive-definite density matrices Z$(x), j = 1, 2, where 

E,(x) = El is constant. It is assumed that E,(x) is uniformly bounded 

from above and below so that 5, and $s are the same vector space B 

with different metrics. The element u = U(X) of 2 is the aggregate 

of the electric field E and the magnetic field H. In [S] the identification 

is made by the identity of u as an element of 2. 

This identification means that the states of the unperturbed and 

perturbed fields are identified if they have the same E and H throughout 

113. But why should one not identify the two states by D and B 

(the electric displacement and magnetic induction), for example,

344 KATO 

rather than by E and H ? These two modes of identification are clearly 

different, for E1 f &(x) implies that the dielectric constant and 

magnetic permeability are different for the two fields. Of course there 

are many other different ways of identification which could claim 

equal right to the above ones. 

It would be futile to try to decide which identification is the “cor- 

rect” one; it is not a mathematical problem, perhaps not even a 

physical one. Thus one must admit that in general there can be many 

scattering theories for a given pair U, , U, of groups, according to 

different choices of the identification operator J. 

Fortunately, however, there are practically not too many different 

scattering theories for the given pair U, , U, , for the difference in J 

is often irrelevant asymptotically, and it is exactly the asymptotic 

behavior of the systems that scattering theory is concerned with. 

Mathematically, two identification operators J and 9 are (asymptotic- 

ally) equivalent if s-lim (J - /) Ul(t) PI = 0. In such a case the wave 

operators obtained by using J and J are the same, as is easily seen 

from (1.2). It can be shown in many cases that all reasonable identi- 

fications are equivalent so that we have essentially a unique scattering 

theory. In particular this is the case with the Maxwell equations 

mentioned above. Intuitively this is due to the fact that the waves 

eventually go to infinity, where E, and E,(x) are assumed to be equal 

asymptotically, and thus the identifications by E and H and by D and 

B are equivalent. 

It might appear, after all, that we have arrived at a rather trivial 

conclusion. But there is at least one positive result of these considera- 

tions. Namely, among all equivalent identifications one can choose a 

particular J which is mathematically most convenient. For example, 

it happens frequently that there is a unitary J on 6, to & among 

equivalent identifications. In this case (1.2) gives 

J-‘W* = y&&y cl&(- t) l&(t) Pl , (1.3) 

where us(t) = J-‘U,( t) J is a unitary group that acts in the same 

space 5, as Ul(t) does. Thus we have reduced the problem to that 

of wave operators of Schradinger type, to which existing results may 

be applicable. This also explains why the wave operators are partially 

isometric in many cases. 

Even if the reduction (1.3) to the case of SchrSdinger type is 

available, it does not necessarily follow that the scattering for the wave 

equations can be handled by the known results. The difficulty is that 

the deviation of 0, from U, is often larger than such a deviation


encountered in the ordinary Schrodinger equations, chiefly because 

one or both of the generators of these groups are usually first-order 

differential operators with variable coefficients. But this is no more 

than a technical difficulty, which could exist in Schrodinger equations 

as well if one considered more general types of equations. 

2. INTERTWINING OPERATORS 

Let .sj, , j = 1, 2, be two separable Hilbert spaces. We use the same 

symbols 11 11 and ( , ) for the norms and inner products in Z& and g2 , 

but there is no possibility of confusion. For each j we consider a 

continuous unitary group U, = (Uj(t)), - CO < t < co. We denote 

by Hj the selfadjoint generator of Ui so that Uj(t) = e-iM*. The 

spectral family associated with Hi is denoted by {E,(X)). 

DEFINITION 2.1. T E I?(&, s2) is called an intertwining operator 

for the pair U, , U, (or for the pair HI , Hz) if the following equivalent 

conditions are satisfied: 

H,TI TH, , (2-l) 

b(t) T = TU,(t), -co

346 KATO 

implies that E, , the support of 1 T 1 , also commutes with H, . Since 

T* is an intertwining operator for the pair U, , U, and T* = V* / T* 1 

is its canonical polar decomposition, it follows similarly that 1 T* ) 

and E, commute with H, . 

Substituting T = V 1 T 1 into (2.2) and using the commutativity of 

I T 1 with Ui(t), we obtain [Uz(t) V - VU,(t)] / T I = 0. Since the ran- 

ge of I T I has closure E&r , it follows that [Uz(t) V - VU,(t)] El = 0. 

Since E, commutes with Ul(t) and VE, = V, we obtain the inter- 

twining relationship 

W) v = vu,(t), -aJ


Proof. Since Ur(t) and PI commute, the intertwining property 

of IV+ follows immediately from (3.1). IV+ = W+P, is also a direct 

consequence of (3.1). The two inequalities in (3.2) are equivalent 

to the two equalities there. Thus all the assertions of the theorem 

follow from Lemma 2.2 except IV+ = P, W+ . 

(2.3) for T = W+ implies that Es(S) IV++ = ?V+E,(S) 4 for every 

Bore1 set 5’ and $ E .!ji . Since I+‘++ = W+P,+, we have 

II &(S) w+4 II < II w+ II II J%(S) PI4 II = 0 

if 1 S 1 = 0. Thus IV++ E P&j2 for each 4 E .fi, so that IV+ = P, W+ . 

DEFINITION 3.3. The wave operator IV,. = W+( U, , Ui; J) is 

said to be semicomplete if E,, = PI . It is said to be complete if 

E,, = Pz in addition. 

In general W+ maps P,$, into P.& . IV+ is semicomplete if and 

only if this map is injective. IV+ is complete if and only if the map 

is injective and has dense range P2!&. 

COROLLARY 3.4. If W+ = W+( U, , UI; J) exists and is semi- 

complete, the absolutely continuous part of HI is unitarily equivalent to a 

part of the absolutely continuous part of H, . If W+ is complete, the 

absolutely continuous parts of HI and of H, are unitarily equivalent. 

Remark 3.5. If T E I@, , &) is an intertwining operator for the 

pair U, , U, , then W,( U, , U,; T) exist and are equal to TP, . In 

particular, if W+ = W+( U, , UI; J) exists, then W,( U, , UI; W+) 

exist and are equal to W+ = W+P, itself. 

4. EQUIVALENCE OF IDENTIFICATION OPERATORS 

Given two unitary groups U, , U, , one could in general construct 

many different wave operators by different choices of the identification 

operator J E B(& , 6.J; see Section 1. But some different J’s give 

rise to the same wave operator. 

DEFINITION 4.1. Let J, 9 E B(Bi , &). We say that J and J are 

(U, , +)-equivalent if 

s;l& (J - J) Ul(t) PI = 0. (4.1) 

(This equivalence is related only to one group Ui; the space $, appears 

only in so far as / and J have range space &.J.

348 KATO 

THEOREM 4.2. Let J, J E B(& , 5,). Then 

W+(U,, UC J) = W+W,, Q; /) (4.2) 

if and only if J and J are (U, , +)-equivalent. Here (4.2) means that 

if one of the two members exists, then the other also exists and equals the 

jirst one. 

Proof. Immediate from 

II W- t) J&(t) p14 - W- 4 JW) Pd II = II (I - J> G(t) f’d II . 

In virtue of Theorem 4.2, W+( U, , Ur; J) actually depends only 

on the (U, , +)-equivalence class to which J belongs. Practically this 

greatly reduces the number of different wave operators that can be 

constructed for a given pair U, , U, . 

THEOREM 4.3 (Chain rule). Let Uj , j = 1, 2, 3, be three unitary 

groups acting respectively in Hilbert spaces & . Let J E B(Jj, , &) and 

J’~B(fiz,5& If W+= W+(U,, Ul; J) and W;= W+(% &;J’) 

both exist and if J” E B($j, , &) is (U, , +)-equivalent to J’J, then 

w; = W+( u, > u1; J”> exists and equals W; W+ . WY is (semi)- 

complete if both W, and Wk are (semi)complete. 

Proof. Multiplying the two defining expressions for W; and W+ , 

we obtain 

Wi W+ = cl&n U,( - t) J’Pa JU,( t) PI , 

where we have used the commutativity of P2 with Uz(t). To prove the 

existence of W; and its identity with W;W+ , it is thus sufficient to 

show that 

St% (/‘Pa J - J”) Ul(t) PI = 0. 

Since J” is (U, , +)-equivalent to J’J, it suffices in turn to prove 

But this is equivalent to 

c&n (1 - Pz) JUl(t) PI = 0. 

s-lim U,(- t) (1 - Pz) JUl(t) PI = (1 - Pz) s-lim U,(- t) JUI(t) PI 

[see (3.2)]. 

= (1 - Pz) w+ = 0


IV+ maps PI& into Paz and W; maps Paz into P&, . If both of 

these maps are injective, the same is true of WI = W;W+ . This 

proves the assertion on semicompleteness. If W+ and W; are complete, 

then W+P$j, is dense in P&j2 and W;P&j2 is dense in Pa3 . Hence 

WJPl& = W;W+Pl$& is dense in P&,: WJ is complete. 

5. INVERSE WAVE OPERATORS 

Suppose W+ = W+(u, , ul; J) exists. If J were invertible with 

J-l E B($a , &), one could consider the wave operator 

w; = W+( Ul 9 uz; J”) 

and expect it to be inverse to W+ in a certain sense. Even when J-i 

does not exist, it is often possible to define such an inverse wave 

operator. 

DEFINITION 5.1. Let J E 45, , -5,) and J’ E IL+&, $ji). J’ is 

called a (U, , +)-asymptotic left inverse to J if J’J is (Ui , +)- 

equivalent to 1 (the identity in Hi), that is, if 

dip (J’J - 1) U1(t) PI = 0. (5-l) 

THEOREM 5.2. Let W+ = W+(U, , U,; J) exist and let J’ be a 

(U, , +)-asymptotic left inverse to J. Then W+ is semicomplete, and 

s;& U,( - t) J’&(t) w+ = Pl , (5.2) 

d&n (JJ’ - 1) U2(t) W+ = 0. (5.3) 

Proof. It follows from the definition of W+ that 

mtt> Pl - 7-w) w+ 9 t-+60, (5.4) 

where N means that the difference of the two members tends to zero 

strongly. Applying J’ to (5.4) and using (5.1), we obtain 

G(t) Pl - Y&(t) w+ 9 (5.5) 

which implies (5.2). (5.3) follows from (5.1) by multiplication from 

the left with J and using (5.4). W+ is semicomplete because W++ = 0 

implies PI+ = 0 by (5.2).

350 KATO 

THEOREM 5.3. Let the assumptions of Theorem 5.2 be satisjed, so 

that W+ exists and is semicomplete. W, is complete if and only ;f the 

following two conditions are satis$ed: (i) J is a (U, , +)-asymptotic 

left inwerse to J’, and (ii) W; = W+( U, , US; J’) exists. When these 

conditions are met, Wi is also complete and 

w;w+ = PI, w+w; = Pz. (5.6) 

Proof. Suppose (i) and (ii) are true. It follows from Theorem 5.2 

applied to the triple U, , U, , J’ that W; is semicomplete. Further- 

more, in view of W+ = P,W+ , (5.2) gives W;W+ = PI. This 

implies that %( Wi) 1 PI.!& . But since the opposite inclusion is also 

true, we have %( W;) = PI& . Thus W; is complete. Since there is a 

complete symmetry between W+ and W; , we have also W+ W; = Pz 

and W+ is complete. 

Suppose conversely that W+ is complete. Then %( W+) is dense in 

P&, and so (5.2) implies the existence of W; , with W; W+ = PI . 

Similarly (5.3) implies that s-lim (Jy - 1) Uz(t) Pz = 0, that is, 

(i) is true. 

THEOREM 5.4. Let W+ = W+( U, , UI; J) exist and be complete. 

Let J’ be a (U, , +)-asymptotic left inverse to J. Then J’ E B($& , &) 

is (U, , +)-equivalent to J’ if and only if J’ is a (U, , +)-asymptotic 

left inverse to J. 

Proof. Since J’ is a (U, , +)-asymptotic left inverse to J, the 

same is true of J’ if and only if (J’ - y) JUI(t) PI N 0, which is 

equivalent to (J’ - J’) U%(t) W+ N 0 by (5.4). But the last relation 

implies the (U, , +)-equivalence of J’ and J’ because W+W; = Pz 

by Theorem 5.3. 

6. PARTIALLY ISOMETRIC WAVE OPERATOW 

For various reasons we are particularly interested in the case when 

the wave operators are partially isometric. 

THEOREM 6.1. Let W+ = W+( U, , VI; J) exist. In orde~ that W+ 

be a partial isometry with initial projection PI , it is necessary and suf- 

$cient that


Proof. Obvious from 

THEOREM 6.2. Let W+ = W+( U, , U,; J) exist. W+ is a partial 

isometry with initial projection PI if there is a unitary operator I on 

liI to B2 which is (U, , +)-epuiwalent to J. 

Proof. (6.1) is satisfied if J is replaced by J, which is permitted 

because J is equivalent to J. 

THEOREM 6.3. Let W+ = W+(U, , U,; J) exist. Assume that J* 

(the adjoint of J) is a (U, , +)-asymptotic left inverse to J. Then W+ 

is semicomplete and partially isometric with initial projection PI . W+ is 

complete if and only if W; = W+( U, , U,; J*) exists and J is a 

(U, , +)-asymptotic left inverse to J*. If these conditions are satis$ed, 

then w: = w;. 

Proof. The first assertion follows because (6.1) is satisfied; in 

fact 

lim II J&(t) P&J 11’ = lim VU- t) J*Wdt> PIA PI+) = (PI+, PI+> 

since J*JUI(t) PI - UI(t) PI . The second assertion follows from 

Theorem 5.3. Theorem 5.3 also shows that W+Wi = Pz . Multiplica- 

tion from the left with W$ then gives Wi = WY because 

W$W+=P,, P,W;= WL,and W$P,= Wz. 

7. APPLICATIONS TO UNIFORMLY PROPAGATIVE SYSTEMS 

As the first application of the foregoing results, let us consider the 

general wave equations discussed by Wilcox [S]. Here the spaces z$. 

consist of complex m x 1 matrix-valued functions 4(x) on R”, with 

the norms given by 

i = L2, (7.1) 

where E,(X) are positive-definite, m x m Hermitian matrices depend- 

ing on x E Rn and where C+(X)* denotes the Hermitian conjugate of 

I#(x). It is assumed that E,(X) = E1 is constant and that E,(X) is 

uniformly bounded from above and from below. It follows that $r 

and $a are the same vector space L? = (L2(P))m with different norms.

352 KATO 

It is further assumed that 

J%(x) -+ 4 3 IxI+m. (7.2) 

Uj is the group in Bj generated by the selfadjoint operator 

Hj = iE,(x)-1 i A, $- , 

k=l 

k 

where A, are constant m x m Hermitian matrices. 

In [S] the wave operators are considered with the identification 

operator J which is just the identity in 2: 

14 =h (7.4) 

As was remarked in Section 1, however, J is not the only possibility. 

Another possible choice would be the operator J given by 

(7.5) 

In other words, $1 E sj, and 4s E 8s are identified under J if 

E,+,(x) = E,(x) r&(x). The identification of the unperturbed and 

perturbed electromagnetic fields in terms of D and B correspond to J 

(see Section 1). 

In most cases of practical interest, however, J and J are (U, , -J-)- 

equivalent, so that there is no difference in the definition of the wave 

operators (Theorem 4.2). In fact, this is true if the system U, is 

uniformi’y propagative (see [S]). 

To see this, let 4 E sj, be given by a function in C;(P) and set 

z+(t) = U&)+. Th en z&(t) = du,(t)/dt = - iU,(t) HI+ exists. Now 

it is shown in [S] that zir(t), as a function on Rn, tends to zero as 

t --+ * co uniformly in each compact subset of P. It follows by (7.2) 

that (J - J) &r(t) = E,(x)-r (E,(X) - EJ r&(t) + 0 in s, . In other 

words, we have (J - J) Ul(t) Hr+ -P 0. When 4 varies over Cc, 

Hi+ varies over a dense subset of %(H,), for Hr is the closure of its 

restriction to C; (see [8]). It follows that (J - J) Ul(t) PI -+ 0, 

t -+ & co, for P,fi, is contained in the closure of %(H,), which is 

equal to %(H,)l. 

There are other identification operators of practical interest. The 

most interesting among them is the one p given by 

&x) = E&)-1’2 E;“#(x), (7.6) 

which is half way between J and 1. f is not only (U, , -&)-equivalent 

to J and J, so that it leads to the same wave operators, but it is zmitary.


According to Theorem 6.2, it follows that the wave operators W, 

(which are identical for all J, J, and p) are partially isometric if they 

exist. The unitarity of p follows easily from the following identity, in 

which we set 4(x) = &x) = &(~)-r/~ Ei’s$(x): 

= s 4(x)* E;“E,(x)-~‘~ E,(x) E,(x)-“’ E;‘2rj(x) dx 

= s (b(x)* J%+(X) dx = II 4 1/12- 

The equivalence of 9 with J and J can be proved in the same way 

as above by noting that E2(~)lj2 -+ Ei/2 as 1 x 1 -+ co. 

As was remarked in Section 1, the existence of a unitary indentifica- 

tion operator f equivalent to J makes it possible to reduce the problem 

to that of Schrodinger type, according to 

f-lW* = s-lim 02(- t) U1(t) Pr , 02(t) = f1U2(t) 1. (7.7) 

We have not proved the existence, let alone the completeness, of 

W, . The reduction (7.7) to the wave operators of Schrodinger type 

raises the hope that some of the criteria deduced for such wave opera- 

tors might be applicable. But this does not seem to be easy, for the 

selfadjoint generator of the group 02(t) is given by 

fi2 = j-1H2j = E;1’2E2(x)-1’2 c A 

(k rc 4 E,(x)-~‘~ E:12, (7.8) 

a rather complicated expression. 

It should be noted that the existence of W, has been proved in [S] 

under rather mild assumptions, but the question of the completeness 

of W, is open. 

8. ABSTRACT DIFFERENTIAL EQUATIONS OF SECOND ORDER 

As the second application, we consider the scattering problem 

associated with abstract differential equations of the form 

$+Au=O, -co

354 KATO 

In this section we construct the unitary group U(t) associated with 

(8.1), and the scattering problem relating to two such groups will be 

discussed in the following section. 

We assume in (8.1) that u = u(t) is an element, depending on t, 

of a separable Hilbert space R and A is a nonnegative selfadjoint 

operator in R. The general solution of (8.1) may be written 

u(t) = (cos tB) u(O) + B-‘(sin tB) C(O), (W 

where B = A1/2. The operator B-r(sin tB) E B(R) is well-defined 

in an obivous way even when B-l does not exist. 

(8.2) is a generalized solution of (8.1). If u(0) E D(B), then 

du(t) 

- = C(t) = - (sin tl3) Bu(0) + (cos tB) C(O). 

dt 

(8.3) 

The given Hilbert space R is not a very convenient one for des- 

cribing the system (8.1). The usual procedure is to choose the pair 

(u(t), G(t)} as an element of a new space 8. Then (8.2) and (8.3) induce 

a transformation 

U(t) MO), 40)) = w>, W)). (8.4) 

The U(t) form a unitary group in $ if 5 is made into a Hilbert space 

with the norm 

II @, 4 II2 = II 21 lle2 + II u II23 (8.5) 

where 11 v I\ is the norm in R and 11 u IjB = II Bu 1) . To be more precise, 

we start with D(B) (the domain of B) equipped with the pseudonorm 

11 u IjB and complete it to a Hilbert space [D(B)], and then define 

8 as the product space [a(B)] x R. We use the symbol [D(B)] always 

in this sense and D(B) in the usual sense as a linear manifold of R. 

It should be noted that in [D(B)] any two elements U, w E D(B) such 

that Bu = Bo are identified. Also [a(B)] in general contains ideal 

elements which are not in R. 

The unitarity of U(t) follows from 

II BW II2 + II WI II2 = II wx II2 + II YO) II29 +-4 E qa W) 

which can be verified easily by (8.2) and (8.3). 

The structure of the group U is closely related to that of the group 

{e-f1B} acting in R (cf. [2]). To bring out this relationship, we first 

note that 

Jt = p(B)] 0 W(B), 63.7)


where X(B) is the null space of B and [s(B)] is the closure in R of the 

range of B. The space .$J = [D(B)] x R thus has the decomposition 

sj = [W>l x Pv910 @I x WV (8.8) 

Now U(t) acts in {0} x ‘S(B) trivially, the latter being invariant under 

U(t). In fact, for z, E%(B) we have U(t) (0, ZJ> = {ta, w} = (0, a), for v 

is identified with 0 in [a(B)]. The behavior of U(t) in [D(B)] x [S(B)] 

can be best described by introducing the transformation {u, w} --f {f, g} 

given by 

= Bu + iw, g = Bu - iw, (8.9) 

where B is the unitary map of [D(B)] onto [S(B)] defined as the unique 

extension of B. As is easily seen, (8.9) is a unitary operator from 

[D(B)] x [S(B)] to [R(B)] x [X(B)]. In this “representation”, the 

action of U(t) is given by 

f(t) = e-itBy(0), g(t) = eitB’g(0), (8.10) 

where {f(t), g(t)) is the transform of {u(t),ii(t)) under (8.9) and B’ 

is the strictly positive part of B. In this sense U(t) is unitarily equiv- 

alent to the direct sum 

e-itB’ @ eitB’ @ 1. (8.11) 

Finally we need a description of the absolutely continuous part 

of the selfadjoint operator H which generates U(t). We denote by P 

and Q the projections of 8 and Jt on the subspaces of absolute con- 

tinuity for H and B, respectively. 

LEMMA 8.1. The following three conditions are equiwalent: 

(i) {u, w} E Psj; 

(ii) fEQ% gEQ% 

(iii) Bu E QR, w EQR. 

Proof. Obviously (ii) and (iii) are equivalent. The representation 

of U(t) by (8.10) s h ows that (ii) is equivalent to (i) if 

1% 4 E Pv31 x [S(B)], that is, if w E [X(B)]. Thus it suf- 

fices to show that each of (i) and (ii) implies w E [S(B)]. Since 

Qst C [X(B)] = S(B)‘-, 1 ‘t is clear that (ii) implies w E [g(B)]. On the 

other hand, (i) implies that {u, w} is orthogonal to any eigenspace of 

H. But {0} x 92(B), being invariant under the U(t), is an eigenspace 

of H. Hence {u, w} 1 {O] x g(B), that is, w ER(B)I = [X(B)].

356 KATO 

9. SCATTERING FOR THE SECOND-ORDER EQUATIONS 

Suppose we have two differential equations of the form (8.1) in the 

same Hilbert space St. We distinguish the two systems by subscripts 

j = 1, 2, thus writing Uj = ui(t), Aj , Bi = A:/“, etc. for the quantities 

discussed in Section 8. We introduce the corresponding unitary 

groups Uj(t) acting in the spaces 5, given by 

aj, = p(4)] x fi = [w-u x [w4)1 + (0) x %(B,). P-1) 

Our object is to construct the wave operators W,( Us , Ui; J) by 

introducing an appropriate identification operator J. 

As was noted in Section 1, there is in general no unique choice of J. 

But since we are dealing with groups U, constructed from two 

equations in the same space R, there are certain J’s that seem to be 

natural. Without going too much into the question how natural 

they are, we shall consider the wave operators associated with them. 

In any case we need some assumptions on the relationship between 

the two operators Aj . We make the following basic assumption. 

Condition 9. I. The Schrodinger type wave operators 

C, = W,(B, , B,) exist and are complete. 

This means that 

where Qj is the projection of R on the subspace of absolute continuity 

for Bj (or for A, , equivalently). C, are partial isometries with initial 

projection Qr and final projection Qs . 

Note also the intertwining property 

Cd, C B&k, C$B, C B&z. (9.3) 

Remark 9.2. Since the Bj = A:la are defined in an abstract 

way, they are not “elementary” operators even when the A, are. Thus 

it might appear that Condition 9.1 is difficult to verify in concrete 

problems. This is not necessarily so, however. In many cases 

W,(B, , B,) exist, are complete and coincide with W,(A, , A,) if 

the latter exist and are complete (the principle of invariance of the 

wave operators); see [2], [IO], and [I], p. 543. 

In this section we choose an identification operator J E B($, , .fQ 

that seems to be mathematically the simplest. A different and physic-


ally more reasonable one will be considered in the next section. Let 

Fi be the projection of a on [‘X(Bj)], so that Fi > Qi . J is defined by 

Jh 4 = Qm$% 4 E 52 > {u, 4 E -51 9 (9.4) 

where lPj is the unitary map of [3)(Bi)] onto [X(Bj)] as the unique 

extension of Bi [cf. (8.9)]. I n other words, J{ul , q} = {uz , w2} if 

and only if &u, = Fz&ul and va = v, . Since the ~j are unitary and 

F, is a projection, it is clear that 11 J 11 < 1. A simple calculation shows 

that the adjoint J* E B(& ,a,) is given by 

I*&, 4 = @?@b, 4 E -5, , 

We can now prove (cf. [2]) 

THEOREM 9.3. If Condition 9.1 is satis$ed, the wave operators 

W,( U, , UI; J) and W,( U, , Uz; J*) exist, are complete, and are 

mutually adjoint partial isometrics. 

Proof. Let x E D(B,) and y ~%t(Bi). Then 

G(t) c% Y> = {Ul(Q 4(t)> 

is given by (8.2) and (8.3) with B replaced by B, and u(O), G(O) by X, y, 

respectively. Thus 

B&t) = + eeitB1(Blx + iy) + 8 eft4(B,x - iy). (9.6) 

If we further assume that x, y E QiR, then B,x f. iy E QI~ too so 

that by (9.2) 

B1ul(t) N 4 eeftsaC*(Blx + iy) + Jj eitB2C~(B,x - iy) 

= 4 e-ftBa(B,Cjyc + i&y) + 3 eftBa(B&x - i&y) (9.7) 

as t -+ f co, where the intertwining property (9.3) of C, has also 

been used. Similarly we obtain 

Hence 

4(t) - 7j- - ’ e-f’Ba(B2Cjgc + i&y) + + e’tBa(B,CTx - iC,y). 

(9.8)

358 KATO 

with 

x* = g (C, + c-) x & + (C, - CJ qy, 

y*=r~(c+-c~)B1S+;(C++C-)y. 

(9.10) 

Here Bily denotes, rather improperly, the element of ID(&) orthogonal 

to R(B,) such that B,(Bily) = y (note that y E R(B,) by assumption). 

Set 

Then by the definition of J 

hence 

JW> lx, Yl = J@l(Q WI = MG @>I~ 

&%W = ~&%W~ %@) = W), 

II Jut> 1x9 Y> - U&) lx* 9 Yi> II2 

= II @z(t), %W> - h&>, ~a&)~ II2 

= II M4t> - %&)) ll”R + II 4(t) - 4*(t) IIt3 --+ 0 

(9.11) 

(9.12) 

(9.13) 

as t -+ & co, for the second term in the last member tends to zero 

by (9.9) and the same is true of the first term because by (9.12) it is 

equal to 

note (9.9) and that B,u,,(t) E [%(B,)]. 

Since the {x, JJ> with the above properties (x E D(B,) n QIR, 

Y E WV 

continuity 

n Ql@ f orm a dense subset of the subspace of absolute 

for Hr (see Lemma 8.1), (9.13) implies the existence of 

IV, = W,( U, , Ul; J), with 

W&, y> = {a 3 Y*) 

(9.14) 

for the {x, JJ} restricted as above. (9.10) does not make sense for a 

general {x, r} E !& . But it can be extended to all {x, y) E I’#, (which 

is equivalent to X, y E QIR by Lemma 8.1) if Bil and B, on the right 

are replaced by &l and 8, , respectively, for the ensuing map from 

P#, to P&, is bounded. Furthermore, (9.14) is then even true for 

every (x, r} E& in virtue of the property C, = C&r . In what 

follows (9.10) should be read in the extended sense so that (9.14) is 

true for all {X, r} E .Eil .


Next we shall show that J* is a (U, , &)-asymptotic left inverse to 

J (see Definition 5.1). Since 

II (1 - J*l) G4 4 II& = II ((1 - ~‘,lwa~,) %O> llsj, 

by (9.4) and (9.5), it suffices to show that 

= 11 %l(l -FlF‘2) &u I/B1 

where ul(t) is as in (9.6). Thus it suffices to show that 

But 


(1 - F,F,) 6’ tBIQ1 -+ 0, t--t-&a. (9.16) 

1 - F,F, = (1 - Fl) + Fl( 1 - F,) 

(1 - FJ FitBIQl = SitE1(l -FJQ, = 0 

because Qr < Fl. On the other hand, it is known that 

(1 - Q2) e-itBIQI -+ 0 whenever C, exist (see, e.g., [I], p. 531.) 

Since F, > Q2 , it follows that (1 - F,) e-itBIQr + 0 and (9.16) is 

proved. 

According to Theorem 6.3, the wave operators W,( U, , U,; J) 

are semicomplete and partially isometric. Since the same is true of 

w,tU, > Usi I*) by s Y mmetry, the assertions of Theorem 9.3 follow 

from Theorem 6.3. 

10. ANOTHER IDENTIFICATION FOR THE SAME PROBLEM 

The existence and completeness of the wave operators 

W,( U, , U,; J) proved in Theorem 9.3 depends on a fortunate 

choice of the identification operator J. Mathematically it seems to be 

the simplest one (Birman [2] uses the same J, rather implicitly). But 

it is a different question whether it is a natural choice from the physical 

point of view. 

Physically it would seem more appropriate to identify {u, V} of .sj, 

with the same pair of & (simple identification). But this is possible 

only when [xJ(&)] = [a(&)]. We shall modify this identification 

slightly to allow somewhat more general situations.

360 KATO 

We introduce 

Condition 10.1. D(B,) C %(B,) and 

* II BP II < II &u II < M II BP II, u E WV, (10.1) 

where m and M are positive constants. 

Such a situation may occur, for example, when A, is the Friedrichs 

extension of a nonnegative symmetric operator A’ in A and A, is 

another nonnegative extension of A’. (For more concrete examples, 

see the following section.) 

If Condition 10.1 is satisfied, we have also [ID(&)] C [P(&)] under 

an obvious identification for ideal elements, and 

* II BP II < II &u II S M II &J II, u E P(&)l C PWI. (10.2) 

This implies that the two norms J] &,, and 11 JIBa in [a(&)] are equivalent 

so that [D(&)] is a (closed) subspace of [D(B,)]. Let us denote 

by G the orthogonal projection of [%I(&)] onto its subspace [%)(B,)]. 

We now introduce the identification operator Jon &1 = [B(B1)] x A 

to 82 = [W&J] x R by 

Jo4 4 = i% 4. 

(10.3) 

J is the projection of $r onto $a regarded as a subspace of Z& . (10.3) 

reduces to J{u, V} = {u, V> (simple identification) if [a(&)] = [a(&)]. 

We shall now show that under a certain additional assumption, 

J is (U, , &)-equivalent to j so that W,( U, , UI; J) exist and are 

equal to ?V,(U, , UI; J). In fact, for x E D(B,) n QIR and 

y E %(B,) n Q1~ we have 

II (i - I, ul(t) 6~ r> II = II (k’F,$ - G) udt) bz 

= II (F,B, - &G) u,(t) IIR , 

(10.4) 

where ul(t) is as in Section 9. But Blul(t) N B,u,,(t) in R by (9.9), SO 

that 

~,B,dt) -~dMti(t) = BGZ&) in R. (10.5) 

Since, on the other hand, the metric I( ((s, is equivalent to (/ IIs, on 

[D(&)], we have 

B,%(t) - &%&) = J&44 in R (10.6) 

provided that the following condition is satisfied:

Condition 10.2. 

SCATTERING THEORY WITfI TWO HILBERT SPACES 361 

II UlP> - %k(t) /I& -+ 09 

t-+&al. 

It follows from (10.5) and (10.6) that (10.4) tends to zero as t --t & co, 

showing that J is equivalent to J. 

Let us now introduce an inverse identification operator 

J’ E ~(~2 9 81) bY 

Y’h 4 = e4 4, {u, 4 6 52 - (10.7) 

J’ is simply the canonical injection of 5, into 5, . We shall show that 

y is a (U, , &)-asymptotic left inverse to J. Since yf{u, o} = {Gu, V} 

for {u, w> E $r , we have 

II (1’9 - 1) Wt) ix, ~1 II = II (G - 1) W 11~1 

with the notation used above. But since z++(t) E a(&), we have 

/I (1 - G) ul(t> IIB, < 11 q(t) - %&) iiBl + 0 

by Condition 10.2. Thus (fJ - 1) Ul(t) PI --+ 0, as we wished to 

show. 

Noting Theorems 5.3 and 9.3, we have thus proved 

THEOREM 10.3. Suppose Conditions 9.1,10.1, and 10.2 are satisfied. 

LetJ,JandJ’b e d$ e ne d as above. Then J is (U, , j-)-equivalent to J, 

so that W, = W,( U, , UI; J) exist .and are equal to W,( U, , UI; J) 

constructed in Theorem 9.3. In particular they are complete and partially 

isometric with initial projection PI and $nal projection Pz . J’ is a 

(U, , &)-asymptotic left inverse to J. Thus W,( U, , Uz; y) exist, are 

complete and equal to W+ . 

Remark 10.4. Theorem 10.3 is somewhat unsatisfactory in that 

Condition 10.2 is not easy to verify directly. Thus it is desirable to 

give some sufficient conditions for it. 

THEOREM 10.5. Under Conditions 9.1 and 10.1, each of the follow- 

ing conditions (all involving convergence in a) is suficient for Condition 

10.2 to be satisfied: 

(4 

(b) 

for each u E D(B,) n Q,R; 

m= 1; 

B,(edit~ - ebifBaC*) II 4 0, t+*=b

362 KATO 

(c) Bleit4e-“tB2Cg4 -+ B,u, 

for each 24 E D(B,) r\ f&l; 

(4 

for each v E a(&) n Q.$; 

(B, - B,) emitB% -+ 0, 

(e) Wl) ’ W2) and (A, 

for each v E X+4,) n Q&t. 

t-+z!c:oO, 

A2) emitBee, + 0 

Proof. Looking at the expressions for q(t) and u,,(t) in terms of 

x,yandx,,y,, respectively, given in Section 9, it is easy to see that 

(b) implies Condition 10.2. Also it is clear that (b) and (c) are equivalent. 

(d) implies (b), as is seen by setting v = C,u and noting that 

=e -itBeB2Cku = B2ewitB~Chu 

as t--t&co. 

To show that (a) implies (c), we first note that (c) is always true 

if the strong convergence +- is replaced by weak convergence. In 

fact, the left member of (c) is bounded for t -+ & co, its norm being 

majorized by 

11 Ble-“tBaC+u 1) < m -’ 11 B2e-itB2C*u I/ = m-l II B&3 II = mm1 II G&u II 

= m-l 11 B,u II . 

Furthermore, it is easy to see that the scalar product of the left member 

of (c) with w E a(&) is equal to 

(eitB1e-itB2C;tu, B,w) -+ (u, B,w) = (B,u, w). 

Since a(B,) is dense in St, this proves the weak covergence. If m = 1, 

the above estimate shows that the weak limit has norm not smaller 

than the converging elements. Thus the convergence must be strong, 

by a well-known theorem. Thus (a) implies (c). 

The same argument shows that (e) implies (c). In fact, writing 

v = C,u we have the estimate 

11 Ble-itBzv 112 = (eitBaAle-itBav, v) - (e-zA2e--PtBsv, v) 

= (A,e, w) = 11 B,v II2 = /I B&u II2 = II C+B,u II2 = I/ B,u II2


as above. Here we assumed that u E D(A,) n QIA so that 

v E D(A,) n Q$, but the general case u E a(&) n QIR can be dealt 

with by a standard method. 

Remark 10.6. Consider the special case B(B,) = D(B,) so that 

P,(4)1 and PwI are the same vector space. Then Theorem 10.3 

is symmetric with respect to W,( U, , U,; _I> and W,( U, , U2; jl), 

for both J and /’ are the simple identification in the same vector space 

!& and .fj, . In order to verify Condition 10.2, therefore, one may as 

well verify the conditions of Theorem 10.5 with the subscripts 1, 2 

exchanged. (For example M = 1 instead of m = 1.) 

11. APPLICATIONS TO THE SCATTERING FOR WAVE EQUATIONS 

We shall apply the foregoing results to the construction of wave 

operators for various scattering problems related to the ordinary 

wave equations. It would be of some interest that we are in this way 

able to deduce some important results without using the Huygens 

principle. 

A. Wave Equations with a Potential (cf. [A) 

First we consider the wave equation with a potential 

a224 

m - Au + q(x) II = 0 

in R3, where q(x) is a real-valued measurable function. In order to 

apply the results of Sections 8-10, we start from the Hilbert space 

K = Lz(R3) and define the selfadjoint operators A, = - d and 

A, = - A + q(x). A s is well known (see, e.g., [I], p. 303), such an 

A, is well-defined under a mild assumption on the potential q(x). For 

example, it suffices that q(x) is the sum of a bounded function and an 

L2-function; then we have %(AJ = B(A,). Furthermore, we assume 

that A, > 0. 

We can now construct the spaces 5j and the unitary groups U, , 

j = 1,2. (The se are identical with the ones considered in [7].) 

For Condition 9.1 to be satisfied, we need stronger restrictions on 

q(x). For example, it suffices to assume that (i) q EU(R~) n L2(R3) 

(see, e.g. [I], p. 546). Another sufficient condition is that (ii) q ELM, 

q(x) is locally Hold er continuous with a finite number of singularities, 

and q(x) = o(I x 1-2--.), E > 0, as ] x 1 -+ co (see Ikebe [II]). It 

580/1/3-8

364 KATO 

should be noted that in each of these cases, not only W,(A, , A,) 

exist but the principle of invariance holds so that Condition 9.1 is 

satisfied. The invariance in case (i) is a consequence of a general 

theorem related to perturbations of the trace class (see, e.g., [I], 

p. 543). In case (ii), it follows from a more general theory recently 

developed by Kuroda [3]. 

Under these conditions the existence and completeness of 

W,( U, , Ui; J) and I+‘,( U, , Uz; J*) follow from Theorem 9.3. 

Since none of the Bi has eigenvalue zero, J identifies {ui , vi} E ~j, 

and {z+ , ~a} E $a if and only if&u, = &u, and zli = va , and J* = J-l. 

But these wave operators may not be very interesting physically, 

for the identification operators J and J* are rather artificial. 

A more realistic identification is given by I of Section 10, which 

identifies the above elements if and only if ui = ua and vi = vo2 

(this / is adopted in [7j). It should be noted that we have 

w4) = W2) in each of the cases (i), (ii) considered above (a con- 

sequence of a stronger property D(A,) = D(A,) which is valid in 

these cases; see, e.g. [12]). But the condition (10.1) imposes further 

restrictions on p(x). It is easy to show that (10.1) is true if p(x) 3 0, 

but it is stronger than necessary. It suffices that the negative part of 

Q(X) is not too strong. We note that 4 EL~/~(R~) in each of the cases 

(i), (ii), so that 

by the Sobolev inequality, where c is a numerical constant. The same 

inequality holds when q is replaced by q+ , the positive and negative 

parts of q. Thus (10.1) is true if 

c II q- lip/* < 1, 

(11.2) 

with m = 1 - c ]I q- /lLsjz and M = 1 + c I] q+ llLs/~ . In what follows 

we shall assume (11.2) so that Condition 

be noted that (11.2) implies A, > 0. 

10.1 is satisfied. It should 

Finally we shall show that Condition 10.2 is also satisfied under the 

above assumptions. This would be obvious if we assumed q(x) > 0 

as in [A, for then (a) of Theorem 10.5 is true. But we shall show that 

(e) of Theorem 10.5 is applicable without such an assumption. 

To verify (e), we may exchange the subscripts 1, 2 in (e), according 

to Remark 10.6. Furthermore, D(A,) = 3(/l,) as mentioned above. 

Since (A, - A,) e--ilBU = (A, - A,) (A, + 1)-l f+~l(A, 

where (A, - A,) (A, + 1)-l E B(R), it suffices to prove 

+ 1) u, 

(A, - A,) e-itE1u -+ 0 in RR, f+ztCO, (11.3)


for each u E 3, , where 9, is a core of A, [i.e., a subset of D(A,) such 

that (A, + 1) 3, is dense in R]; note that A, is absolutely continuous 

so that Qi = 1. (Equation (11.3) is analogous to, but different from, 

a convergence proved by Wilcox [13] in which B, is replaced by A,). 

We take as 9, the set of all linear combinations of u = U(X) E R 

which have Fourier transforms of the form 

22(p) = const. pypTp3”p”,” j p 1-l e+I, (11.4) 

where aj > 0 are integers. It is easily seen that DD, is a core of A, 

(cf. [I], p. 300). s ince e&@ is simply the multiplication by e--illPl in 

the p-representation, w(t) = emilBIU for u with (11.4) can be computed 

explicitly by inverse Fourier transformation. The result is 

w(t) = w(t, x) 

= (1 +& + (X 12 CC~le,fi~fii 

Br, (11.5) 

[ (1 + it;+ I X 12 1 

where ca,ag, are constants and the sum is taken over the indices 

& 3 0 such that /3i + & + & = 01~ + a2 + ~/a . What is important 

for our purpose is that ru(t, X) is a bounded function of x E R3, with 

the bound going to zero as t + f co. The proof is simple and may 

be omitted. Since A, - A, is the multiplication by q(x), which is in 

L2(R3), it follows that (11.3) is true. 

Thus Theorem 10.3 is applicable: IV, = IV&( U, , U1; J) and 

I#‘,( U, , U2; y) exist, are complete, and are adjoints to each other. 

IV, are isometric, for PI = 1 in virtue of Qi = 1. But we do not know 

in general whether or not they are unitary. The nonunitary case will 

occur if A, has a nonzero singular part. Such a possibility is excluded 

in case (ii) stated above (see [II]), but the question is open in case (i). 

In any case this is a question about the Schrodinger operator A,. 

B. Scattering by Obstacles: Zero boundary condition (cf. [4], [.5]) 

We now consider the wave equation 

a%4 

--Au=O. 

at2 

(11.6) 

The unperturbed equation is (11.6) considered in Rn, and the per- 

turbed one is the same equation considered in a domain J2 of Rn, 

which is the exterior of a bounded open set S2,. We assume that 

the boundary 82 is sufficiently smooth.

366 KATO 

The unperturbed equation can be written in the form (8.1) with 

A = A, = - d as in the preceding paragraph, with the Hilbert 

space R = L2(Rlt). To describe the perturbed equation in a similar 

form, we have to specify the boundary condition on X! Here we 

choose the zero boundary condition, which makes - A into a selfadjoint 

operator A; in R’ = L2(s2). To fit the present problem into 

the framework of Section 9, we enlarge this space a’ to R = R’ @ R”, 

where R” = L2(ln,), and accordingly the operator Ai to 

A,=A;l@Ai, where A,” is the selfadjoint operator in s” defined 

by - A and the zero boundary condition. In this way we have two 

selfadjoint operators A, , A, acting in the same Hilbert space R. 

A, is absolutely continuous so that Qr = 1. Since it is well known 

that Ai has pure point spectrum, the absolutely continuous part of 

A, must be a part of A;1 , and Q2R C a’. Since the results of the preceding 

sections are essentially related only to absolutely continuous 

parts of the operators, the part Ai does not appear at all in our final 

results. But it is convenient to include it as a part of A, so that 

A, acts in the same space fi as A, does, as our general theory 

requires. 

It is obvious that both A, > 0; none of them has the eigenvalue 

zero. Thus Bi = Aila and the unitary groups Uj in the spaces 81 are 

well defined. We also define Bk = Ai1J2, B,” = Ai112. 

Condition 9.1 is satisfied for the pair B, , B, , as is seen from the 

result, due to Birman [14], that (A, + 1)-k - (A, + 1)-k belongs 

to the trace class for sufficiently large positive integer k. In fact this 

implies that not only the Schrodinger type wave operators W,(A, , A,) 

exist and are complete but the invariance principle holds (see Remark 

9.2). 

It follows from Theorem 9.3 that the wave operators W,( U, , U1; J) 

and W,( Ul , U2; J*) exist, are complete and mutually adjoint 

partial isometries. Here again, however, the identification operators 

J, J* are rather arbitrary and may not be interesting physically. 

More reasonable identifications are given by J and J’ as before. 

In order to be able to define them, we have to verify Condition 10.1. 

It is easily seen from the definition of A, , A, that both I] B,u II2 and 

11 B,u ]I2 are formally equal to the Dirichlet integral JR” I grad u I2 dx. 

But a(B,) is smaller than D(B,) owing to the presence of the boundary 

conditions on 8JJ = 2X$,. More precisely, D(B,) = 5@(P) whereas 

?D(B,) = P(G) @ .W(52,), where @ denotes the completion under the 

Dirichlet norm of the space of smooth functions with compact supports 

(see Deny and Lions [15]). Thus ~~(23,) C D(B,) and 11 B,u ]I2 

is a restriction of )I B,u l12. This means that Condition 10.1 is satisfied


with m = M = 1. It follows also that & = [X$&J] x Jt is a subspace 

of a1 = [ID( x R (including the metric). Note that 

of which the second summand is contained in (1 - P,) $ja and is of 

no interest in scattering theory. 

Since m = 1 in (lO.l), Condition 10.2 is also satisfied by Theorem 

10.5 (a). It follows that Theorem 10.3 is applicable; W,(U, , Ur; J) 

and W,(U,, U2; Y) exist, are complete and mutually adjoint partial 

isometries. 

Let us recall that J is simply the projection of 8i onto its subspace 

8, and J’ is the canonical injection of 8, into br . These identifications 

coincide with the ones implicitly used in [4] and [.5]. 

Although we have deduced the existence and completeness of the 

wave operators for the scattering for wave equations from the cor- 

responding Schrodinger wave operators, we have not touched upon 

other more delicate problems. For example we have not proved that 

A;1 is absolutely continuous, a result proved by Shizuta [Id] (see also 

[5]). But again this is a problem related to the SchrSdinger operators. 

C. Scattering by Obstacles. Other boundary conditions (cf. [5J) 

Let us consider the problem of the preceding paragraph with the 

zero boundary condition replaced by other conditions. First we 

consider the Neumann condition a~/& = 0. 

The corresponding operator A, is now defined as the direct sum 

$4 @A”,, where the summands are the self-adjoint operators in Ji’ 

and R”, respectively, constructed from - d with the Neumann 

boundary condition. The operator A, for the free wave is the same 

as before. 

Again A, 3 0 and B, = Ai/” can be constructed. B, has an eigenvalue 

zero, but it occurs only for the inessential part B”, . Condition 9.1 

is satisfied, again by the trace condition due to Birman [Z4]. Thus 

Theorem 9.3 holds true, but it may not be interesting. 

Regarding Condition 10.1, there is a great difference from the case 

of the zero boundary condition. This time 1) B,u II2 is an extension 

(instead of a restriction) of Ij B,u 112, for it is the sum of Dirichlet 

integrals taken over Q and Sz, separately, D(B,) being the direct sum 

WQ) 0 BL(Q,,), w h ere BL denotes the Beppo-Levi space (see [15J) 

[which means that u E D(B,) can be completely discontinuous across

368 KATO 

the boundary between IR and @,I. It follows that Condition 

satisfied with the subscripts 1, 2 exchanged, with m = M = 1. 

10.1 is 

By Theorem 10.5 (a), Condition 10.2 is also satisfied with the 

subscripts exchanged. Applying Theorem 10.3 with the subscripts 

exchanged, we see that W,( U, , U,; 9’) and W,( U, , U,; 9) exist, are 

complete and mutually adjoint partial isometries. Here 9’ is the 

projection of & onto $jl , which is now a subspace of J& , and p is the 

canonical injection of $r into sjB . Other remarks similar to the ones 

given in the preceding paragraphs apply also in this case. 

Finally we consider boundary conditions of the third kind 

au/&z + u(x) u = 0 on aJ2. We denote the corresponding selfadjoint 

operator in R’ = L2(sZ) by Ai . If we assume that o(x) > 0, 

then A; > 0 and it is associated with the nonnegative quadratic 

form 1) Bju ]I2 which is the sum of the Dirichlet form and the surface 

integral Jan u(x) 1 u I2 dS, where B; = Ai1i2. For convenience we 

define A: = Ai as above as the selfadjoint operator in a” = L2(Q,,) 

with the Neumann boundary condition, and set A, = Aj @ A:. 

Again Condition 9.1 is satisfied by Birman’s criterion, and Theorem 

9.3 holds true. 

The form 11 B,u /I2 = 11 Bju /I2 + (I B& II2 associated with A, has 

domain 3(B3) = BL(Q) @ BL(Q,) = a(B,), since the surface integral 

is bounded with respect to the Dirichlet integral if u(x) is sufficiently 

smooth. Thus we have D(B,) 3 D(B,) and Condition 10.1 is satisfied 

with the subscripts 1, 2 replaced by 3, 1. But the corresponding 

constants are such that m < 1 = M if u(x) > 0 for some x E 8Q. 

Thus we do not know whether or not Condition 10.2 is satisfied. 

To avoid this difficulty, it is convenient to use the chain rule 

(Theorem 4.3). We consider three groups, the unperturbed group 

U, , the group U, considered above for the Neumann boundary 

condition, and the group U, for the third-kind boundary condition 

under consideration. Accordingly, we introduce the following 

identification operators. Jzr E B(& ,a,) and Ji2 E B(& , &) are the 

Jand J’ considered above for the wave operators for the pair U, , U, . 

Similarly we define Jal E B($j, , $a) and Jr, E B($& , $i) for the pair 

U, , U, . For the pair U, , U, , we define Ja2 E B(Jj, , a,) and 

-723 = hi1 E W, 9 52) as simple identifications between fj, and $a , 

which are identical as a vector space. In fact we have a(B,) = Ad 


II B,u II < II B,u II < M’ II B,u II > u E a(B,) = 3)(B,). (11.7) 

It follows that not only Condition 10.1 but also Condition 10.2 

is satisfied for the pair U, , U, , again by Theorem 10.5, (a). Hence


W,( U, , U,; JS2) and W+( U, , Ua; J2a) exist, are complete and are 

mutually adjoint partial isometries. Also we know that the same is 

true of w,( us , UC Jzl) and W,(u, , us; j&J. 

But 331 = 332321 9 both members being equal to the canonical 

injection of 5r into &. Thus it follows from Theorem 4.3 that 

W+( U, , Ul; y3r) exist, are complete and are partial isometries. 

Furthermore, since JrsJai = 1 (identity in $Q, J1a is a (U, , &)asymptotic 

left inverse to J3r . It follows from Theorem 5.3 that 

W+( U, , U3; Jra) exist, are complete, and are equal to W+( U, , Ul; 

33,>*. 

REFERENCES 

1. -TO, T. “Perturbation Theory for Linear Operators.” Springer, Berlin, 1966. 

2. BIRMAN, M. SH., Existence conditions for wave operators. Izv. Akad. Nauk 

S.S.S.R., Ser. Mat. 21 (1963), 883-906; Am. Math. Sac. Transl., Ser. 2. 54 

(1966), 91-117. 

3. KURODA, S. T., An abstract stationary approach to perturbation of continuous 

spectra and scattering theory. J. Anal. Math. (to be published). 

4. LAX, P. AND PHILLIPS, R. S., Scattering theory. Bull. Am. Math. Sot. 70 (1964), 

130-142 (also, forthcoming book “Scattering Theory,” Academic Press, New 

York). 

5. SHWK II, N. A., Eigenfunction expansions and scattering theory for the wave 

equation in an exterior region. Arch. Ratl. Meek. Anal. 21 (1966), 120-150. 

6. SCHMIDT, G., Scattering theory for Maxwell’s equations in an exterior domain 

(Preprint, Stanford University). 

7. THOE, D., Spectral theory for the wave equation with a potential term. Arch. 

Ratl. Meek. Anal. 22 (1966), 364-406. 

8. WILCOX, C. H., Wave operators and asymptotic solutions of wave propagation 

problems of classical physics. Arch. RatZ. Mech. Anal. 22 (1966), 37-78. 

9. PUTNAM, C. R., On normal operators in Hilbert space. Am. J. Math. 73 (1951), 

357-362. 

10. KATO, T., Wave operators and unitary equivalence. Pacific J. Math. 15 (1965), 

171-180. 

11. kEF3E, T., Eigenfunction expansions associated with the Schroedinger operators 

and their applications to scattering theory. Arch. Ratl. Mech. An&. 5 (1960), l-34. 

12. KATO, T., Notes on some inequalities for linear operators. Math. Ann. 125 (1952), 

208-212. 

13. WILCOX, C. H., Uniform asymptotic estimates for wave packets in the quantum 

theory of scattering. J. Math. Phys. 6 (1965), 611-620. 

14. BIRMAN, M. SH., Perturbations of the continuous spectrum of a singular elliptic 

operator under the change of the boundary and boundary conditions. Vest. 

Leningrad. Univ., Ser. Mat., Meh, Astron. 1 (1962), 22-55. 

15. DENY, J. AND LIONS, J. L., L-es espaces du type de Beppo Levi. Ann. Inst. Fourier 5 

(1953-54), 305-370. 

16. SHIZUTA, Y., Eigenfunction expansion associated with the operator -A in the 

exterior domain. Proc. Japan Acad. 39 (1963), 656-660.


On Structure Spaces and Extensions of C*-Algebras 

ROBERT C. BUSBY 

Department of Mathemutics, Oakland University, Rochester, Michigan 48063 

Communicated by Irving Segal 

Received July 11, 1967 

1. ~TR~DUCTION 

In [I], we discussed the extension problem for C*-algebras and 

raised the following question: If A and C are C*-algebras with 

HausdorB structure spaces, and if the C*-algebra B is an extension 

of A by C (see [I] for definitions), when is the structure space of B 

Hausdorff ? It is the purpose of this note to answer this question for 

the case when B (and therefore also C) has identity. In the process, 

we will make some remarks about the structure space of a C*-algebra 

with identity. In particular we give a characterization of the interior 

of the set of closed, Hausdorff points in this space, and use this result 

to reveal some structure properties of certain C*-algebras with 

identity. 

2. NOTATIONS AND PREREQUISITE RESULTS 

If A is any C*-algebra, Prim (A) will always denote the structure 

space of A, that is the set of all primitive ideals of A, with the Jacobson 

or hull-kernel topology. We will denote the center of A by C(A). 

If A and B are subsets of a C*-algebra M, we will denote by (A : B) 

the set of all x in M with XB + Bx C A. If A and B are closed, two- 

sided ideals in M, so is (A : B), and we will refer to (A : B) as a 

quotient ideal. The algebraic facts concerning quotient ideals are to 

be found in [IO], p. 8. The proofs are given there for a commutative 

ring, but commutativity is not needed. If A and B are C*-subalgebras 

of M with A C B, then (A : B) is a C*-subalgebra of M. 

Finally, we shall let M(A) d enote the double centralizer algebra 

of a C*-algebra A, and refer the reader to [I] and [S] for details 

concerning this C*-algebra. We wish only to remark that M(A) can 

370

STRUCTURE SPACES AND EXTENSIONS 0~ C*-ALGEBRAS 371 

be considered as the idealizer of A (that is the C*-subalgebra (A : A)) 

in the enveloping Von Neumann algebra of A, the latter being canonically 

isomorphic to the bidual, A**, of A. 

An essential tool for us is a result of Dixmier ([4], Theorem 5). 

This result does not explicitly mention double centralizer algebras, but 

using the above remark concerning these algebras the reader can easily 

verify that it is equivalent with the following: If A is any C*-algebra, 

the set Cb(Prim (A)) of all continuous, complex, bounded functions 

on Prim (A) can be canonically identified with C(M(A)). In its final 

form, this identification can be described as follows: If P E Prim (A) 

and x E C(M(A)), then (P : A) E Prim (M(A)) ([3], Section 3.2) and 

the image R of x in M(A)/(P : A) is a multiple,f,(P) : 1, of the identity. 

Thus P+fJP) d e fi nes a complex valued function on Prim (A). 

Dixmier shows in [4] that fz is in C,(Prim (A)) and then uses a fundamental 

result of Dauns and Hofmann ([2], III, 3.5, 3.9 and III, Sec. 5) 

to show that every function in Cb(Prim (A)) can be realized uniquely 

in this way. As a corollary, we get a result already proved in [2], III, 

Sec. 5, namely that if A has a unit, C&Prim (A)) is canonically isomorphic 

with C(A). 

3. HAUSDORFF POINTS IN THE STRUCTURE SPACE OF A 

(?-ALGEBRA 

In [5], Dixmier defines a point x in a topological space x to be 

HausdorfI if given any pointy in X, not in the closure of x, there exist 

disjoint neighborhoods of x and y. We shall always let S(X) represent 

the set of closed, HausdorfI points of X, and s(X) the interior of S(X). 

s(X) was investigated in [5] and [6] for the case where X = Prim (A) 

for some C*-algebra A. We will also consider this case and give a 

characterization of S(Prim (A)) when A has identity. 

LEMMA 3.1. If A is a C*-algebra, then 

C(A) = A n C(M(A)) 

This lemma is a consequence of [4], Theorem 8, but we prefer to 

give a direct proof. 

Proof. If x E C(A) and T E M(A), then for all y E A, 

(Tx)y = T(q)= T(yx) =(Ty)x = x(Ty) =(xT)y 

and so TX = XT. Thus C(A) C A n C(M(A)) and the other contain- 

ment is obvious.

372 BUSBY 

If A and B are C*-algebras with A a closed, two-sided ideal in B, 

we know that Prim (A) can be identified with an open subset of 

Prim (B) by making P E Prim (A) correspond to (P : A) E Prim (B) 

(see [3], Section 2.3 and [I], Lemma 6.1). We will usually make this 

identification. 

COROLLARY 3.2. If A is a closed, two-sided ideal in a C*-algebra B 

and if Prim (A) is dense in Prim (B), then C(A) = A n C(B). 

Proof. By the definition of the Jacobson topology and the embed- 

ding described above, we see that the closure of Prim (A) in Prim (B) 

is the set of all J E Prim B with J3 r)lGPrim(A) (1 : A). The latter 

intersection is just [(nlcPrim(A)I) : A] ([IO], p. 8) which is (0 : A) 

([3], Section 2.9.7) the annihilator of A in B. Since in this case 

Prim (A) is dense in Prim (B), (0 : A) C nicPrimB J = 0 and so by [I], 

Proposition 3.7, B can be considered to be a subalgebra of M(A). 

The result now follows easily from Lemma 3.1. We remark in passing, 

that while the inclusion A n C(B) CC(A) clearly holds in general, 

equality does not hold in general without the denseness condition. 

We now come to the main result of this section. 

THEOREM 3.3. Let B be a C*-algebra with identity, and A a closed, 

two-sided ideal in B with Prim (A) Hausdor#. Let X be the maximal 

ideal space of the commutative C*-algebra C(A). As usual we consider 

Prim (A) to be an open subset of Prim (B). Then 

(1) X can be embedded as an open subset of Prim (A). 

(2) If WY B) re P resents the set of points of Prim (A) which 

are closed and Hausdorff in Prim (B), and S(A, B) is the interior of this 

set, then with the above embedding, X = $(A, B). 

Proof. (I) Since A is an ideal in M(A), C(A) = A n C(M(A)) 

is an ideal in C(M(A)). By the identification of C(M(A)) with 

C,(Prim (A)) mentioned in Section 2, we see that C(A) can be identi- 

fied with a C*-algebra of continuous functions on Prim (A). By [3], 

Proposition 3.3.7, each of these functions belongs to C,(Prim (A)), the 

set of all continuous, complex functions on Prim (A) which vanish at 

infinity. It is well known that a closed, two-sided ideal in C,(Prim (A)) 

consists of all functions vanishing on some closed set, and therefore 

the maximal ideal space of such an ideal corresponds to the comple- 

mentary open set. Since C(A) is isomorphic with an ideal in 

C,(Prim (A)), (1) follows.

STRUCTURE SPACES AND EXTENSIONS OF C*-ALGEBRAS 373 

(2) We will consider X as an open subset of Prim (A), and 

continue with the notation previously used. We first show that 

S(A, B) C X. It was pointed out in [6], Lemma3.2, and Lemma3.3, 

that every point in S(Prim (B)) has a neighborhood system 

composed of sets which are closed and compact in Prim (B). If 

I E S(A, B) C S(Prim (B)), th ere is a closed, compact neighborhood V 

of I in Prim (B), such that I’ C S(A, B). 

In the usual way, we can find a continuous function 

g : Prim (B) -+ [0, l] such that g(x) = 1 and g = 0 outside P (even 

though Prim (B) is in general not Hausdorff, g can be defined in the 

Hausdorf? space V first, set equal to zero on the rest of the space, and 

shown to be continuous). Since B has identity, C(B) z Cb(Prim (B)j, 

and so g = fZ for some element x of C(B). Since g vanishes outside 

$(A, B) CPrim (A), we have that x E npE,,rim~(B) P = A (see [3], 

Theorem 2.9.7), where Prim, (B) is the set of all P E Prim (B) such 

that P 1 A. Thus x E C(B) n A C C(A). Since g(l) = f,(1) # 0, 

I E X, and since I was an arbitrary in &A, B), S(A, B) C X. 

To complete the theorem, we need to show that X C &A, B), that 

is we must show that for every I in X and J E Prim (B), I and J have 

disjoint neighborhoods It is enough to consider points J in the closure 

Prim (A) of Prim (A), since if I E X, J E Prim (B), but J $ Prim (A), 

then I and J are already in disjoint open sets Now Prim (A) is the 

structure space of some C*-algebra containing A 

-- 

(see [3], Section 

3.2.2), and since we need only to consider J in Prim (A), we may 

assume Prim (A) = Prim (B). Let I E X. If J E X, then since X is 

Hausdorfl’, and open in Prim (B), I and J have disjoint neighborhoods. 

If J $ X, choose an element x E C(A) such that x + 0 (mod I), i.e., 

f#) j- 0. Since by Corollary 3.2 C(A) = A n C(B), fz has an extensionfZ 

to all of Prim (B). By the definition of X, Jj, is zero outside X, 

so j$(J) = 0. Th e sets U, = {J’ E Prim (B) / f,(,(J’) > 8 fZ(l)} and 

US = U’ E Prim W I .&I’> -c hf#>l are disjoint neighborhoods of 

I and J, respectively, and so I E ,$A, B). This completes the proof 

of the theorem. 

Remark. An interesting consequence of this theorem is that the 

set of points in Prim (A) which are in &Prim (B)), is completely 

independent of B, as long as B has identity. 

DEFINITION 3.4. We will say that a C*-algebra A is central if 

C(A) separates the points of Prim (A), that is if Pi and Pz E Prim (A) 

and Pl n C(A) = Pz n C(A), then Pl = Pz . (See [9], Section 9)

374 BUSBY 

COROLLARY 3.5. (to Theorem 3.3.) If B is a C*-akebra with 

identity and A is a closed, two-sided ideal in B, then Prim (A) consists 

entirely of closed, Hausdogpoints in Prim (B) if and only if A is central. 

Proof. If A is central, then the correspondence P-+ P n C(A) 

is a homeomorphism of Prim (A) onto the maximal ideal space of 

C(A) ([9], Theorem 9.1). The corollary then follows. 

COROLLARY 3.6. In any C*-algebra B with identity, the unique 

closed, two-sided ideal of B corresponding to $(Prim (B)) (which 

exists by [3], Proposition 3.3.2) is the largest central ideal in B. Following 

the notation of [6], Section 4, we denote this ideal by L(B). 

COROLLARY 3.7. With notation as above, ;f J ~Prim(B) then 

J E S(Prim (B)) if and only if th ere is a central ideal I in B, with I $ J. 

COROLLARY 3.8. With B as above, the following are equivalent: 

(1) Prim (B) is a HausdotJg space. 

(2) All primitive ideals in B are central. 

(3) There are two primitive ideals, J1 and Jz , which are central. 

(4) B is central. 

Proof. 

(1) S- (2). Follows from Corollary 3.5. 

(2) =+ (3) is obvious 

(3) 3 (1) The only difficulty here is that, a priori, J1 (say) 

could be in the closure of Jz . But since J1 is central, and JS is a point 

in the complement of (x}, Jz is closed. Now Prim (B) is the union 

of two sets, both in &Prim (B)), and so (1) follows. 

(1) * (4) also follows from Corollary 3.5. 

(4) 3 (1) is just Theorem 9.1. of [9]. 

We now refer the reader to [3] Chapter 10 and [A, Chapter 1 for 

the definition and properties of a continuous field, (9, 0), of C*-alge- 

bras over a Hausdorff space X. We will use Definition 10.1.2 of [3]. 

If, for each x E X, F(x) (the fiber over x) is a full matrix algebra of 

some dimension, (depending on x) we will refer to (9, e), or simply 

to 8, as a continuous field of matrix algebras. If condition (iv) of [3], 

Definition 10.1.2 does not hold we will call (9, 8) a continuous 

prefield of C*-algebras. 

THEOREM 3.9. Let A be a separable C.C.R. algebra with identity 

(see [3], Chapter 4, where they are called h’minal (?-algebras). 

Then there is a compact Hausdorff space X, and a continuous prefield 8 

of matrix algebras over X such that A g 8.


Proof. Since A has identity, Prim (A) is compact (see [3], Proposition 

3.1.8, we do not assume HausdorfI when we say compact). It 

was shown ([S], L emma 2) that if A is separable, with compact 

structure space, then S(Prim (A)) is dense in Prim (A). Again let 

L(A) be the ideal corresponding to $Prim (A)). ThenL(A) is a central 

C.C.R. algebra. Since L(A) is C.C.R. with HausdorfI structure space, 

we know that there is a continuous field of C*-algebras, (9, fQ, over 

S(Prim (A)) such that for each I E Prim (A), F(I) is the algebra of all 

compact operators on some Hilbert space HI , and I,(A) is isomorphic 

with the algebra of all sections 5 E 8 which vanish at infinity (see [3], 

Theorem 10.54). Since I,(A) is central each g(I) has center and so 

must be a full matrix algebra. Now M&(A)) s 13. In fact the proof 

of [I], Theorem 3.15, can easily be modified to include the case under 

discussion. Also by [I] Theorem 3.15, since S(Prim (A)) is dense in 

Prim (A), A can be considered to be a subalgebra of M(L(A)) = 19. 

It is then easy to see that A satisfies (i)-(iii) of [3], Definition 

This completes the proof. 

10.1.2. 

We refer to [3], Section 4.3 for the definition of G.C.R. (postliminal) 

C*-algebra, and to [3], 4.7.12 for the definition of generalized 

trace class (G.T.C.) 

we prove: 

algebras. As a final application of Theorem 3.3, 

THEOREM 3.10. If A is a G.C.R. algebra with identity, the 

following statements are equivalent: 

(i) A is G.T.C. 

(ii) If I is a proper, closed, two-sided ideal in A, there is a proper 

closed, two-sided ideal J 3 I with J/I central. 

(iii) If I is a proper, closed, two-sided ideal in A, there is a proper, 

closed, two-sided ideal J 3 I, such that C(J/L) # {O}. 

Proof. Dixmier has shown in [6-J Prop. 4.2. that if A is G.C.R., 

then A is G.T.C. if and only if for all closed, two-sided ideal I of A, 

W/I) # {O} (using p revious notation). This together with Corollary 

3.6. shows that (i) o (ii), and obviously (ii) 3 (iii). Finally, (iii) + (ii) 

follows easily from Theorem 3.3, and the proof is complete. 

4. APPLICATIONS TO EXTENSION THEORY 

If A is a C*-algebra, and M(A) is its double centralizer algebra, we 

will denote the quotient M(A)/A by O(A). An extension of A by C, 

where A and C are C*-algebras, is the equivalence class of some short

376 BUSBY 

exact sequence, 0 -+ A -+ B ---t C -+ 0, of C*-algebras. For a fixed A 

and C, the set Ext (C, A) of all extensions of A by C was shown in [Z] 

to be in one-to-one correspondence with the *-homomorphisms 

from C to O(A). We construct a canonical element of E, , the extension 

associated with a given *-homomorphism y : C ---f O(A), by defining B 

to be the subset of the product algebra M(A) x C consisting of all 

(m, c) with r(m) = y(C), where r : M(A) -+ O(A) is projection. B 

is easily shown to be a C*-algebra, and the maps f : A -+ B and 

g : B --t C given respectively by f (u) = (a, 0) and g(m, c) = c com- 

plete the definition of the canonical representative of E, . It is easy 

to see that B has identity if and only if C has identity and y( 1 c) = 1, tR) . 

In fact if the latter condition holds, (1, 1) is an identity for B, and on 

the other hand if (m, c) is an identity for B, then since g is surjective, 

c must be an identity, 1 c, for C. Clearly m is an identity for rr-ly( C) C A 

and by the definition of M(A), any identity for A must be lMcA) . 

Hence (m, c) = (1, 1) and ~(1) = lata) = y( 1,). 

DEFINITION 4.1. Let the image r(C(M(A)) be called the projected 

center of O(A) and denoted by C,(O(A)). We now answer the question 

of when Prim (B) is Hausdorff (A, B, and C as above). 

THEOREM 4.2. Let A and C be C*-algebras, let C have identity and 

let 0 -+ A L B -% C -+ 0 be the canonical representative of the exten- 

sion E,, associated with a *-homomorphism y : C --f O(A) with 

y( 1 c) = l,,ta) . Then Prim (B) is HausdorJff if and only if: 

(1) Prim (A) and Prim (C) are Hausdorff; 

(2) A is central; 

(3) Y(W)) C C,(W)). 

Proof. Suppose (l)-(3). E ver 1 ea in Prim (B) is either of the 

y ‘d 1 

form (P : A), P E Prim (A), or g-l(Q), 9 E Prim (C) (see [3], Propo- 

sition 2.11.5). Since Prim (A) is HausdorB, A is central and B has 

identity, Theorem 3.3 shows that all ideals in Prim (B) of the form 

(P : A), P E Prim (A), are closed, Hausdorf? points. It remains only 

to show that if or = g-‘( Q,), and Dz = g-I( a,), Q, and Q, in Prim (C), 

then Dr and az have disjoint neighborhoods in Prim (C). Since 

Prim (C) is Hausdorf?, there is an element x E C(C) with x E Qr and 

x $ Q, . In fact C,(Prim (C)) = C(C) by the identification previously 

discussed, and some continuous function must separate 52, and Q, . 

Since y(C(C)) C C&O(A)), there is an element m in C(M(A)) with 

(m, X) E B. It is easy to see that (m, x) E C(B) and (m, x) E or, 

(m, x) $ D, . Therefore since (m, x) can be thought of as a continuous


function on Prim (B), we see that D, and 0s have disjoint neigh- 

borhoods in Prim (B) and so Prim (B) is HausdorE 

Conversely, suppose Prim (I?) is Hausdorff. Condition (1) is then 

clear and (2) follows from Theorem 3.3. Now denote r-l(C,(O(A)) by 

K, and consider K n C(C). This is easily seen to be a C*-algebra 

of continuous functions on Prim (C) (the usual identifications being 

in force). This algebra contains constants since 

r(l) = lo(a) ~~,W)). 

Prim (B) is Hausdortf, and so for any @ = S-l(Qn,), i = 1, 2, 

L$ E Prim (C) there must be an element (m, c) E C(B) with (m, c) E DI 

and (m,c)$ On,. Thus there is a c E C(C) such that c E L?, and 

c 4 Q, . This property together with the properties of containing 

constants and being a C*-algebra, imply in the usual way using Stone- 

Weierstrass Theorem, that K n C(C) = C(C) and condition (3) holds. 

This completes the proof. 

RRFERRNCR.S 

1. BUSBY, R., Double centralizers and extensions of C*-algebras. Trans. Am. Math. 

Sot. (to be published). 

2. DAUNS, J. AND HOFMANN, K. H., Representations 

Memoirs Am. Math. Sec. (to be published). 

of rings by continuous sections. 

3. DIXMIER, J., “Les C*-algebras et Leurs Representations.” Gauthier-Villars, 

Paris, 1964. 

4. DIXMIER, J., Ideal center of a C*-algebra (to be published). 

5. DIXMIER, J., Points s&pare dans le spectre dune 

(1961), 115-128. 

C*-algebra. Acta Sci. Math. 22 

6. DIXMIJIR, J., Traces sur les C*-algebras. II. Bull. Sci. Math. 88 (1964), 39-57. 

7. FELL, J. M. G., The structure of algebras of operator fields. Acta Math. 106 

(1961), 233-280. 

8. JOHNSON, B. E., An introduction to the theory of centralizers. PTOC. London Math. 

Sot. 14 (1964), 299-320. 

9. KAPLANSKY, I., Normed algebras. Duke Math. J. 16 (1949), 399418. 

IO. NORTHCOTT, D. G., “Ideal Theory” (Cambridge Tracts in Mathematics and 

Mathematical Physics, No. 42). Cambridge University Press, Cambridge, 1963.

JOURNAL OF FUNCTIONAL ANALYSIS i, 378 (1967) 

Erratum 

Vol. 1, No. 1 (1967), in the article, “An Extension of a Theorem 

of L. O’Raifeartaigh,” by Irving Segal, pp. l-21 : 

Page 1, insert, “Received April 1 I, 1966.” 

Printed in Belgium 378

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