Extension Theorems of Hahn-Banach Type for Nonlinear Disjointly ...

Extension Theorems of Hahn-Banach Type
for Nonlinear Disjointly Additive Functionals
and Operators in Lebesgue Spaces
AND
\-T(‘T0R J. R~IZEI.*
(,‘ahrn~‘~ie~lWellor2 U&e&y, Pittsburgh, Pennsylvauiu 15113
Commtmicatrd by thr Editors
Received October 28, 1975
I& (Q, 7, wz) be a nonatomic separable finite measure space. Every continuous
functional N on L=(m), 1 :; p :. co, which is disjointly additive in the sense
N(u ~-~ 2’) = N(u) + N(v) whenever zw = 0, is known to be representable by an
integral with a nonlinear Caratheodory kernel. Such functionals share several
regularity properties with continuous linear functionals. Here we study the ques-
tion of whether every continuous, disjointly additive functional defined on a closed
subspace of L’(m) possesses an extension to L?‘(m) with these same properties.
This question has applications to the study of nonlinear functionals on Sobolex
spaces. It is shown that for a class of subspaces, including those of finite co-
dimension, such an extension always exists, but there are also closed subspaces
not possessing this extension property. Analogous results are obtained for
disjointly additive mappings from closed subspaces of Ln(wz) into L’(m) and for
functionals defined on subspaces of L”(m). The techniques depend heavily
on the utilization of Lyapunov vector measures.
I. INTRODUCTION
Disjointly additive functionals defined on Banach spaces of measurable
functions forming a complete lattice have recently been studied by several
authors [2, 10-121. Such functionals, which are involved in manv nonlinear
x Research partially supported by the U.S. Sational Science Foundation under Grant
YIPS 71-02776-A03.
303
Xl1 rights of reproduction in any form rcscrved. ISSS 0022-1236

di&rcntial and integral equations, share several regularity properties \vith linca~-
functionals [6]. Hence the question naturally arises ai to whether a Hahn
Banach type extension theory is also available for disjointly additive functionals.
It turns out that a full analog of the Hahn-Banach theorem does not hold in
this case. More precisely, there exist closed subspaces of LJ’ and continuous,
disjointlv additive functionals defined on these subspaces which do not l~osscss an
cxtcnsion to the entire space 1,” preserving these properties. On the other hand,
the results to follow do provide a significant partinl analog of the Hahn Hanach
theorem.
1Vc shall say that a subspace of LJ’, 1 ‘_ p ’ ._ ,a, possesses the d.a. P.\‘te?1Sion
proper!\’ if every continuous, disjointly additive functional on the subspace
has an extension to L” preserving these properties. The continuity to which WC’
refer here is with respect to 1,” norm if I --. p -: sci and with respect to bounded
convergence in measure if p W. A characterization of those subspaces of L”
which possess the d.a. extension property is highly desirable, in view of the
fact that disjointly additive functionals defined on the entire space Z,” have been
extensi\-cl!- studied and many of their properties are known. In the present
paper we initiate a study of this problem by giving certain gcnerai conditions
which guarantee that a subspace l~ossesses the d.a. extension propcrt!. Out
method, which is to a large extent constructive, proves the existence of extensions
by developing an integral representation of the functionals. A result concerning
the existence of extensions from a certain subspace J&’ of finite codimension
was proved in [S], where it was used in order to obtain a characterization of a
class of disjointly additive functionals on Sobolev spaces. The proof given in [8]
utilized the special features of the subspace J&’ considered there.
Regarding disjointly additive operators, we show that the same conditions
which guarantee that a subspace possesses the d.a. extension propert>- for
functionals are also sufficient in order that the subspace shall have the analogous
property for mappings into U(m). However, these mappings are assumed to
satisfy an additional condition, namely, that the image of a function vanishes
\vhere\-er the function vanishes.
The plan of the paper is as follows. In Section 2 we state the extension theorems
for functionals. In Section 3 we discuss the relation between the class of suhspaces
under consideration (which we call “rich” subspaces) and certain associated
Lyapunov measures. This relation plays a crucial role in our treatment of the
subject. In Sections 4 and 5 we develop an integral representation for functionals
defined on linear manifolds of simple functions. This development employs
a Radon-Nikodym type theorem for functionals, obtained in [9]. Using this
representation we prove the extension result for functionals on subspaces of L”;
the proof occupies Section 6 (for p ~~ co) and Section 7 (for I 1 p -:. m).
An extension theorem for disjointly additive operators is stated and proved in
Section 8. Finally, in Section 9 WC‘ give examples of subspaces of I,” kvhich
do not possess the d.a. extension property.

EXTENSIOK THEOREMS FOR FUNCTIOK.4LS 305
2. DEFIKITIONS AND STATEMENTS OF REX-LTS
Let (Q, T, vz) bc a nonatomic separable finite positive measure space. Denote
by I,~‘(rn), 1 ’ p :< CO, the Lebesgue spaces of real functions with respect to this
measure space. In this paper the terms “null,” “ax.,” etc. will refer to the
measure vz except if otherwise stated.
.4 linear subspace .)4 of Z,‘(M) will be called a rich subspace if it satisfies the
following conditions.
(A) ,// is za”-closed.
(IS) c A! separates sets, i.e., for every nonnull set E in 7, there exists a function
.f in JY such that f is nonnull and vanishes outside R.
A linear subspace A ofP(m), 1 s p -C cc, will be called Grk if J/Y n 1,‘(m)
is a rich subspace of L=(m).
‘I’he notion of a rich subspacc of LZ(~z) is closely related to the notion of a
thin set in 1,l(m) introduced by Kingman and Robertson [3]. Indeed, if j ii is a
xx-closed subspace of LW-(m) then J is rich if and only if A’ , its annihilator
in L’(m), is a thin set.
\Te note that if .A! is a closed subspace ofL”(nz), I I - p < x3, then ../i n Z,‘(m)
is rc*-closed. Indeed, if ~2’~ is the annihilator of ./2’ in L’l(m) (where (1 .‘p) --
(1:9) I) then ,fl is the annihilator of ~2’~ in ~,J~(Hz). (‘Iearly, ever\ element
in the zc*-closure of A! n L7(m) annihilates ,//. . and hence belongs to i/.
~1 closed linear subspace of L”(nz), 1 -I p < or,, which is of finite coclirnension
is necessarih a rich subspace. The same statement holds also with rcspcct to
J,‘(w~) if “closed” is replaced by “zc*-closed.” This is a consequcncr of tlic’
Lvapunov theorem [7]; a proof will be given in the neat section.
‘In LV(m) we shall consider, in addition to the norm and weak” topologies,
the (bin)-topology which is defined as follows. A sequence in I;l(~~~) is (bn~)-
convergent if it is bounded in norm and if it converges in masurc’. .-I set is
closed in the (Dm)-topology if and (ml\- if it is scquentiallv closed with respect
tn (bm)-convergence.
Given a real function f on R, the set K(f) {t E f2 :-f(t) ,’ 0) will hc called
the strict sup@~ of ,f. Tl’e shall say that t\vo functions .f, R arc disjoint if
K(f) n AI(‘F) ‘C
\Yc are now ready to state our main results.
rI’HEOREbI 2.1 . I,et A/ he a rich s&pace of I,x(m). Let S he a fumtional
012 _ //‘ such that
(i) :X7 is disjoidy additic.;e, i.e., &V(,f + ,F) -7 :\-(.f) !- >Y(,y), pro7+/efi that
f, g ape disjoint;
(ii) 9 is (hm)-rontinuous.
Then there exists an extension of AT toL*(nz) zuhich presevees properties (i)‘Nw/ (ii).

306 MARCUS AND hIIZE1
\\*c conjecture that this result remains true if L’(m) is replaced b! /.“(N).
1 < p : ; 2, provided that it-is continuous with respect to theI,” norm. I-lowevei-,
at present we have only the following more restricted result.
THEOREM 2.2. Let ,i’i be a closed subspace of L)‘(m), 1 p . ,z , of finite
codimension. Let A be a functional on ,N such that
(i) S is disjointly additive;
(ii) X is continuous with respect to the 1,” norm.
Then there exists an extension to Lo’ which preseraes properties (i) and (ii).
A function H: Q ;C R -+ R is called a Caratheodory function if I[(., a) is
measurable for every a in R and H(t, .) is continuous for almost every t in Q.
A function H as above is normalized if H(t, 0) ~~ 0, a.e., in 0. If f is a real
measurable function on D then the function Hf defined b\
I-&f(t) =- Wt, f(t)), vt t 8, (2.1)
is measurable. For I &p :G xz we denote by Car” the family of Caratheodoq
functions [H) such that Hf is in Z,i(m) whenever f is in L”(m). If g is a function
in L”(IIz), 1 -; q .;i CD, then the function H, : Q :‘. R + R defined b!
Wt, a) aldt), qt. a) E Q I\ R, (2.2)
is in Car”, where (lip) -; (l/q) I.
C’ombining the results of the first two theorems with a known characterization
of disjointly additive functionals on T,” spaces [2], wc obtain the following more
detailed result.
1‘IIEOR13i\I 2.3. I,et ,!/ be a subspace of Ll’(m) and let ,\: be a functional on &.
Suppose that .// and ,1’ satkfi, the assumptions of Theorem 2.1 ;f p -XT, arrd
those of Theorem 2.2 if I p _ : m. Then there e.*.ists a normalized Sfunc?ion
II: B ~: R - R helorging to Car” such that,
The fuuction II is unique module the family of functions .[I{,! : g t A/ j
\Yc note that if ‘q E &tip, then
(3.3)
.r, H,f dm 0. yf E .A’. (2.4)
As before ,‘I1 denotes the annihilator of i // in L’I(m), where (l/p) -{-- (1 /q) I.

EXTENSION THEOREMS FOR FL'NCTIONALS
3. RICH SUBSPACE~ AND LYAPUNOV MEASURES
The notion of a rich subspace ofL%(m) is closely related to Lyapunov measures.
This relation plays a crucial role in the derivation of our results. In this section
we discuss the above relation and state certain results concerning Lpapunov
measures that will be used in our proof.
If zq is a set in 7 we denote 1~~ 7.j the family of T-measurable sets contained
in A. The characteristic function of a set .-1 will be denoted by x,,
Let S be a Banach space and p: 7 --f X a vector-valued measure. Then p is
said to be a Lyapunov measure if the range of p on 7A is closed and convex, for
every Ag in 7. The following characterization of Lyapunov measures is known
([3, 51; see also [4]).
PROPO~ITIOS 3.1. p is a Lyapunov measure if and only if for every set PI in 7
zuhirh is uot p-null there exists a real bounded r-measuvahle function ,f satisfyiq
f is not /L-null, K(f) (I A, and
fdp -=o.
-n c
This result extends the following theorem due to Lyapunov [7].
If ,Y is finite dimensional and TV is nonatomic, then p is a Lyapunov measure.
An important property of Lyapunov measures is expressed in the so-called
“bang-bang” principle, which is stated below. This property follows directI\
from the definition of Lpapunov measure.
PROPOSITION 3.2. Let TV be a Lyapunov measure. Iff is a -r-measurable function
such that 0 1-Y f < 1, then there exists a set A in 7 such that Jn f dp -= ~(9).
Another property of Lyapunov measures which can be derived directly from
the definition is the following.
PROPOSITION 3.3. Let TV be a Lvapunoz measure and let B be a set in T which
is not IL-null. Then there exists a a-akebra for E, sa?l Z;. , such that
(i) ZEC~;
(ii) FE ZE 3 p(F) = pE(F) p(K) where 0 ::i p,(F) :g 1;
(iii) (IA Z;: , 14, with pt defined as above, is a separable, nonatomic measure
space.
A a-algebra for E possessing properties (i)-( iii will be called a Blaclzeaell
)
a-algebra for E with respect to CL. This class of u-algebras was first considered
bv Black&l [I] in the case of finite-dimensional vector measures.
‘Th c relation between rich subspaces of L-(m) and L\;apunov measures is
described in the following proposition which is a simple consequence of
Proposition 3.1 (see [9; Sect. 21).
307

308 MARCUS AND MIZEI.
PROPOSITION 3.4. Let ~2’ be a rich subspace of L”(m). Then. there e.rists a
vector measure p: 7 + I1 such that
I. TV has bounded total variation (which we denote by j p I);
II. TV is absolutely continuous with respect to m;
III. p is a I.yapunov measure;
IIT. &? =: {f ELm(m) : Snf dW = O}.
Conversely, if TV: 7 ---+ l1 is a Lyapunov measure satisfying conditions I and II
and if AS? is de$ned by IV, then Jd is a rich subspace.
A measure p satisfying the conditions of Proposition 3.4 will bc called a
Lyapunov measure associated with JY.
Given a Lyapunov measure, one can construct related measures which arc
also of Lyapunov type. The f o 11 owing proposition describes two such
constructions.
PROPOSITION 3.5. Let TV: 7 - l X be a Lyapunov measure which is absolutely
continuous with respect to m.
(a) If gl ,..., g,. are in Ll(m) and p: 7 + Rk :< S is the measure given bzl
then j.2 is a Lyapunov measure.
F(E) .-z (lEgl dm ,... , .kg, dnz, p(E)), VE E 7, (3.1)
(b) If u is a p-integrable function and af 1/U is the indefinite integral of u z&h
respect to p, then vl, is a Lyapunov measure.
For the proof of part (a) see [9; Sect. 21 and for the proof of part (1~) see
[4; Chap. V, Sect. 21.
We now prove the following statement which was mentioned in Section 2.
LEMMA 3.6. (a) If J&’ is a closed linear subspace of LY(m), 1 ; p < z,
such that A? is of $nite codimension, then At is a rich subspace.
(b) If .Ad is a w*-closed linear subspace of Lx(m) of Ji ni t e codimension then A
is a rich subspace.
Proof. Under the above assumptions the annihilator of JZZ’ in L”(M), (1 /p) +
(l/q) :: I, is finite dimensional. i2s before, denote the annihilator by k’ .
Clearly, by assumption (a) or (b), the annihilator of .~&‘l in Ln(m) is precisely JZ.
Let (ql ,..., pk) be a basis for J&‘~ and set

EXTENSION THEOREMS FOR FLJNCTIONALS 309
Then p is a nonatomic finite-dimensional vector measure on 7. Hence, bp
Lyapunov’s theorem [7], it is a Lyapunov measure. Therefore, by Proposition
3.1, .A+’ is a rich subspace.
We conclude this section with an additional lemma concerning certain
decompositions of functions in relation to a Lyapunov measure /L on 7.
Let f be a measurable simple function,
f 2 aixE,, I:‘, ,..., E, disjoint sets.
i=l
Let Y be a positive integer. Since p is Lyapunov, for every set E, there exists a
partition (Ef,, ,..., Ei,r] of Ej into sets of equal p-measure, i -: I...., II. Set
,tj -= Fl a&,,, , j -= l,..., r.
The set of functions {fr ,...,fr) is called a ~-uniform decomposition off.
LEMMA 3.7. Let f, g be two measurable simple functions and let p be Lyapunov.
Let Y be a positive integer. Then there exist ~-uniform decompositions off and g,
sa?) {fi ,..., f;~ and {g, ,..., g,), respective$f, such that fV , g,’ are disjoint whenever
u j VI.
Proof. Let
.f I= i %XE, ,
i-1
.A’ =: tl bm, 1
where [E, ,..., E,,) and {F, ,..., FJC) are partitions of 8. Set
‘Thus,
w;,j == E, n Fi , j := 1 ,...I 12; j --= I ,...) k.
Let {rqi ) 6qj )...) W{‘} be a partition of IV,,, into sets of equal p-measure. Set
jq” =z (j qj ) Fj” = 5 rqj
,--1 i-l
for i I ,..., n, i 1 ,..., k, I, : : I,..., r. Further, let
Then (.fr ,..., fJ and [,q, ,..., ~~1 satisfy the requirements of the lemma.

310 MARcI .4ND MIZEL
4. INTEGRAL REPRESENTATION OF FUNCTIONALS DEFINED ON .4 SPACE OF
SIMPLE FCJNCTIONS
Given a subspace &Y of L%(rrz) we denote by

EXTEIiSION THEOREMS FOR FUSCTIONALS 311
-wxc; -- XVJ) -wxw,,, - xwn.J) -i- W(Xw,,, -- x&J),
W(x v,’ - XVLf)) = W(xw,,, - xw,,,)) t w4Xw2,, - xw,,Jb
In view of the fact that N is even, the last two equalities imply (4.4).
Let y,, 1 N E R, be a set function on 7 defined by,
rl,(r-) ~~ W(Xvl - XV,))> v I,. E 7, (4.6)
where [I’, , I ilj is a partition of I r into sets of equal /L-measure. In view of (4.4),
y. is well defined. The disjoint additivity and (bm)-continuity of iV imply that
y(, is a signed measure which is absolutely continuous with respect to nz. Moreover
y,, is the zero measure and y,, y (, , v’n t I-?. (4.7)
Thus, by the Radon-Nikodym theorem, there exists a function I-I: Q :< R - R,
which satisfies (4.1) and is even in its second variable, such that
yo(c-) 1. H(., u) hl, VuER and Vf7r:7. (4.X)
. 1’
Now, let fE Y,, (
f C aixrL ,
i-l
I, ,..., I 771 disjoint sets.
Let :I-;., , f .!,?] be a partition of 1-i into sets of equal p-measure, and set
This completes the proof of the lemma.

312 MAKCUS AND MIZEL
The proof of the theorem for odd S relies on the following Radon Sikodyn
type result which was established in [9].
PROPOSITION 4.3. Let ,//I’ be a rich subspace of L=(m) and let ‘1 be an odd
functional on ~?j[ uhich is disjointly additive and (bm)-continuous. Then there
exists a function G in Ll(m) such that
Let A! be a rich subspace of L%(m) and let p be a Lyapunov measure associated
with A?. A functional X on my,/ is said to be t”-invariant if, for ever!. pair of
functions f, g in .y,[ such that
we have N(f) == X(g).
t4.f ‘(4) P.(‘TW), VaER, (4. IO)
Using Proposition 4.3 we shall reduce the proof of the theorem for odd ;V to
the case where N is, in addition, p-invariant.
LEMMA 4.4. Let ,k/ and JV be as in Theorem 4.1 and suppose that S is odd.
Then there exists a function G: Q K R --f R such that
and
G(., a) EL’(m), tin E R,
G(., a) r= -G(*, -a), Va E R,
(4.11)
,V(ah) =~. 1. G(., a) 17 dm, Qh E &,, and VaER. (4.12)
I “0
Furthermore, if p is a LJ,apunov measure associated with & and fi is a Sfunrtional
on 9’;( defked by,
then fi is p-invariant.
Proof. Given a real number a, set
A,,(h) = h:(ah), Q’h E cl’,, . (4.14)
Then A! and A,, satisfy the conditions of Proposition 4.3. Hence, there exists
a function G(., a) in Ll(m) such that (4.12) holds. Since A,, mu I 1 ,, . one can
select the function G in such a way that (4.11) holds.
It remains to be shown that the functional 2%’ defined by (4.13) is ~-invariant.

EXTENSION THEOREMS FOR FUNCTIONALS 313
Let f, g be two disjoint functions in x,/ satisfying (4.10). Then f - g is a linear
combination of &sj,A functions in CC/~ . Hence (4.12) and (4.13) imply that
Since ,%T is odd and disjointly additive, this implies that iI’ Lq(g).
Kow let J; g be two functions in P,, satisfying _ (4.10) but not necessaril!disjoint.
Let
range(f) = range(g) -:- (ur ,..., a,,)
j-l(q) = s, ) ‘p(q) _: 7’; (i I,..., n),
Wi,; == Sj n 7; (i,j -- l)...) 72).
Let i Jr-,, i , rr’(j] be a partition of Wj,j into sets of equal p-measure, and set
\Ve similarly define S’; , T9!’ , f”, g”. Then each of the pairs f’, g” and f”, g’
consists of disjoint functions in ,y(, satisfying (4.10). Hence, by the preceding
argument,
Ayf’) = Iv(f) and qf”) = !Q’).
Summing up the two equalities we obtain
qf) = 19(g).
This completes the proof of the lemma.
The next main step is the proof of the existence of a kernel for [L-invariant
functionals.
LEMMA 4.5. Let A’ be a rich subspace of Lx(m) and let p be an associated
LTapunozq measure. Let .@ be a functional on P,, satisfying the folloz&g conditions.
(a) AT is odd and disjointly additive;
(b) .f is ~-invariant;
(c) lirn,+,,, 6(~; a, b) = 0, Va, b > 0 where
6(6 a, b) sup{1 .$(,f)l :f E .V;, , m(K(f)) ,< E and range(,f‘) (7 (0, n, -b)].
Then there exists a kernel Fl for N.

314 MARCUS AND MIZEL
Note that condition (c) is weaker than (bm)-continuity. We defer the proof
of this lemma to the next section and proceed now with the completion of the
proof of Theorem 4.1. We need one additional result which was obtained in
[9; Theorem 3.11.
PROPOSITION 4.6. Let & be a rich subspace of Ll(m). T’hez sp C,,( the
linear span of rZfl) is dense in A![ with respect to the Lr norm.
The existence of a kernel for N follows by the following argument. If ;V is
odd and fi is defined by (4.13) then l%r satisfies conditions (a))(c) of Lemma 4.5.
Therefore the function H : R + R, defined by H G 2 I^r with G as in
Lemma 4.4 and l? as in Lemma 4.5, is a kernel for N. If N is even, the existence
of a kernel is given by Lemma 4.2. As previously mentioned, the existence of a
kernel in the cases where X is either odd or even, implies its existence in the
general case.
In order to prove the uniqueness statement in Theorem 4.1 we have to show
that any function H: Sz x R + R satisfying (4.1) such that
i’ H(f) dm 0, Vf E ?I > (4.15)
R
is, essentially, of the form H, for someg E A-. For the definition of H, . ~~(2.2).
Let H be a function as above and set
H,,(., a> = [H(., a) - H(., -a)]i2,
H,(., a) ~~ [H(., a) -I- H(., -a)]/2.
(4.16)
Let p be a Lyapunov measure associated with AC. Let U E 7 and let {Ur , U,jbe
a partition of U into sets of equal p-measure. Then,
s
by (4.15) and (4.16),
H,(., a) dm : Jb, Hk, 4 dm -t ju. Hk, -a)
u 2
elm
Thus,
1
z = 2 [I, fJ(., 4 dm + Ju,, HC.9 -4 dm]
1
+ i [jul H(*, -a) dm -i- ju2 H(., a) dm] == 0.
Zi,(., a) Mom 0, Va E R.
Next, (4.15) and (4.16) imply, in particular, that
jD H,(., a) h dm 0 Vh E CT,, , Vu E R.
(4.17)
(4.18)

Hence, by Proposition 4.6,
EXTENSION THEOREMS FOR FIJNCTIONALS 315
j-QH,,(.,a).fdnz =0 Vf’fJZ, \JngR.
Thus I{,,( ., u) E ~@2’ L for every real a.
\Ve claim that, for U, J-E 7, if p(U) = C+L( Jj7) for some a: >. 0, then,
( H,,(., a) dm == a J Ii,(., a) dm, V’n E R. (4.19)
“U Y
If CY is a rational number then (4.19) is a simple consequence of (4.18).
If 01 is not rational, let (01~) be an increasing sequence of rational numbers
converging to 01. Let ZU be a Blackwell u-algebra for U with respect to p and
let (U,] be an increasing sequence of sets from ZU such that
Then,
g1 1.;, 1: C’, p( U,) =: (cxn/a) p(U) = o!,p( I’).
lu H,(., a) dm = ~1, / H,,(., a) dm.
71
v
Taking the limit as n + co we obtain (4.19).
Let a, b be two positive numbers and let U be a set in 7. Let {U, , U,> be
a partition of U such that
Then, by (4.15), (4.16) and (4.19)
Thus,
0 = j- H,(qU1 - Zyu,) dm
“R
cL(Ul) = (V(a - 6)) CL(U).
-1 %(.,4dm-J %(.,h)dm
Ul u2
(W + 6)) S, fw, 4 dm - (4~ + 4) -1, H,(., 4 dm.
bH,(‘, u) - UH,(‘, 6) := 0, vu, b > 0.
Since H,,(., -a) := --H,,(., a), th e a b ove equality holds for all real a. Thus
Ho = H, . (4.20)
Since ZI,,(., u) E A-, it follows that g E di. Finally, (4.17) and (4.20) imply
that H = fi, . It should be noted that this equality means that H(., u) ==
H,(., U) = ug(.) as elements of Ll(m), for every real a. This completes the
proof of the theorem.

316 MARCUS AND MIZEL
5. ~-INVARIANT FUNCTIONALS; PROOF OF LEMMA 4.5
PROPOSITION 5.1 [9; Sect. 41. Let A’ be a rich subspace of L=(m) and let X be
an odd, disjointly additive functional on .y,[ If f, g, h are disjoint simple functions
such that f - g and f - h belong to Y/J , then
iv(f - g) = - N(f - I/) - X(g - I?). (5.1)
The same statement holds if YJ. is replaced by

EXTENSION THEOREMS FOR FUNCTIONALS 317
Next we claim that, for every triple of positive numbers, a, 6, C,
Since the equality is symmetric with respect to a, b, c, we may and shall assume
that c 3.: mas(a, b). Given a set S in 7, let (S, , S,} be a partition of S into sets
of equal p-measure. Further, let (Sr’, $1 be an (a, h) partition of S, and let
S,’ be a subset of S, such that
By (5.5)
p(&‘) = (ub/c(a 4- h)) /L(S$).
Ya,b(w r= ; Y,,,(S),
These equalities, together with (5.2), yield,
For the last two equalities we used the fact that &’ is odd in conjunction with
Proposition 5.1.
Now let c,, be a fixed positive number, and set
By (5.6) and (5.7),
yn = Ul%) Yo.c,, 1 vu T- 0. (5.7)
YdS) = @a.(S) - qqq (5.8)
Let h E fFi, , h = uxu - 6xr, ) U n If :-= c‘(, n, h positive. By (5.2) (5.5),
(5.7), and (5.8), we have,
The measure rU is absolutely continuous with respect to nz. Hence, by the
Radon-Kikodym theorem, p,, possesses a density function in Ll(m) which we

318 MARCUS AND MIZEL
denote by I?(., a), a > 0. 1fT:e define A(., 0) = 0 and Z?( ., -u) ---J?( ., CZ)
for n > 0. The function Z?: Q Y R - R defined above satisfies condition (4.1).
Furthermore, by (5.9),
8(h) ~-~ f A(h) dm, Vh t cf,, . (5.10)
-Q
By disjoint additivity the above equality holds also for every function .f which
is a sum of disjoint functions in c?/! .
Let f be an arbitrary function in q”Il and let fr ,..., fy and Jr ,..., .f!. hc as in
Proposition 5.2. Thenf, - fi is a linear combination of disjoint functions in 6/l
andfi is a sum of disjoint functions in 8,/ (z’ mu l,..., I’). Thus,
:?(fJ = Jo fi(f;,) dm,
Since fj and f+ are disjoint functions in z,( and % is odd, these two equalities
imply,
#(fi) j-> @j-J dm, i ~- I,..., r.
Finally, by disjoint additivity this implies,
Thus Z? is a kernel for 1x7.
Ag(f) == [ Z^i(.f’) dm.
* R
6. PROOF OF THEOREM 2.1
Let A and N be as in Theorem 2.1 and let H be a kernel for 1%;‘ in the sense
of Theorem 4. I. The function H defines a mapping H from the space .4r of all
measurable simple functions into Ll(m), defined b>
H(f)(t) == Wt, f(t))> VffEY, V’tEB. (6.1)
Our next objective is to show that this mapping has a disjointly additive (blr~)-
continuous extension to the entire space L”(m). This will be accomplished
through several lemmas. In all of these p denotes a Lyapunov measure associated
with A&‘.
LEMMA 6.1. Suppose that ( fn) is a bounded sequenre of simple functions such
that m(K(f,)) -+ 0 zuhen n - cc. Then,
(6.2)

ESTENSION THEOREMS FOR FIVVCTIONALS 319
Proof Let {fn’, fi] bc a p-uniform decomposition of fit , 11 I) 7,....
Let Al be a bound for the sequence {fn). Then, there exists a function I/G with
K(/I~) C K(.fE) such that 17; mm:- M(xu -~ xr; ) and
n 7,
‘l’his follows from Proposition 3.2 b>- considering the positive and ncgati\Te
parts of /$,iM. ‘The functionf,,’ - 11: belongs to ,y,, and itsL/ norm is bounded
b!- JI. Since m(K(f,)) --) 0, it follows that the sequence [fri’ ~~ /I:;: is (/VU)-
convergent to zero. Thus, by the continuity of A7 and Theorem 4. I,
Since I{(-, .I/) and (I(., -,11) belong to L’(m), vve have
when II + x. ‘I’herefore, noting that f,,’ and /zt are disjoint, eve deduce that
Similarly,
[ H(fiL’) h + 0.
. .‘.’
1:) H(J‘3 dm - 0.
Since,f,,‘, f,: arc disjoint, we ohtain (6.2).
IXM~IA 6.3. Let if,!) be a bouncieci sequence of simple functioms in I,‘(w/) such
fhnt {.f,,: coneeyes in menswe to a fitnction f in A. Then,
Proof. For any function h in L’(m), set
@I inf(h ~;- m(t E Q : A(t) ’ ~5;).
h :- 0
The metric d(., .) defined by d(k, y) U(/I --- ,y), generates the topolog!
of convergence in measure.
Let f,! fn -. .f (72 I, 2,...) and set
(6.3)
(~6.4)
4ftJ ~~~ an > -d,, [t E 9 : ‘f,,(t) . n,,], R,, =- Q”,‘:.d,, , n I, 2 ,,.._ (6.5)

320 MARCW AND MIZIX
Then the following assertions hold.
Without loss of generality, we may and shall assume that urr :. I for all rr.
By Proposition 3.2, there exists a function /I,,* of the form
such that Sn*, Z’,,” are disjoint subsets of B,, and
a,,(~~ .
II XT.,,*)
By Proposition 3.5(a), the measure (m, p) : 7 + K )I l1 is a Lyapunov measure.
Hence, there exist subsets S,, , I”,, of Sn*, T,,“, respectively, such that
(7% P)(S,J ~~~~ 44 I*)(&,“‘),
(w cL)( 7’4 a,,(m, p)(T,,"), 77 I, 2....
Set II,, xr -~~ xs, . Then, by (6.7) and (6.8),
. I/
(6.7)
(6.X)
Nest, let ;\I be a bound for the sequence [.f,,l. Again, by Proposition 3.2,
there exists a function g, of the form 2,16(,yCT,, - x,,,~) such that U, , I Vri are dis-
joint subsets of A,( and
Set, p,, f/n sn (n I, 2 ,... ). Then, by (6.6) (6.7), (6.9). and (6.10) CT,! is
a bounded sequence in L”(m) such that
From (6.11) it follows that {P)~} is (bm)-convergent to zero. ‘Ihus {I;, ~~ v,!I is
(bm)-convergent to ,f. Further, by (6.1 I), (fn - v,,)

EXTENSION THEORERIS FOR FUNCTIONALS 321
But H(fn ~~ 4(t) H(fn)(t) for all t in Q:,K(~I,J. Therefore, (6.12) and (6.13)
imply (6.3).
LEMMA 6.3. Let {fill be a (bm)-convergent sequence of simple functions. Then
{H(f,,)\ conz!erges in IdI(
I’Yocf. Let f he the (bm)-limit of if,,:. First vvc consider the cast where ,f is
in A. Denote
‘Then,
where
Clearly,
Thus, b\
1i.j I’ I H(.fi) - H(fj)l dft2 [ (H(xT,j) -- H(l2j.j)) C/II?, (6.14)
3’ i2 - R
Lemma 6.2,
(bnl)-Jim gi,j : (bm)-lip h,,i mm f.
Z,J"" 1,J~ )Z
By (6.14) and (6.15), limi,j,7j Z,,j 0.
Nest we consider the general case where f is not necessarily in A!. By Proposi-
tion 3.5(b), the indefinite integral off with respect to p is a Lyapunov measure.
Hence, there exists a partition of Q, say (9, , Q,}, such that,
Set f” = f(xr), -- x~I,) and fiL” = fn(xal - xs+ a =--~ 1, 2,.. . . Then f-% E . c’/
and :f,l”‘) is (bm)-convergent to f *. Thus, by the first part of the proof, {H(f?! “)I
converges in Ll(m). Since H(f,)xo, = H(f?z*)xn, this implies that (H(f&J
converges in Lr(m). Similarly, one shows that [H(fJxa2} converges in Z,‘(m).
This completes the proof of the lemma.
~XhIMA 6.4. The mapping H: YA L’(m), defined by (6.1), possesses nn
extension 2: L”(m) ---f L’(m) which is disjointly additive and continuous ,with
respect to the (bm)-topology of L7-(m) and the norm topology of Ll(nz). Furtllermore,
we have
fl(fXE) S(f)XE 7 VftL-f(m), VEE 7.

322 MARCITS AND MIZEI.
I’roqf. Let f be a function in La(m) and let (f,J be a sequence of simple
functions converging (om) to f. By Lemma 6.3 , [H(f,,)J con\:ergcs in Z,‘(~z).
Furthermore, the limit of [H(fJ) c 1 oes not depend on the particular choice of
the approximating sequence (fni. Set
X(f) L’-‘,)I; H(f;,). (6.16)
Ohviousl~-, iff is a simple function then &‘(,f) H( /‘). ‘l’hus .A’ is an extension
of H.
Let f, ‘y be two disjoint functions in I,~(nz). Then we can select sequences
{fn>, {slli converging (6~2) to f and R, respectivcl!,, such that K(f,,) m’- K(,f)
and K(R,!) c-1 k-(g), ?z 1, 2,.... Since H is disjointly additive,
.m(.f 7~ R) lirn H(.f,, I- ,T,,) lim H(f,,) lim H(g,,)
,X(f) ! AQ).
If (:,,I is a sequence in L--(m) converging (bm) tof we can select a sequence of
simple functions (.f,l. such that {fn) converges (6~) to .f and
!,ll_lj JXfn) -- Wx’n); L,(,,,) 0.
Thus, by (6.16), [X(gJ) converges to %(,f‘) in l,l(m).
The last statement of the lemma is easily verified in view of (6.16) and (6. I ).
The proof of Theorem 2. I is now essentially completed. Set
%,4’(f) I . X’(f) dm, Vf’fELn(m), (6.17)
-5?
with It// as in Lemma 6.4. Then -4’ is a (bm)-continuous disjointly additive
functional on I,=(m). By Theorem 4. I, .Af coincides with N on - .‘f(( By
Proposition 4.6, Cyjfl is dense in A’ with respect to the I,’ norm. Hence, by the
continuity of ,Y- and N, it follows that A”(,f) Al’ for everyf in A’. Thus .A”
is an cstcnsion of IV possessing the properties stated in Theorem 2.1.
7. PROOF OF THEOREMS 2.2 AND 2.3
Let 1 --: p < c/u and let :A’ be a subspace of L”(m) of finite codimension.
Let (v, ,..., F,() be a basis of ML, the annihilator of A! in Z,“(m), (I !p) (1 /q) 1.
Set
p(E) -: (.lE y, dm ,..., lE v,$. dm), V’B t 7. (7.1)
Since p is a nonatomic finite-dimensional vector measure. it is ;I I,yapuno\-
measure [7].

EXTENSION TfIEOREMS FOR FUNCTIONALS 323
Let :Y be a functional on A’ satisfying the conditions of Theorem 2.2. As
before, we shall denote by 9”( the space of simple functions belonging to A’.
N restricted to 9’,/ satisfies all the assumptions of Theorem 4.1. Thus there
exists a kernel H for :V !9,1c . By (6. I), H generates a mapping H: 9 4 Li(m).
We shall show that this mapping has a continuous, disjointly additive extension
to the entire space L”(n2).
IJEhlRlA 7.1. There exist R disjoint sets iz r, saj’ El ,...l l?,L , such that
[p(EJ,, ,., p(E,)) are linearly independent vectors in R,b .
P~ooj-. In view of the fact that y~r ,..., ‘pi. are linearly independent as elements
of L”(O), it is not difficult to see that the range of ,U over 7 is k-dimensional. Let
Fi ,..., F,, be sets in T such that {CL(&) ,..., p(FB)) are linearly independent. Let
FT.1 B,F; , i =~ I,..., k. Let Q denote the set of multiindices {CX (pi ,..., 1,;):
‘xi 0, I for i =: I ,..., k). Set IX’, ~mm nf.-, F;
’
r(
L
where Fi ,, em F; . Then
Fi z= u r/v, , Fi,, :m= u IV-, , i -= 1 ,..., k.
AB Q &E Q
x*=0 ,x,:-l
Let (kl

324 MARCUS AND MIZEI.
LEMMA 7.3. Let {f?,) be a sequence of simple functions such that .f,, - + 0 i/z
I,“(m). Then,
!jz lo H(fn) dm 0. (7.2)
Proof. Our assumption on {fn) implies that safn cI~ --+ 0. Hence, by Lemma
7.2, there exists an integer n, and a sequence of functions {h,f,,;),,, , each of
which is a difference of characteristic sets, such that
.I’ fil dp
n
[ h, dp for 72 .‘; 12” and m(K(h,)) --f 0. (7.3)
- R
By Lemma 3.7 we can construct p-uniform decompositions of fiL and h,, , say
{fn’, f3 and {h,‘, hi’J such that K(fn’) n K(h,“) K(f,!J n K(h,,‘) G. Set
g, 7 fn’ - h,‘, n 2: n, . Then g, E q(, (n > n,,) and g, ---f 0 in L”(m). Ry
Theorem 4.1 and the continuity of AT, we obtain
Wg,) j3, W,) dm, n ;;- I+,
However, the properties of h,,’ imply that
Since f,,’ and 11,’ are disjoint we have,
Hence, by (7.4),
Similarly, one shows that
km= [ H(-h,‘) dm : = 0.
’ ‘R
H(g,) == H(fn’) -+- H(--A,‘).
i+i 1 H(fi) dm = 0.
u
and lim N(g,) :=I 0.
?1 .> rl
Since H(fJ = H(fn’) -t H(fi), th e p roof of the lemma is completed.
LEMMA 7.4. Let {.frL} be a sequence of simple functions which converges in
L”(m) to a function f in A’. Then,
Proof. The assumption on (fn} implies that,
(7.4)
l,‘fl .i, H(fn) dm -= iv(f). (7.5)

EXTENSION THEOREMS FOR FUNCTIONAW 325
Let n,, be an integer such that for n 2 n, , snfn d p is sufficiently small in norm,
in the sense of Lemma 7.2. Then, by Lemma 7.2, there exists a sequence of
functions
such that
{JzJ,~>~~ , each of which is a difference of characteristic functions,
Let grl fn -.- h, , n 3 n, . Then {g,,]n3no _C y n,, ; !;1E N(g,) = iv(f). (7.7)
Set C,, K(k,). Since {fVL] and (‘n) are convergent sequences in P(m),
(7.6) implies that (fnxc,) and (g,xc,) converge to zero in D’(m). Therefore,
bv Lemma 7.3,
lirir !; H(g,) dm = 0,
rt
;i .r, H(fJ dm == 0.
71
However, in Q\,C,, , g, coincides withf,? and hence H(g,) coincides with H(f,).
‘lherefore, (7.7) and (7.8) imply (7.5).
LEMMA 7.5. Let {fn} be a sequence of simple functions converging in L)‘(m).
Then {H(f,J} conz?erges in Ll(m).
Proof. Using Lemma 7.4, this result follows by the same argument as was
employed in the proof of Lemma 6.3.
LEMMA 7.6. The mapping H: 9 --f Ll(m) generated b}l the kernel H, possesses
an extension 2: L”(m) +Ll(m) which is disjointly additive and continuous with
respect to the norm topologies. Furthermoue,
WfXE) = *(f)XE Vf fz L=(m), VE E 7.
(7.8)
PYOO~. Cising Lemma 7.4, the proof is the same as that given for Lemma 6.4.
LEMMA 7.7. Cy/t is dense in A? with respect to the Lt’ norm.
Proof. Let f be a function in A! and let {fn} b e a sequence of simple functions
converging to f in L”(m). Then sfi fn dp + 0. Let n, be an integer such that
for n -3 n, , SD fn dp is sufficiently small in norm, in the sense of Lemma 7.2.
Then, by Lemma 7.2, there exists a sequence of functions (hJnanO, each of
them being a difference of characteristic function, such that
Thus fn - 11,~ E .‘f,, , 11 2: n, , and f,, - k, ---t f in Ll’(m).

326 MARCUS 4ND MIZEI,
To complete the proof of Theorem 2.2, we set,
.4qf) = 1 Z(f) dnz, Vf ELqn), (7.10)
- I?
with s’? as in Lemma 7.6. ‘I’hen JV‘ is a continuous, disjointlv additive
functional on P(m). By Theorem 4.1, M coincides with N on q,/[ . By Lemma
7.7 and the continuity of IV and A’“, it follows that the two functionals coincide
on A’. Thus, J1/’ is an extension of N possessing the properties stated in Theorem
1.2.
Note that the fact that p is finite-dimensional has been used only in order
to prove Lemma 7.2. Thus the proof yields a slightly more general result than
the one stated in Theorem 2.2.
We turn now to the proof of Theorem 2.3. Let N and A’ be as in Theorem 2. I
(respectively, 2.2) and let A’” be an extension of X to the entire space Lx(m)
(respectively, L”(m)), such that .A’” is disjointly additive and (bm)-continuous
(respectively, continuous in the L” norm). Then, by 12; Theorem 31, there
exists a normalized function H: Q x R - R belonging to Cam (respectively,
Car’)) such that,
(7.11)
for every f in L”(m) (respectively, I,“(m)). In particular, (2.3) holds. l:inallp,
the uniqueness statement in Theorem 2.3 is an immediate consequence of the
parallel statement in Theorem 4.1.
8. EXTENSION THEOREMS FOR OPERATORS
In this section we consider operators of the form 2: A? + Ll(m), where .A
is a subspace of L”(m), I < p S< SO, and Z possesses the following properties.
(a) &? is disjointly additive;
(b) 2 is “local,” i.e., K(G@Cf‘)) C K(f) for every f in A;
(c) J? is continuous with respect to the norm topologies, if I p -< IX,
and with respect to the (bm)-topology in A and the norm topology in L’(m)
ifp = rx).
For operators of this type we have the following extension result, which is
parallel to Theorems 2.1-2.3.
THEOREM 8.1. Let X he an operator possessing the properties stated above.
If 1 < p < co, suppose that AC is of finite codinzension. If p ‘7: suppose only,
that A? is a rich subspace of L’(m). 7’1 ien .w‘ possesses an e.\.tension to the fWirf~

EXTENSION THEOREMS FOR FUNCTIONALS 327
space which satisfies conditions (a)-(c) on L”(m). Fuvtheumow, there exists a
normakeri~function H: Q i: R + R such that H belongs to Carl’ and
The jicwtion IT is essentially unique.
2 (f) T H(f), VfEE.J/. (8.1)
1;~ the proof of the theorem we need several lemmas.
I,IGCJ.~ X.2. Let A’
odd, rli~joi~rt~~~ ndditive,
be a rich subspace of L7 (m). Let X: -G$[ ---f Ll(m) be an
local operator. I,et f, g be functions in .‘, additive, local operator. Then there exists a function II: Q x R 4 R
such that
amI
Ht., a) EL’(m),
H(., 0) == 0,
Va E R
(8.4)
JrCf) =- H(f), Vf E Y,, . (8.5)

328 MARCUS AND RIIZEI.
Proof. The odd and even parts of X each satisfy the assumptions of the
lemma. Therefore it is sufficient to prove the lemma in the case where .iy’ is
either odd or even.
Suppose that Z is odd. Let a bc a real number and let C; be a set in 7 such
that there exists a simple function g with K(g) n 1: 8:. such that
where p is a Lyapunov measure associated with ~ti. Then we define
H,(., a) =- Af(UX~ ~- &-o! . (8.6)
In view of Lemma 8.2 the right-hand side of (8.6) does not depend on the
choice of the function g. Thus H,( ., u) is well defined. If F is a measurable
subset of l!, then H,( ., a) is also defined; the function g -~ a~~.,,,. has the
properties required in order that H,(., a) be defined. Moreover, we have
Thus,
H,(., 4X,J = *qaxLT - R)XV = S(axr. - (g - ~Xu\v))Xr~ .
H,(., u) = I{,(., a)xv. (X.7)
As a consequence of (8.7), if { I’, , L’,} is a partition of U, we have
II,(., u) =- Hul(*, a) -+ El&(., a). VW
Now let U be an arbitrary set in 7 and let (C; , U,} and {I’, , I~VT) be partitions
of U into sets of equal p-measure. ‘Then HVt(., u) and Hvi(*, a), i 1. 2, arc
defined. We claim that
This is proved as follows. Let Wj,; mm= Uj n lTj (i, j -- 1, 2). Let i lI’P’,j ( II’: ;j
be a partition of W,,j into sets of equal p-measure. Set
Note that U,’ u U,’ = Vi’ v ET2’ and denote this set by C:‘. Similarly
U: v UG = I,‘: v I/‘; and we denote this set by 0’“. We also have p( I -‘) jl( U”)
so that H,,(., a) and H,-(., u) are defined. By (8.8),

EXTENSION THEOREMS FOR FUNC’l’IOSAI,S 329
Summing up and using again (8.8) we obtain,
Zf,:(., u) : H”f,(., u) = Hq(., u) + Hu,(*, u) =- HQ(., u) -I- If&, a).
This proves (8.9). In view of (8.9) we define I{,(., u) for an arbitrary set 1. in 7
as follows. Let CL.‘, CT”} be a partition of c’ into sets of equal p-measure. Then
FZ,.(., a) and H,,-(., u) are defined by (8.6). Set
H,(., u) =-= If,,,(., u) + H,-(., 0). (8. IO)
Since the right-hand side does not depend on the choice of the partition, fI,J., (1)
is well defined, and it is consistent with the previous definition.
If 1. is a measurable subset of CT, let CC”, V”{ and (W’, W’“; be partitions of TV
and I ’ I. into sets of equal measure. Set 1,” ~mz I” U II/‘, 7:” I”’ u W”. Then,
Zf,(., u)x,, = H,,(., a)xv + H,,,(., a)x,,
=: H,f(., U)Xv’ -1 Hp(., U)X[.”
=~ I~,,(., a) -+ Hv,,(., u) H,>(,, u).
For the second equality we used the fact that H,,(., a) vanishes outside I_”
and 11,.,( ‘. 0) vanishes outside C”‘. The third equality follows from (8.7) since
V, Z”’ are sets for which H,, , H,,J may be defined by (8.6). Thus (8.7) and
hence (8.8), remain valid for arbitrary sets in T.
Set ZI( ., u) : H,(. , u) for every real a. By its definition, N(., a) ill
for every a and N(., 0) 0. Furthermore, by (8.7)
Let .f be a function in Y,/ ,
H,( .) a) == H(., u),Qr VC: E 7 and Vu t Ii. (8.1 I)
(El ,..‘) fi,,) a partition of &Q.
Let (I:‘,., , Ei,?j be a partition of Ej into sets of equal measure and set
Then,
fj ~==: Cl UiXE,,, , j -~: I, 2.
HCft) = f H(., (ti) XE,,~ = f IfE,,,(., Ui)
,=-1 1. 1
(8.12)

330 MARCUS AND MIZEL
Here we used (8.1 I), (8.6), and the fact that SF is local and disjointly additive.
Similarly, we obtain
WtJ Wfd (8.13)
By the disjoint additivity of H and .X, (8.12) and (8.13) imply (8.5).
Next we consider the case where 2 is even. Let U be a set in 7 and let [I .i, L.,],
(E,‘, CT,‘> be two partitions of li into sets of equal p-measure. 1Ve claim that
By the disjoint additivity of X and (8.15),
Jwx u, - XUJ) = Je(Xw,,, - xw,,J -- m4Xw,,, - XWJL
=@Mx U,’ - X$‘)) =- x(u(x~l,, - xw,,2)) -t- ~~MXW,,, -- xw,,,)).
In view of the fact that X is even these equalities imply (8.14). Let i. and
{Cj’r , Uz} be as above, and set
ffu(-, a) = JWXU, - XUJ), Vu E R. (8.16)
By (8.14) the right-hand side of (8.16) d oes not depend on the choice of the
partition. Let H(., u) HsL(., u) for every u. Then
I{,(., 0) =- ff(., a)xc , Vl-~r, VaER. (8.17)
Indeed,
additivity,
if CC”, 1,. “‘] is an arbitrary partition of I - then, by (8.16) and disjoint
]I,.(., u) FI,.,(., 0) L- H,.-(., u), Vu E R.
In particular, we have
I-i(., 0) = HA., a) + ffc,,,-(., n), Vu E R.
Since X is local, (8.16) implies that K(1l,-(., 0)) 1; L.. ‘l’his relation together
with the above equalitv viclds (8.17).
. I

l\jow, let f be a function in F,/ ,
ESTESSION THEOREMS FOR FUNCTIONAM 331
f -f, aiXE, , (B, , . , En: a partiti(jn Of O.
Let .fr , ji be defined as in the previous part of the proof. Then,
2?(f) = A?(fl) -I- j’t(b2) -= 2qfJ -+-- X(-“/J
-L Z(fl -fJ == i 2@,(XE,,, .- X&J)
i-l
It is also clear that H satisfies (8.4). Th us the proof of the lemma is complete.
\I’e are now ready for the proof of Theorem 8.1. Let ;\: he a functional on J?’
defined bv
Y(f) = 1 Z(f) dwl, Vf E A!.
-R
Then L\’ satisfies all the assumptions of Theorem 2.1 if p = ‘CC or Theorem 2.2
if 1 Z< p < cc. The function 15 of Lemma 8.3 is a kernel for N in the sense of
Theorem 4.1. Let H: .Y -Ll(m) he the operator generated by 2,
H(f)(.) = H(.,f(*)), VfE9.
By Lemma 6.4 if p =m= x) or Lemma 7.6 if I < p < so, H possesses an extension
X”: L”(m) --+r,ym),
which is disjointly additive and continuous. The continuity is understood with
respect to the norm topologies if 1 VfEL"(WL), VE E 7.
In view of the above properties of ,X *, it follows, by [2, Theorem 3.31 that
there exists a normalized function N*: D ‘-: K -+ R, which belongs to Car”,
such that
X”(f) =- H”(f), VfEf,“(772). (8.18)
Hence, in particular,
H(f) = H”(f), Vj.E 9. (8.19)
Applying (8.19) to the constant function with value n, where a is a real number,
we get
f~f( ‘, lJ) _~ Il”( ‘, CJ), Vu E R. (8.20)

332 MARCrS I AYD i WIZEL J
Of course, the equality is understood as equality of elements in Z,i(m). By
(8.20) and (8.3,
W.f) = H”(f), Vf E .y/, . (8.21)
Since CY’,l is dense in A! with respect to the norm topology (Proposition 4.6
and Lemma 7.7) the continuity of F and H* together with (8.21) imply,
*(.f) = H*(f), Vf E dti. (8.22)
This proves the existence of a function H” with the properties stated in Theorem
8.1. The essential uniqueness of H* is obvious. Therefore the proof is completed.
9. SUBSPACES LACKING THE d.a. EXTENSIOK PROPERTI
The main tool in our construction is the following result.
LEMMA 9.1. Let N be a (bm)-continuous disjointly additive functional on
La(m) and let H t Car”(Q) be a normalized Caratheodory function which represents
N
N(u) : f H(t, u(t)) dm, u EL”(m). (9.1)
-a
Let q, E L%(m) be bounded az’ay from zero and let a, , a, be real numbers satisfving
a, < a, . Set
F(w) -~- 1”’ lV(ae,, ~-~ w) da == J”’ (/ H(t, aa, + w(t)) dm) da, zc EL”(m).
- “1 0, n
(9.2)
Then the functional F is G-differentiable everywhere in L%(m).
Remarks. By G-differentiability of F at a point w we mean that the real
valued function X ‘++ F(w + AZ) is differentiable at h -: 0, for each x ~l;~(m).
Relation (9.1) implies the uniform (bm)-continuity of N on balls of L%(m).
Hence not only is the existence of the integral in (9.2) assured, but F is also
(bm)-continuous.
Proof. The existence of a kernel HE Car”(Q) representing :\’ via (9.1)
follows from [2]. In order to prove that F is G-differentiable at points zc E L”(m),
we consider the following integral whose existence will be shown below,
_ o,,. v”(t)-’ H(t, .x) dm”.
c
Here I),,. ((t, X) EQ 1: R : a,v,,(t) w(t) ‘ .A+ _ a,v,,(t) + w(t):, xnd m” is
the product of the measure m with one-dimensional Lcbesguc measure. \Vc

EXTENSION THEOREMS FOR FUNCTIONALS 333
claim that in fact the integral in (9.3) is equal to F(w). For notational simplicity,
set
alvo + zu == fI , a+,, + I(! == f2 .
Then by- the Tonelli-Fubini theorem,
JD ,“&-I H(t. ,x) nrrl* = 1 [j’““’ q(t)-1 H(t, x) dx] dm
t< ‘s? fl(l)
= f [ i”‘H(f, aDo(f) j zu(t)) du] dtrz
-0 ‘a2
= ia’ [ (_ H(t, q,(t) -+- w(t)) dm] da. (9.4)
‘q -R
Here the applicabilitv of the Fubini-Tonelli theorem follows from the relation
H E Car”(Q), since the latter ensures the existence of an element R :-:= gLC E L’(m)
such that
H(t, x); :

334 MARCUS AND iVIIZEL
Sow let ~2’ CLV-(m) be a subspace possessing the properties
(a) ,&’ contains a function uu bounded away from zero,
(b) whenever f, g E JZZ are nonnull, then .f. g is nonnull.
Let G be an arbitrary (bm)-continuous functional on ~2’. Put
where
Xf,(u) = aG(w), (9.8)
20 = u - av() ) a = (1’ uv,, dm) \,(J v()2 dm).
Clearly, NO is (bm)-continuous on JY and, vacuously, disjointly additive (here
we use (b)).
Finally, consider the following functional F,, defined on the subspace /2” C i ti
consisting of functions za such that j ZU’Z’~ dnz mm: 0.
F,,(w) = (.n2iVo(av, + w) da = (a, - a,) G(w),
. “1
w E J&i?‘. (9.10)
Supposing that ATO has a (hm)-continuous, disjointly additive extension to all
of L”(m), it follows by Lemma 9.1 that F,, is G-differentiable everywhere in .J
with respect to functions in ~2’. However, the functional G was arbitrary so that,
in general, this differentiability property fails to hold. Hence, in general, ATO will
not possess an extension of the type sought.
As a consequence of the above, a class of subspaces lacking the d.a. extension
property can be constructed, for instance, by taking the uniform closure of any
linear manifold consisting of harmonic functions (and including constants) on a
domain Q C R”. Here m is n-dimensional Lebesgue measure.
REFERENCES
1. D. BLACICWELL, The range of certain vector integrals, Proc. Amer. Math. Sot. 2
(1951), 390-395.
2. L. DRENOWSKI AND W. ORLICZ, Continuity and representation of orthogonally additive
functionals, Bull. Acad. Polon Sci. Ser. Sci. Math. Astronom. Phys. 17(1969), 647-653.
3. J. F. C. KING~IAN AND A. P. ROBERTSON, On a theorem of Lyapunov, 1. London
Math. Sot. 43 (1968), 347-351.
4. I. KISJVANEK AND G. KNOWUS, “Vector Measures and Control ‘l’heory,” Sorth-
IHolland, 1976.
5. G. KNOWLES, Lyapunov vector measures, SIAM J. Control 13 (1974).
6. M. A. KRASNOSEL’SKII, P. P. ZABREIKO, E. I. PCSTYI.NIK, AND P. E. SOBOLIXSKII,
“Integral Operators in Spaces of Summablc Functions,” Nordhoff, Leyden, 1975.
7. A. LYAPCNOV, Sur les functions-vecteurs completement additives, (Russian, French
Summary), Izo. Akad. Nauk. SSSH Ser. Mat. 4 (1940), 465-478.
8. M. MARCCX AND V. J. MIZEL, Representation theorems for non-linear disjointl!
additive functionals and operators on Sobolev spaces, Trans. Amer. swath. Sot.,
to appear.

KXTEXSION THEOREMS FOR FLWCTIOXA1.S 335
9. AI. ~~.AINYY AND V. J. L~IZEL, A Radon-Sikodpm type theorem for functionals, J.
Fnnctio~7ul =Innlysis 23 (1976), Z-309.
10. I’. J. ?VhZEL, Characterization of non-linear transformations possessing kernels,
Cnnnd. J. Math. 22 (1970), 449-471.
Il. V. J. MIZEL AND K. SUNDARESAN, Representation of vectorvalued non-linear functions,
Tvnns. AMY. Muth. Sm. 159 (1971), I I l-127.
12. Ii. SUNIIARESAN, ;Idditive functionals on Orlicz spaces, Studio Math. 32 (1969).
269.-276.

JOLTRWAI 01: FUNCTIOKAL ANALYSIS 24, 336-352 (I 977)
Measure Space Automorphisms, the Normalizers of their Full
Groups, and Approximate Finiteness
\\‘e deal with the normalizer 2 ‘[T] of the full group [Tj of a nonsingular
transformation ‘T of a Lebesgue measure space in the group of all nonsingular
transformations. We solve the conjugacy problem in .1”[T]/[T] for a measure
preserving and ergodic ‘1’. Our results show that a locally finite extension of
a solvable group is approximately finite.
1. IXTRODLTTI~X
Let (S, CL) be a Lebesgue measure space, and let .-1 be the group of auto-
morphisms of (X, p), where we mean by an automorphism a bi-measurable
transformation that leaves p quasi-invariant. The full group [C] of a I’ E .-/
consists of those Q E .d with the property that f.a.a. ,Y F X- the Q-orbit of x is
contained in the C-orbit of x. We consider an ergodic T E .rl’ and we aim to
elucidate the structure of the normalizer .N[T] (R E .d : R[T]R .’ [T]j of
[T] in .c/. .k“[T] contains exactly those automorphisms R E J/ that cart-v f.a.a.
x E A- the T-orbit of x onto the T-orbit of Rx. If, for an R E .N[T], there is an
n E N such that R” E [T], then define the outer period p(H) of 12 as the smallest
such II. If, for all n EN, R” r$ [T], then say that R is outer aperiodic and set
p(R) 0. A countable group 9 C .d is called approximately finite if there
exists a rV t .-/’ such that, f.a.a. x E X, the Y-orbit of s coincides with the
U-orbit of s. In Section 2 we shall prove that every R t ./t’[T] generates together
with T an approximately finite group. In Section 3 we are concerned with the
conjugacy problem in .N[T]/[T] f or an crgodic and measure preserving T.
We briefly describe our conclusions. For this, we recall that every R E N[T]
has a module such that, with v an invariant measure of T,
(c/R-‘v/ch~)(s) mod R, f.a.a. .x t .\-.
ISSS 0022-1236

MEASURE SPACE AI-TOiVIORPHIBRIS 337
If V(S) c: a, then mod R = 1 for all R E J"[T], but if ~(-1~) m, then cverq’
positive real appears as a module. If mod R f 1 then R must be outer aperiodic.
Let us say that R, R' rN[T] are outer conjugate if R[T] and R'[T] arc conjugate
in . 1‘[7’]/[T]; in other words, if there is a P t [T] and a ,O E ,t ‘[?“I such that
Ii’ QRPQ-I. The outer period, and outer aperiodicity, as well as the module,
arc invariants of outer conjugacy, and it turns out that these in\ ariants are
colnplete.
Our results have a number of applications. In particular, it follo\vs tllat all
locally finite extensions of solvable groups are approximately finite. \\-v comment
on these applications in the last section of the paper.
There is a close connection between the topics that wc deal with hc and
certain developments in the theorv of von Neumann algebras. -It the origin
of this is the group measure space construction, that produces out of an crgodic
mcasul c preserving automorphism T an approsimately finite t\‘pc II factor.
Evcrv clement of N[T] induces an automorphism of this factor. I:or the auto-
morphisms of a von Neumann algebra, one also has the notion of outer conjugac!-
where two automorphisms R and R' arc called outer conjugate if R is conjugate
to a product of R' by an inner automorphism. Outer conjugate elcnlcnts of
. t ‘[ 7’1 ~icld outer col?jugate factor automorphisms. Hence the problem that we
deal with here can be viewed as the measure theoretic analog of tilt outcl
conjugac!, problem for the automorphisms of the approsimatcl!. finite t! pc II
factors. (‘onccrning the outer conjugacy for the automorI7hisnls of the :lpprosi-
mately finite type II factors, see [2, 31.
:\lso the methods that WC employ here are similar to those encountcrcd in the
theor!, of x-on Neumann algebras. Thus we use unit systems (also called al-rays)
to prove outer conjugacy. Thcsc unit systems are particular instances of s! stems
of matrix units. Rloreover, we shall arrive at the approximate finiteness theorem
by the use’ of an ultraproduct. This is similar to the LBC of the ultraproducts
of \-on Scrnnann algebras that were described in [ 11. For finite \UII X;c~umann
algebras these ultraproducts were considered by AIcDuff [14]. SW also the
remark of Dixmier and Lance [6].
2. ~1PPROSIXI.4TE FISITENES:;
Let (-I-, /L) be a Lebesgue mcasurc space, ,L(-\-) 1. \Vith a fret llltratilter UJ
on N, we form the ultraproduct (99, pczJ) of the measure algebra of (.\-, p).
Thus we consider sequences A,, C X, n E &I, identifying two such sequences
(--l,Jni.~ and (L3,F’)r,,~ if
‘,iT> p(-37, .A .A,‘) 0.
The resulting set of equivalence classes is :+. This #J carries Boolean operations.
Here (S),,,,d represents I E .P and ( .;),trbq represents 0 E 99. If (A,,),,, j c .J s: -&‘,

338 CONNES AND KKIlXEK
then (S - &JnGN E 1 ~ a E iyAw, and if (H,s),,,k E B E .P, (B,,‘),,sk E 8’ E 99,
then (B, n B,L’) E B n i?. p is given by
P(J z- iftt #+L), (~~rr)rt&! EAES+ .
(W, ~(1’) is a measure algebra. Every automorphism II: of (,Y, p) induces an
automorphism c’lU of P bv
( cA,),EN E l,.~,JA, (AnLv t A- E .&P .
(‘onsidcr now an ergodic automorphism T of (-I-, CL). L\‘e denote by .H

YIEASURE SPACE AUTOMORPIIISMS 339
We remark that for all a E gU , thefa EL,(S, CL) that is given b!
fA m: weak*-lim xa, ,
12 ‘W
is T-invariant, and therefore a constant. It follows that
and also that
p*,(A) =: weak*-lim xa,, , (A,),& E a E ?g,, ,
n-iw
P&J, = pi 44, MJEN EA-ES, 0 > L’ -I% 4X) = 1. (3)
For R E .,Y[T], Rw leaves &Jo, invariant and we denote the restriction of P
to S?‘,,, by R,,, . By (3), R, is a pw-preserving automorphism of gn, .
(2.2) LEMMA. Let (An)nsN E a E dc9,, , and let B C AY. Then
Proof. From (2),
,,(A) P(B) = fi; &% n B).
ifez ~(-4n n B) -= ‘,i; j-x Xa,XE? 4-l = .c, CL&f) XB 4 ~~ ,A4 P(B). Q.I:.D.
(2.3) I,EMMA. Let R E .N[T] be such that for some ff E .H(,., , a + 0,
Then R,,, == I .
R,i? = i?, B c a. (4)
Proof. U:\;C’e assume that there is a c C 1 .--- d, f? +: 0 such that R,,,C C 1 - e,
and we proceed to arrive at a contradiction.
be such that
Let (Av)nEv E a and let (C,),t.,N E 6
RC,< C S - C,; , REN. (5)
By lemma (2.2) there exists a subsequence I
PL(BJ ) MJn) ,4~1).
12 EN.
(BnLtw represents an element B of 8, fi ~/- 0, and fi C A^. From (5)
RB, C S ~ B,, , TlEiw.
Hence R,.,B C 1 - B, contradicting (4) QED.
(2.4) LEMMA. For R E .N[T], R2,,, I ;f and on.[~s if R E [T].
(2)

340 CONNES AND KRIEGER
Proof. Let Y(a), Ii E N, be an increasing sequence of finite subgroups of [Y]
that have uniform orbit size, and whose union is uniformly dense in [T]. Let d
be the uniform metric on ./lr[?“]. If R,,, =: 1, then there is a K, E N such that
inf d(R, S) -: 1.
StJ(k,)
Indeed, otherwise one could find for every hen; a q(k)-invariant set R(k)
such that
(L@(k) A RB(k)) 2: 1, kEN> (7)
(see [7, pp. 137-1391). By Lemma (2.1), (B,),,N would represent an clement
2 -f 0 of -CZU , and by (7) one would then have R,$ =& a. That R E [T] follows
from (6). Q.E.D.
The proofs of the following two theorems are quite similar.
(2.5) THEOREM. Let T be an ergodic automorphism of (X, p), and let R E . V[T]
be outer aperiodic. Then T and R generate an approximately finite group.
Proof. To prove the approximate finiteness of the group 9 that is generated
by T and R it is enough to construct a sequence B(k), K E N, of finite s&groups
of [Y] such that
9(k)xC!qk + 1)x, Yx : : yN Y(k) .v, f.a.a. .I’ E zY
(see [Ill). A construction of such a sequence by an inductive procedure will
be possible, once one has established that, for all L E N and for all ? 0, there
is a finite subgroup SC!? of [9] such that
jL t
-Lii.iCL
n {X E s: TiRjs E XS:. ‘-. 1 --- ?l,
and for this it suffices to know that for every E :, 0 the finite subgroup ,Z of [Y]
can be chosen such that
To produce such an 2, let )Ur E N,
p({x ES: Tzc E 2x}) 3:. 1 -- t,
p({x E X: Rx E 2.~)) : I t.
i
(6)
(8)
.;II ;- 2EC1. (10)
By Lemmas (2.3) and (2.4) one can apply the Rohlin tower theorem to R,,, , and
hence one has an P E g

C‘hoosc a representative (F,,),,N of fi such that
MEASURE SPACE AIJTOMORPHIS~IS 341
F n n R”‘F n = c: , 0 < m CL -If, I? E N.
Let then F he one of the F, such that
and
For
by (13),
(13)
(14)
/L(n) > 1 - ; E. (15)
Let 6 I- 0 be such that for iz C X, p(A) < 8 implies that 4Mp(R”‘A) < E,
0 < m < M. Choose then a FE [T] that is the identity on S -F, that is
periodic on F, and that is such that, with ,F the group generated by T;,
Denote by R the element of [!g] that is the identity on S ---- n, that has period
period M on n, and that is such that
The finite subgroup 2 of [Y] that is generated by T and J? satisfies (8) by (14)
and (15) in conjunction with (16) and th e c h oice of 5. SF also satisfies (9) by (10)
and (13). QED.
(2.6) THEOREM. Let 7’ be an ergo& autonzovphisnz of (-I-, p), and let r be a
jinitegroup. Let there begiven for every y E ran R(y) E .N[T], such that R(y) $ [T]
for y + e, and such that
R(y’) R(y) = R(yy’) mod[T], y> yt E r.
Then 7‘ generates togetlzer with the R(y), y E r, an appvosimatelv finite group.
_
Proof. In order to prove the approximate finiteness of the group 3 that is
generated by T together with the R(y), y E r, it is enough to construct for
every E ‘-. 0 a finite subgroup .Z of [Y] such that

342 CONNES AND KRIEGER
For this, let L E N be such that
p c .,L
-Li!z {x E x: R(y--1y’) R(y).v = R(yf)7%) > 1 - r /-le. (19)
)
Lemmas (2.3) and (2.4) allow an exhaustion argument that yields an FE.%- 0 be such that for A C X, p(A) < 6 implies that 4 1 I’ 1 /L(R(Y)A) < E,
Y E r. Choose then a 17 E [T], that is the identity on X -F, that is periodic
on F, and that is such that, with 7 the group generated by 7”‘,
Order r,
p i -L2 p(F) - 6. (23)
r -= iy,,! : 0 -g nz c f i:.
Denote by I? the element of [g] that is the identity on X - D, that has period
1 r 1 on D, and that is such that
I%x -~ R(y,,,)x, f.a.a. .Y EF, 0 K m < / r !.
For the finite subgroup Z& of [q] that is generated by F and l?, one has, by (21)
and (22), in conjunction with (23), and by the choice at 6, that
Thus A? satisfies (I 7). By (19) (22) and (23). .?P also satisfies (18). Q.1X.D.

MEASURE SPACY AUTOMORPIIIS~I~ 343
(2.7) COHOLLARI.. Let 7’ be an ergodic automorphism of (A-, p), and let
R E ~ I’[ T]. Then T and R generate an approximately finite group.
Z’roof. This is Theorem (2.5) and a particular case of Theorem (2.6).
Q.E.D.
3. OUTER CONJUCAC~
M-e describe now unit systems (compare [12]). For this let (-\-, p) be a Lebesgue
measure space. A unit system of A C S, p(A) :- 0, consists of a partition
(JJc,iiR of .-J together with a collection C:(W’, o), W, W’ t B, of isomorphisms,
that is, bi-measurable and nonsingular bijections,
such that
C-(W’, w) : .-f(w) - A(w’).
LT(Q, w) : = I) LyWN, w’) LT(w’, w) = 1’(w”, w), w, w’, w” E f2
TT’e indicate such a unit svstem N bv the notation
a: = (Q, ‘4, 1-I(.), c;(., .)).
Assign to a permutation 7~ of Q the automorphism C(r) of d that is given b!
P(%-)x = V(Trw, W)K, f.a.a. x t .-l(w), w E -0,
and denote by Y(a) the group of these l;(n). 1Vith Q,, C B, and
B ~-: (J A(w),

344 CONNES AND KRIEGER
Given a countable group X of automorphisms of (X, p), vvc ~;a! that an
isomorphism c: =1 + B, A, B C X is associated to Z, or is an X -isomorphism,
if C:x EXX, f.a.a. .x E A. We say that an isomorphism is a 7’-isomorphism if it
is associated to the group that is generated by an automorphism 7’. -4 unit
system is called a system of Y-units (resp., of T-units) if its isomorpliisms are
X’-isomorphisms (resp., ‘I’-isomorphisms).
\\‘hen constructing our models for outer conjugacy we need to consider
product spaces. The cylinder sets in a product space S n,,, 12, , \vhcrc
the Q, are finite sets, will bc denoted by
Given such a product space the term odometer group will refer to the group of
transformations of S that is generated by the I’,,, , where. with 7 a cyclic
permutation of Qn,,, ,
(C~2’),, z-= T,,?,,, > if 12 = Vz,
2’, , if n 2 fn, llEN, WlEN.
Denote by Qn the set of functions q~: Z, -+ Z_ with finite support and such
thdt v(O) 2 1. Set
T(cp) = [(j, k) ; 1 i k -,- v(i), j E sqp lpj,
and define for h > 0 measures q,,,, on r(v) b!
We shall have to consider product spaces
For an isomorphism 1,‘: A + B, A, R C I’, that is associated with the odometer
group of such a product space (IT, v), we set f.a.a. .x (.in 3 k”>),d* E --I
For isomorphisms c:: A4 -+ B, I-: B + C, .-I. B, C‘C I-, that arc associated
with the odometer group of the product space, one has a cocy-cle identitj
/ll~rT(,Y) h(T(x) f hr,( lk), f.a.a. x E .-I
Let q+, denote the element of @,, that is given by-
T,,(O) v,,(l) 1, di) 0, .i 1

MEASURE SPACE AYI’O~IORPHIS~ld 345
(3. I) Ll3lX~. Let q+ appear infinitely often as an entry, in a sequence
(y(n))!,, -_ E cD,,‘~. Then the elements I,: in the full group of the odometer poup of
(1’9 v) -== Jj (r(dfl))l 4dd.A x ., 0,
?iEhl
such that 11~. --= 0, form an eyodic group.
Proof. Let Y stand for the odometer group of (I, 1’). Let B C 1, V(B) ‘., 0.
FVe show that there exists a 11 C B, V(D) > 0, and a Y-isomorphism IV that
maps 11 into B such that h, = 1. For th is, we can assume that X 51 1. i5-e first
find an .Yt N and an
a E n r(dTl))
l(n (I -+ (X/2))(1 -+ A)-’ v(Z(a)). (3
Then we choose an 112 > N such that v,(M) :: v. . For
then, hy (25)
(‘(0) =: {x E c: X,,f = (0, I)], C( 1) = {x E c: x,\, z-r (I, I)),
v(C(0)) > i( I $ X)-r v(Z(a)), v(C(l)) > $I( I + A) l v(Z(a)). (26)
With 1. denoting the automorphism of (E;, V) that changes the Mth coordinate,
set
D = C(0) n FC(1).
From (26) r,(Z)) > 0, and we can choose for V the restriction of C to 11.
If now R C 17, 0 < v(E) < I, then we have by the ergodicity of the odometer
group sets z4 C E and B C Y - E, v(A), v(B) > 0, and a J E Z, together with
a 9’-isomorphism W: iz + B, such that h, = J. A repeated application of the
preceeding remark yields a set D C B and a 8-isomorphism I’ that maps D into
B and such that h, -- -1. For the restriction E of I’W to W-lD one has then
11,. 0. QED.
\ve set
QD ={g’E@“:cp(j) =o,j-;:pp:, PEN,
and we fix now for all p E Z,. sequences (~(p, YZ)),,~ E Qip’u’ such that every
element of Qp appears infinitely often as an entry in (q(p, n)),,h . Then we
build the product spaces

346 CONNES AND KRIEGER
and, with K the counting measure on Z,
Let Y{, stand for the odometer group on (S, , v,,), p E Z_ , and let cg,, stand for
the odometer group on (XA , vh), X ‘, 0. With 7 the translation of Z I~!- one, set
ZD = (Cm E [Yr,] : h,(x) =~ 0 mod p, f.a.a. $2: t S,,), pez .
As is seen from Lemma (3.1), the ;I”, are ergodic. We make on the !B,,.,, , the
analog definition of Iz, and set
.i@ I),l,m = (C E [9P,1,K] : h,(s) = 0 mod p, f.a.a. x E Xp,&, PEZ-,
2P O,i\,W = {C: E [Ya,J : h,(x) ~- 0, f.a.a. x E Xc,,,n,YOj, A- 0.
These &‘n , ,, , o. are ergodic as well. Further, set
ZP = {R E [??,I : AR(x) == I mod p, f.a.a. s E X,1,
9 Il,l,m = (R E [9,,r,J : AR(x) I mod p, f.a.a. x E X9,1,m}, /lEZ. )
2 O,h,cc = lR E [~~‘,,A,,1 : h&4 1, f.a.a. xEX,, ,,,, ,I, A > 0.
An exhaustion argument based on Lemma (3.1) shows that these &‘,, and
92 o,i\*m are not empty. From the cocycle identity, one further has that
R’R-1 t ZD , R, R’ E S$
R’Rpl E y%;,,h,m , R, R’ E 9?p,A o. ptz-,, h >o.
Given an ergodic v-preserving automorphism T of (X, v), and given R E .,1 ‘[T],
let, for A, B C X of positive measure and for k E Z, flA,,([T]R”) be the set of
isomorphisms C: A - B such that for some T-isomorphism I-: A4 + R m”B,
U = RxV. If here .4 and B are understood then we write simply $([T]R”‘).
(3.2) LEMMA. Let T be an ergodic v-preserving automorphism of (-id, I)), let
R E M[T] and let 4, B C X, i t Z, such that v(B) := (mod R)” v(A) 0. The-n
yA,R([T]Ri) 77 ,^.
Proof. An exhaustion argument based on the ergodicity of 7’ yields this.
Q.E.D.
(27)

MEASURE SPACE AUTOMORPHISMS 347
(3.3) LEMMA. Let T be an ergodic v-preserving automorphism of (X, v), and
let R EM[T]. If U E $,,,([T]Ri), ~~~ fD,,-([T]Ri), then VUE $A.C([T]RZmm’).
Proof. Use R E N[T]. QED.
In the sequel we use, for a = (& , R,)l~~,,,~, , a’ = (j,,‘, h,,,‘)l&,,,L,, E I‘(p)“,
the notation
C (a’ - a) = C CL -LA.
l- 0. Then there exists, with some NE N, a unit system (In”, -4, A(.),
Z-( ., .)) and subsets A,; C r(q#’ with
and
U(a’, a) E f([T]Rc(‘+“)), a, a’ E IJq+,).V.
Proof. One first chooses an appropriate NE N and a partition (4(a))aEr(u,,~.~
of iJ and then applies Lemmas (3.2) and (3.3). QED.
(3.6) LEMMA. Let T be an ergodic v-preserving automorphism of (S, v),
V(X) =m I, and let R E N[T]. Set p = p(R). Let Q be an automorphism of (X, V)
such that
{T”lRz~ : m, 1 E Z} = {Qj.x : i E Z), f.a.a. ,x E X. (28)
Then there exists for
a unit sirstem
all A C X, v(A) > 0, and for all E > 0, I E N, a y E @?, and
a =z (I-(p), A, A(.), U(., .))
such that
I-T(a), a) E d( [ T]Rx(a’-a)), a, a’ E r(9)
v ,J, {x E -3: QAi.X
E 9(a) x}
i 1 > (I - c) v(A). (29)

348 CONNES AD;D KRIEGER
Proof. With some L, E N there is a system p of G-units,
such that
B = ([I, Lll, -4, We), f 2E-lL,L, ,
v(C(Z,)) = L,‘V(A), 1 :: IO /. By (30) and (31), we have satisfied
(29).
Q.E.D.
(3.7) THEOREM. Let T be an evgodic v-preserving automorphism of (X, v),
v(X) : I, and let R E JV[ T]. Set p p(R). There exists an isomorphism
(31)

such that
and
Pvoqf. We denote
On C-y,, , v,), one has the unit systems
>IEASI!RE SPACE AUTOMOKPHIS?vE 349
T;[T]V-1 == F,, ,
(r&h x-1, > Z(.), PC.5 .)>, ?zEN,
where for (I, a’ E r,(n) the P(a’, a) leave all coordinates beyond the nth coordinate
unchanged, n E N. The proof relies on an inductive construction of a sequence
a,3 of unit systems
such that
(32)
(331
L’,(a’, a) E $([T]R“(“‘-“I), a, a’ E T,,(n), n E N, (34)
and where 01~~~~ refines 01,) . One requires also that the a-algebra of (S, 11) is
generated by the -d,,(a), a E F,,(n), n E N, and that
To obtain such a sequence of unit systems one constructs inductively the unit
systems n,,o,) with some increasing sequence n(k), E N, k E N, where the a,,(Ic+ll
is produced by refining c+~(,~) , by means of a suitably chosen unit system plc .
To make a suitable choice of the /3,( one appeals now to Lemma (3.5) in order to
ensure the generation property, and one appeals to Lemma (3.6) and Corollary
(2.7) in order to ensure the validity of (35). Both lemmas are used in conjunction
with Lemmas (3.2) and (3.4). 0 ne uses at this point the hvpothesis that ever\
p) E Dr, appears infinitely often as an entry in (&, n)),!,% . -
The generation property yields an isomorphism
such that
and by the refining property of the 01, one has
VUn(u’, u)T’-1 = P(a’, a), a, a’ E r&z), n E N. (36)

350 CONNES AND KRIEGER
By (34) and (35) we can cover X by an increasing sequence E(n) C S, n E N,
such that one has a decomposition of the sets A,(a) n E(n),
A,(a) n E(n) = w qn, n’, u), a t r,(n), tt t N
(lr’El‘,,(?i~:~:(~‘~0)=0nrOd 73
such that
It follows from this and from (34) and (35) that (32) holds. Similarly, we have
from (34) and (35) that we can cover X by an increasing sequence Z)(n) C S,
n E FV, such that one has a decomposition of the sets A,(a) n D(n),
A,(u) n D(n) =: u qn, a’, a), a E r,(n), tt E N
{a’Er,,(n):C(a’-a)=1 mot1 7’)
such that
Rx = U&z’, a) x, f.a.a. s E n(n, a’, a), a, (2’ E FP(n),
It follows from this and from (34) and (35) that
and (33) is shown.
c (a’ - u) -= I mod p, n E N.
(3.8) THEOREM. Let T be an evgodic v-presevzin,n automorphism of (X, v),
v(X) = a, and let R E .N[T]. Set p = p(R), A : mod R. Then there exists
an isomovphism
such that
Proof. The proof is achieved bp the same means as the proof of Theorem (3.7)
Q.E.D.
(3.9) COROLLARY. The outer period and the module form a complete set of
invariants for the outer conjugucy of the elements in the normalizer of the .full
gvoup of an ergodic measure preserving automorphism.
Proof. This is from Theorems (3.7) and (3.8) and (27). Q.E.D.

4. ~WPLICATIONS
\\‘c want to describe next some examples for Corollary (3.9). There arc first
the examples of infinite product type. Let X.:,. E N, k E N, and set
(-I-: 1.) lvN ({I >...: iV,.:,P,J> Pk(?l)
)7y
I<
1
> 1 .I II : . -I-,( , I7 E Pd.
Let .F be the odomctcr group on (-Y, v). .% is an ergodic hyperfinite group that
leaves 1: invariant. The infinite products are defined b!
(Rx),, = xl, -r 1 mod LVa , k E N, s : (s,,.),~~~~ ES.
and we have that R E JV'[.FJ. If supkEN S,, =-- SC, then R is outer aperiodic,
and hence by Corollary (3.9) all such R arc outer conjugate. One can prove
directly in this instance that R and .% generate a hyperfinite group. Infinite
products induce infinite product automorphisms of the hyperfinite II, factor.
Further, there art’ the Bernoulli shifts and the Markov shifts. ‘L’ake, e.g.,
the pz-shift. Set
351
and define an .FrS as the odometer group on (1’,, , p,(). ‘The n-shift ,S;, .v\,
(J,~, ljliz , ~3 E II-, is then in ~ $“[.&I, By a similar construction, all Hernoulli
shifts and all mixing nlarkov shifts can be viewed as elements of ..1 ‘[‘/‘I, \\-here 7’
is crgodic type II, (compare [4, 51). All these shifts arc outer aperiodic, and
therefore th& are all outer conjugate by (‘orollary (3.9), nnc 1 a 1~ bo outer conjugate
to the outer aperiodic infinite products. Furthermore, the iloncornnilitati\.e
Rcrnoulli shifts and the noncommutative Markov shifts that these shifts induce
on the hvperfinitc II, factor arc all outer conjugate, and also outer conjuqte to
the outer aperiodic infinite products. These noncommtitati\.c shifts arc’ not
corl.jugatc if their entropy dialers [4, 51.
(4.3) ~OKOI.l.AR\-. II%ere exists a I’ E [&] such t/rot PS hns discwff spf7Yrtrm.
\Ye conclude with an application to the problem of appro\;imatc finiteness,
which has been prominent in this theory since its creation by Dye [7. 81. Let
us say that a countable group is approximately finitc if all its action 1~~ auto-
nmrphisms of a Lebesgue measure space arc approximately finite. 1:~ the proof
of the following corollarv one must, when distinguishing cases, in addition to
Theorems (2.5) and (2.6), also use an argument of the sort as gi\.cn in [I 3,
Sect. 5, Lemma 5.41, and one must carry out an ergodic decomposition. It ~vas

352 CONNES AND I,‘.Yterlsiotls ?f
approximatel\~ jinile groups bj, solvable groups are appro.rimate[v jhite.

The Theta Transform and the Heisenberg Group
RI(.IIAIW TOLIMIEKI
1. INTRODUCTIVE
In this paper we continue the program of [7] of establishing a “good” function
theory for the Ileisenberg manifold. The point of view hcrc is seemingly very
different and has been motivated by the aim of building a function theory
which would reprove some of the known results involving automorphisms
of the Heisenberg manifold. At the same time, our tools unify under this theory
some ideas of Hardy in abelian Fourier analysis regarding the growth of an
P-function and its Fourier transform. The main idea we consider is a variant
of the complex Fourier transform which we call the theta transform and is
more natural from the theory of the Heisenberg manifold. Surprisingly, perhaps,
it also appears, as indicated by the Hardy theorem mentioned abol-c and the
results on the Hermite functions, to be more natural for some problems in
abelian harmonic analysis as well.
J,et .Y be the three-dimensional Heisenberg group,
and r the discrete subgroup of A. consisting of those matrices
with .x, -\v, at Z. The homogeneous space of right cosets r’X is a compact
manifold admitting a unique probability measure invariant under right translations
rr ~+ CY~, y E 1V. This measure will be assumed throughout. Th c ,pr.c : c~ *
I’ 1\T is called the Heisenberg manifo/d.
Let LP(~‘I,-V) be the Hilbert space of square integrable functions on f’ -1..
( qnngilt c 1477 b, .\cadenuc Press, Inc.
.\II righrs of reproduction in my form rescrvcd. ISSS 0012-1236
353

354 RICHARD ‘~OLIRlII

THE THETA TRANSFORRZ 355
The original functions 19[:]( z, r ) are the Jacobi theta functions, argument u”,
period 7 and characteristic [z]. They have appeared before in the theory of
the Heisenberg group (see [2, 3, 81). In [7] a kind of Fourier analysis of the
Heisenberg manifold was initiated using theta functions. Here, their role is
different and stems from functional equations satisfied by the theta function
with respect to the automorphisms of the Heisenberg manifold. This analysis
is contained in [2]. We repeat those facts which are necessary.
An uutomorphism of r\N is defined to be a group automorphism A of n-
satisfving /l(r) == I‘. Clearly /Z induces a measure-preserving diffeomorphism,
still denoted by d, of Ti,X onto itself. Up to inner automorphism by the
normalizer of r in N we shall describe the automorphisms of r\!&‘. Let d
(f i) E S’&(R) the group of all 2 x 2 matrices with real coefficients having
determinant 1. Then A induces an automorphism of 1Y according to the formula
Then the group of automorphisms of r;N, modulo the inner automorphisms,
is given by those iz E S&(Z), &I ; (:: i) with nc :-~ bd 0 mod 2. \Ve denote
this subgroup of S&(Z) by L.
Our main concern is the action of L on P(r’l,N) defined by the rule
(AF)(x, J’, z) = F(d(x, y, 2)), ,4 EL, F E P(r:l\-).
Clearly I, acts as a group of isometries of S?(r’,,lY) and as a group of unitary
operators of Hi .
The general problem of this work is an investigation of the eigenvalue one
space of an A EL. In particular we shall use our methods to reprove the following
result found in [I].
'THEOREM A. If 9 EL is hyperbolic, i.e., has real positive cigenvalucs,
then A acts evgodically on QV.
It is sufficient for our purposes to define the notion of A acting ergodically on
l‘~tiV as follows: If FE zyrjx) and F = F 0 il, then F : constant ax.
To interpret in our language the Hardy result we repeat the following
observation of A. Weil and found in [2]. The space of functions Hi can be
characterized as those functions on N of the form
wherefE 9’“(R). Moreover the map

356 RICHARD TOLIMIERI
is a unitary isomorphism between P(R) and If1 . Letting 0 = (s :) we have
where f denotes the Fourier transform off. The Hardy result can be given by
the following statement.
'I‘HEOREM B. Let CJ -= (-I: i) and F IV(j) E H, in the eigenaalue one
space of (T. Then if /f(x)! c< C evJ:’ , s E R, C constant, zce have f (x) 7 C”r r;,1.3,
C’ constant.
2. THE THETA TRANSFORM
The complex Fourier transform (see [6]) relates the spaces P(O, ‘KJ) to a
subspace Hz(A) of the analytic functions of h the upper half-plane. \\‘c shall
describe H2(k) below. Hence there is precedence for relating P-theory to
analytic function theory. We shall consider a variant of the complex Fourier
transform more suited in dealing with the Heisenbcrg manifold.
For F E HI define
IfF = W(j), then
a(F, 7) =- 1 qx, J, 2) #(x, ?', 1, T) dxyc, im(7.) > 0,
. T\.v
a*(F, T) == .I;;,F(x, y, z) $i*(x, T, a, T) dx~2c, im(7) > 0.
x(j, T) -1 a(F, T) = j-J(t) eniT” dt,
a*(j, T) = a*(F, s-) = ./R/(t) * t * e*iTt’ dt.
The maps FM a(F, 7) and F tt a*(F, T) are clearly well defined. \Pe call
these transforms, theta tvansjorms. We want to show a(F, T) and a*(F, T) are
analytic for T E 6. When F E P(r\N) this is obvious. Also, if F, is a convergent
sequence in 59(T’,,X) then a(Fn , T) and a*(Fn , T) are both uniformly con-
vergent on compact sets on /I since # and +* are both bounded when 7 is
restricted to a compact set. Hence the class of functions FE P(EX) for
which both a(F, T) and a*(F, T) are holomorphic in /: is closed in Ll(r!JV)
and since this class contains C’(T\iV) it must be L’(T’,,!Y). In particular we
have the following result.
THEOREM I. FOV FEH~, a(F, T) and c?(F, T) are analytic in I.

THE THETA TRANSFORM 357
It is convenient to consider the analog of even and odd functions in P”(R)
for Hr . The matrix (-i -y) considered as an automorphism of LP(r’t,!Y)
decomposes this space into the orthogonal direct sum
the first space being the eigen one space, the second the eigen minus one space
of the automorphism. Since (-i -7) p reserves each of the 11,‘s. This decom-
position can be carried out separately in each [Z,, as well and we get
Under the “L\7eil map,” W: P(R) - II, it is easv to show that ffr’ cor-
responds to the even functions and HIa to the odd- functions of P(R). In
particular, #(x, JJ, x, T) E H,’ and #*(.r, J, z, T) E Hi”.
The first result which makes the transforms useful is contained in the next
theorem. It is of some interest to note that if the density of the Hermite functions
in P(R) is assumed, then a proof could be based on this fact. M’e shall indicate
later how to do this. We shall, however, remain true to the spirit of the complex
Fourier transform in our official proof.
'THEOREM 2. For FE HI , F :: 0 almost everywhere if and only ;f OI(F, 7)
ol*(F, T) :-- 0 for all T E d.
Pro?f. \\‘rite F Fl I Fg where I;; E 1fi” and P2 t 11,‘). Then CX(F, T) --=
ct(Fl , 7). Assuming cx(F, T) 5-1 0 in h we get
where F1 W(fi). A change of variable gives
where we set 7 :mm .w 1 i. However, it is not difficult to show (fi(s11’)/s112) e nc
is in 9(0, 1) and P(i, CO). Hence the uniqueness of the Fourier transform
gives (fi(s’~2)/S1/2) emii’
almost everywhere.
0 and fi(t) : 0 almost everywhere. Clearly Fl =- 0
In this same way assuming u*(F, T) :: 0 in /; gives Fi LC 0 almost everywhere.
Since the converse is trivial we are done.
C'OROILARY 2.1. ~h~functions {$(x, y, z, T), #*(A-, y, x, 7)]Ttd are dense in HI.

358 RICHARD TOLIMIERI
COROLLARY 2.2. The vertor space spanned b? thefunctions {t”e-nt2: M 0, I....:
is dense in S2(R).
Proof. From Theorem I it is simple to show that the derivatives of 01. !x”
with respect to 7 can be evaluated by differentiating inside the integral in the
definition of pi. 01*. Hence
where CL(T) ol(F, T) and F m-m W(f).
Assume jRf(l) Pe rrf’ df == 0 for all n .:z. 0. Then &“)(i) =- 0 for all II 0
and since a(~) is analytic LX(T) 2-1 0 in L. Hence f must be an odd function.
In the same way using OI’~ we can show that assuming JR!(t) P+le ii*” dt 0
for all 72 3 0, for f t P(R),
follows.
implies f is an even function. The corollary casib
The Hermite functions are determined by orthonormalizing the functions
tne-nt2 n 2% 0 in P’(R). Hence we have proved the classical fact that the
Hermite functions are dense in P”(R). If we had begun by first assuming
this result the reasoning of the corollary could be traced backwards to prove
Theorem 2. More important, however, is that our development does not place
the Hermite functions at its center but at the system {t7Le-nt2}. In essence we
have associated in a natural way with a functionfE -P(R) an analytic function
a(f, T) in L and the inner products of f with the system {t”e nt2) in Y2(R)
determining the coefficients of the expansion of a(f, T) at 7 - i.
We shall require some elementary facts concerning HP-theory of the upper
half-plane. For a reference see [4]. A function /L?(T) analytic in /; is said to be
in W’(L), p 3 I, if
A function P(T) analytic in d is said to be in H”(d) if it is bounded in A. Re
accept the following two well-known results of HJ’-theory. First, whenever
/3 E HI’(R), I < p < co, /3 has nontangential limits at almost all points of
the real axis and this boundary function is in 9p(R). Second, /3 E W2(b) if and
only if we can write
P(T) = ijT G(t) e -PnifT dt,
where G E Y2(0, CD). Moreover i /3 /‘L == / G ~Z .
THEOREM 3. For FE HI we can zcrite

THE THETA TRAXSFORM 359
zohew x1 T’ (I ’ (A) und a2 E W(d). The function i+(F, T) 1 10s nontangential limits
at uln~ost trll points of the real axis which zce cm ~LV+P
N(F, .x) = N1(F, s) -t- a!,(F, x),
E’roc$ ‘1’0 prove this result for ci we may without loss of gencralit~- assume
that F is c\xn. Define
where F lV(f). Clear13
we will ha\-e LX&T) E P(A) as soon as we know that f(P)j~~,~ t P’i(l, &CYX).
This follows from
Since we have assumed F is even,
cd(T) r= 247) -+ 244
and we have verified the result for 01.
In general a function analytic on A of the form JR f (t) teziT(’ dt need not
necessarily he as nice as the sum of an H=(A) and an N’(A). However, we can
restrict f in an obvious manner so that a*(J, 7) is a good function on 1.
TIIEOKI;RI 4. Let F -~= W(f) E HI where tf(t) E Pz(R). Then

360 RICHARD TOLIMIERI
Proof. The proof follows immediately by the previous theorem since
a”(f, T) =- a(t .f, T).
1f-e note that the subspace of HI considered in ‘Theorem 4 is a dense subspace.
3. hI:TOMORPHISMS OF r:n;
‘The importance of the functions # and $” in the study of the automorphisms
of I’:,iL’ comes from the functional equations satisfied by these functions. The
following crucial results (see [2]) are contained in the following thcorcm.
where
(b/d) denoting the generalized Legendre symbol.
l+oof. Statement (a) is contained in [2]. 1’0 prove (b) we first note that
K((i F), T) : 1 and K(((y ‘,), 7) (jj~)i;‘?. Also it is quite simple to show
that if (b) holds for a set of elements in L it holds for the group genrratcd by
this set. Since (i :) and ((y ‘,) generate L we will have verified (1~) once we
show it to be true for these two matrices. The computation for (i f) follows
by direct substitution. For (-T ‘,) we note that the effect of this operator on
$*(x1 y, u”, T) is analogous to the Fourier transform, i.e., $*((_y :,)(.v. j’. r), 7)
$ni* C g(y -1 n) $fllTl.r, where g(y) (yeniT’( ‘)^. It is a simple excrcisc to show
that g(y) = (i/7)’ jz( 1,‘~) y~~(--~/~)1J*. Hence (b) holds for these two matrices
in L and for all of L by our remarks above.
The following corollary is of central importance in our applications. Its
proof is a simple change of variables argument along with Theorem 5.
COROLLARY 5.1. Let F E HI and A EL. Suppose F = F 0 A. Then
@, 7) K-(4 T) 4F, ‘A(T)),
,“(F, T) = &?(A, 7) ol*(F, ‘,4(~))3
whew K*(il, 7) = (I i(bT -I- d)) R(.-l, 7).

TIIE TfIEI‘A ‘lxANSFOK~1 361
For our first application to the theorem of Hardy, stated below in a slight]!
more general form than given in the Introduction, we begin by considering
the functions

362 RICHARD TOLIMIERI
stated above /F, 1 is constant. In particular I F, is constant. \!.rite ZE;
E;’ -+ F,” where F’ is even and F,” is odd. Since ((i _~f) centralizes --I, 1~0th
F,’ and Flo are fixed by A. Hence i Flo 1 is constant by the ahot-e reasoning.
However, F,O(O, 0, 0) - F,“(O, 0, 0) = 0 thus E;” Go 0 almost e\-ervlvhere.
Put IX(T) = LU(F, , 7). Then U(T) K(A’, T) ~(‘A’(T)) for all 7 E /: and integers 1.
Let ur , uz (ur < up) be the fixed points of L4 in R. The restriction of the boundar!
function a(~) to the interval [z~i , Q] is in F[u, , u.,] n -4a*[u, . z~j. Also z4
maps this interval onto itself, and
Hence
a(s) = lqd’, .x) cd(‘A’(x)).
by a change of variables and Holder’s inequality, where A’ (y; :;I). Since
A is hyperbolic, the integral on the right can be made as small as we like.
Hence U(X) =m 0 almost everywhere on [ZQ , ~‘1. A standard Z-l”-fact implies
this can only occur if a(~) : 0 for all 7 E d. By Theorem 2, F,l 7 0 almost
everywhere. Hence Fl := 0 almost everywhere.
Since 1 E;, 1 is constant, F,, is bounded measurable. It follows that I nz = n* -1 0, fixed bv .4, we ha\;e F,,, -~I 0 almost c\-er\ where. IVe
first proT-e the following lemma..
LEMMA 1. Let m ~~ n2, n >~ 0. Then the multiplicity of H,,, is 11” and there
exists a direct sum orthogonal decomposition of H,,, into W-invariant and i~~~rducible
subspaces each of which is inaariant under some power of A.
Proof. Let (T E GL,(R) given by u = (F 4:) with determinant of 6 denoted
by / g I. Then g acts as an automorphism of A by the following rule
Let 07,? =m (ii’ 6:) and A,,, :~ {(a/m, b/m, z): a, b E Z, z E R). In [2] it was shown
that a direct sum decomposition of H,,, could be given as follows

THE THETA TRANSFORhI 363
where J u,,,(H~), Lk’,,G(6) == G(h- ‘8) for G E EZ,,, , h t A,,,,‘n;‘(Lz) and 6 E -1..
‘I’hat this all makes mathematical sense was verified in [2] as well. It is clear
that A‘1 acts on A,,,jo;zl(Lz) since iz commutes with o,,, and hence some finite
power of -3 acts by the identity on A,J~;~(Lz) since /l,,,/~;;l’(Lz) is a finite group.
Let &-1k be this power. Take GE&J given by G = YA(G” c CT),!), where
G”tH 1 . Then G o &4 ~~ 9’-1(A)(G* i\ d u,,,). Hence for arbitrary GE YA J,
G, A-1’; E 9, J since 4-l(X) = h and G” : a4 E IT1 . The lemma is proved.
We return to the proof of the theorem. Take F,,, E IT,,, , ~rz n2 , 0, and
consider the projection of F,,, on ZA(J), A t .!l,,,,‘a;S1(La) and J -= a,,,Hl Call
this projection F, . Then F, 0 Al, = F, for some k by the lemma. ?;ote .4’, is
hyperbolic. Write Fh =-: x\(G” 1 a,,,) with G” E ffl Then gA(G” .-1”’ 0 a,,,)
PA(G* c a,,,). It follows that G* 0 1 JA =~ G* and hence G” 0 almost everywhere
by the first part of the theorem. Thus F,, = 0 almost ever!;where and
F,,, 5: 0 almost everywhere. The theorem is proved.
A final word on these matters. To prove that if FE C(r”,X), .3 EL and
F 0 A F implies F is constant almost everywhere is almost automatic from
the discussion at the beginning of the theorem and the result in [2] that every
continuous function in H, must vanish at some point.
The author wishes to thank J. Neuwirth of the University of Connecticut for intro-
ducing him to the uses of H/‘-theory to various problems in analysis.
REFERENCES
1. I,. ALW~ANDEH, Ergodic automorphisms, Amer. Avutk. Moxt/&~ 77 (1970), l-19.
2. L. ATSLANDER AND R. TOLIILIIERI (with the assistance of H. RACCH), Theta functions, iu
“.Xhelian Harmonic Analysis,” Springer-T’erlag Lecture Notes, New York/Berlin,
1975.
3. I’. CARTIEH, Quantum mechanical commutation relations and theta functions, Proc.
Symp. Pure Math. Amev. Math. Sm. 9 (1966), 361-383.
4. I

JOURN.4L OF FUNCTIONAl. ANAT.YSIS 24, 364-378 (1977)
A Dunford-Pettis Theorem for L’/H” ’
I. CNOP ANI) F. L)ELBAEK
The spaces in the title are associated to a fixed representing measure 111 for a
fixed character on a uniform algebra. It is proved that the set of representing
measures for that character which are absolutely continuous with respect to HI
is weakly relativel?; compact if and only if each m-negligible closed set in the
maximal ideal space of Lm 1s contained in an nz-negligible peak set for H’
J. Chaumat’s characterization of weakly relatively compact subsets in L’IH”
therefore remains true, and L’,IH”- is complete, under the first conditions. In
this paper xve also give :t dirrct proof. From this WC’ obtain that L,‘jH”~ has the
Dunford-Pettis property.
This paper has origins both in harmonic analysis and in the theory of classical
Banach spaces. Kahane [ 121 and 1LIoone!- [I 31 proved theorems on the weak
completeness of the integrable functions on the unit circle, 1,i. Amar ]2] has
given a different proof of 1Iooney’s theorem using a property of the peak sets
for the algebra I3< of bounded analytic functions on the unit disc, obtained
by Amar and Lederer [3]. This method of proof was used by C’haumat [4]
to obtain a characterization of weakly relatively compact subsets in the quotient
L1,iH7 ‘, and for a class of uniform algebras (in the sense of Gamelin [7]) with
the same condition on the peak sets. Several steps in his proof do not rely on the
algebra structure, but follow from facts in the theory of linear spaces, as in
Grothendieck [lo]. A recent result by Dor and Rosenthal [5, 141 also implies
part of the characterization, since the quotient is weakly sequentially complete
(see [4, 131 or Havin [I I]).
In this paper it is proved that the condition on the peak sets of the algebra
is equivalent to the following condition: The set of representing measures
absolutely continuous with respect to the measure m is relatively weakly compact
in Ll(m). This condition does not relv on the algebraic structure of the spaces
and is in manvcases casicr tovcrifv since it is difficult to get a complete description
of all peak sets.
364

A DUNFORD-PETTIS THEOREM FOR L1/Hal- 365
We then proceed to give a direct proof of the weak sequential completeness
and Chaumat’s characterization, under this condition on the set of representing
measures. A series of applications is listed in Section 4.
The authors feel indebted to J. Chaumat for using his unpublished results
and the referee for his remarks and comments, especially for his simpler proof
of (i) (iii) in Section 2 which replaces the original argument involving
convergence in HI’ (p -z: 1) and the holomorphic functional calculus.
In the sequel, A will be a uniform algebra on a compact separated space X,
ch is a nonzero multiplicative linear functional on rl, and m is a fixed (positive)
representing measure for 6, on S:
2
4(f) =- {f An for all f in .I.
The Hardy- spaces HJ’ are defined with respect to m for I < p -< o; in particular,
H” =- H7(m) is the weak-star (i.e., a(L=(m), L’(m))) closure of A in L”(m).
We denote by N the set of elements in Ll(m) which annihilate H” with respect
to the bilinear form
Ll(m) x L”(m) + @,
and we have the isometric isomorphism
(.fY 8 - J f‘c d?
(Ll/N)* s Ha.
In the classical case, X is the closure of the unit disc D, A the algebra of con-
tinuous functions which are holomorphic in D, 4 is the evaluation at the origin,
m is the Haar measure on the unit circle, N”- is the algebra of bounded holo-
morphic functions on D, and :\: is the algebra of complex conjugates of H’
functions which vanish at the origin.
Let :lfd> be the convex set of all representing measures with respect to 4.
A representing measure m in Md is dominant if every p in M+ is absolutely
continuous with respect to m; it is a core (strongly dominant) measure if every p
in M,, is smaller tan c m, where c is some positive constant; it is an enveloped
measure if for every sequence U, of nonnegative continuous functions on X,
such that s u, dm converges to 0, there is a subsequence u,, and corresponding
elements f,,. in A such that
4(fk) - 1,

366 CNOP AND DELBAEN
Core measures are enveloped, and the converse holds if n/r is finite-dimensional
[71.
The existence of an enveloped measure in Md does not imply that IVZ~ is
finite-dimensional (see [I 5]), but we have the following result of Gamelin and
Lumer [8, Lemma 11.1.31: If m is an enveloped representing measure for 4,
then the set
{p E MC/> ( p absolutely continuous with respect to m)
is weakly relatively compact in the set of all measures on X (or, equivalently,
in Ll(m)).
If m is a measure on S, the Gelfand transform is an isometric isomorphism
of the algebra L”(m) onto C(A), where J” is the maximal ideal space of l,V(m).
Each continuous linear functional v on LX(m) extends into a measure c on A’,
defined by
In particular, m has an extension &z and measures absolutely continuous with
respect to m extend into measures absolutely continuous with respect to rir,
and conversely.
A closed set F of A is a peak set for I$= if there is a function R in Hz such that
Rlt- 1,
&)I < 1 for all x outside F.
THE~RE~M. The follozcing conditions are equivalent:
(i) The set
is weakly Felatively compact.
{p E Mb 1 p absolutely continuous with respect to m)
(ii) Each representing measure for Hm on A is absolutely continuous with
respect to rk
(iii) Each closed set E in A??, of h-measure 0, is contained in a peak set
for Hz of 6measure 0 (see [4]).
Proof. Since (i) w (ii) [8, Lemma 1.3.101 or [7, p. 1021, we only have to
prove (iii) = (ii) and (i) * (iii).
(iii) 3 (ii) To prove the absolute continuity, it is sufficient to consider
closed sets in A’: if A is not closed, then
@(A) = sup@(E) 1 EC A, E closed}

A DlJNFORD-PETTIS THEOREM FOR L1/Hu i 367
and if &(A) is zero, then Gz(E) is zero and since the set E is closed, G(E) is zero.
and the supremum is zero also. For a closed set E, with h(E) zero, there is a
peak set F of ril-measure 0 containing E:
F -- j l(l)
where f is in II* with ij’~ -< I on . ti’\,F. ‘Therefore, and using that I; belongs
to M 0, Ref’> 1 on Q,f in HZ/
there exists a sequence U, in Re C$ such that
s
Uj 3 0 for all j,
uj 9 >l on 0, for all j,
uj d@z - 0;
passing to a subsequence, we may even assume that Cj”=, j uj d& is finite.
Therefore, the increasing sequence of functions ~1, $ I defined by

368 CNOP AND DELHAES
A
in Kc H’ is such that for IZ ‘3 k,
on 0, and
?‘“I ’ k
converges. By the monotone convergence theorem, its pointwise limit 2’ hclongs
toLl(v?L) and u is finite almost everywhere. Since for all z with real part at least I,
we have
the functions
satisfy
and are uniformly bounded. Any .f in the weak star closure therefore belongs
to I$ and satisfies
Finally, set
Since fur all k 1:: n,
on Q,, and
!.f-- ; A.
F :~- f-‘(O).
F 3 I? and the function 1 ~ f peaks on F. ‘This set has measure zero since it is
contained in the set where 2; is infinite.
3
A function g in flT operates on Hz-, and thus defines an element 715’ in I,‘/iV

the mapping
A DVNFORD-PETTIS THEOREM FOR I,‘,‘ll’ ’ 369
is continuous, injective and has a dense range. Also, for a bounded sequence
in H”, convergence in Lp and norm convergence of its image in Ll/A’, coincides.
LriivthlA 3.1 . Tlfe uGt ball of II” is complete for the metric
induced by I“/ .\:.
cf(“f,

370 CNOP AND DELBAEN
exhausts the unit ball of H7 which, by Lemma 3.1 is complete for the norm
induced by L1/A7. By Baire’s theorem, there exists n,, , go’ in F )(,, , and a positive
real number 6,, such that if g is in the unit ball of HTa and
we obtain, for g in the unit ball of II= with
LEMMA 3.3. If {&I, is a sequence in L’llV which converges weakly to zero,
l U a positive number, t’ mm: (C l “)/2, and n,, is as in Lemma 3.2 (or larger),
there is a positive seal number 6 such that fog f in the unit ball of HZ zcith
Proof Suppose {fr,) I) is a sequence of functions in the unit ball of H’
If Tfi, tends to zero inLl/A’, then thell-norm off,, tends to zero. We now use that
is weakly compact in Ll(m). By [8, lemma 1.3.101 or [7, p. 1011, this set is the
set of representing measures on the maximal ideal space of r,=(m), for 4 on F-i’,
and therefore [7, p. 321:
will tend to zero as n goes to infinity, by the compactness. The same will hold
if 'f,, is multiplied by the large positive number
26,,’ log t

A DCNFORD-PETTIS THEOREM FOR LIIHu '- 371
(6, as in Lemma 3.2). Therefore, if the assertion in the lemma is false, we can
find a real positive number l < 1, a subsequence {$n,}l; (n,,, 2 n,) of {c#~}~~ , a
sequence of functions (fn,ti: in Ha, uniformly bounded by 1, whose norms in
Ll/AT tend decreasinglv to zero and such that
together with functions z+ in H” such that
This estimate implies that the functions e “A are bounded bv 1 and, by the
Cauchy-Schwartz inequality, tend to 1 in L1(7?z):
If g,, is any function in the unit ball of I1’L:
.-: (2 - 2 exp (-.I’ uI, drn))l”
1~ T(ep”kgg, - g,)l; 5: / e--“kgc, - g,, 11, < Ii e-‘ric . ..- 1 i, ;
therefore T(epUILg,, -+ fri,. - g,,) tends to zero in L1/AT. ITsing the function go
obtained in the previous lemma, we prove that
if
II e-g,, + (1 -

372 CNOP AND DELBAEN
Therefore, if k is sufficiently large:
and, by Lemma 3.2:
We obtain:
1 7’(eetlkg,, -1 (1 --~ C) fn, - go) 1: $3,
I(+,, , (1 - c)fn,)l G I(&, , e+&J + i(+,, , e-““l,a, i- (1 - e)fn,)l
which contradicts our assumption.
i _ &I =: E --- t2,
I(A, > fn,)l -c 67
LEMMA 3.4. If {4n>n is a sequence in L’iN which is weakly Cauchy then for
every E, there is a positive real number 6 such that for all f in the unit ball of H” with
we have for all n:
Proof. If there is no positive real number 8, such that
if Iz, p ;> n,, and
(4, ~ Q,, ,f 11 < t
~g Tf~

.t DCNFORD-PETTIS THEOREM FOR L’,‘ffJ 373
if 72, p .‘: n,, . In particular, for all n P-- rl,, :
Finally, we can find 6, such that if
we have for n

374 CNOP AND DELBAEN
4
THEOREM 4.1. If the conditions of the theorem in Section 2 hold, then L’IA: is
weakly sequentially complete and the following are equivalent, fey a subset K of LlIN:
(i) K is relatively weakly compact;
(ii) For each positive number E, there is a positice number 6 such that, fey f
in the unit ball of HZ with
I 7y ~ .I 6,
we have
sup{!($,f): CfJ E K; < ‘;
(iii) For each positive number E, there is a natural number M such that for
each $ in K, there is a g in Hc with
/ g !jr, :c M
(,$A ~~~ l)y (# < E.
(iv) K is bounded and there is no (infinite) sequence {$,), in K which is
interpolating, i.e., a sequence for which the map
[&j,, : Hz -> P
sending f onto the sequence I($, , f )) ,) , is surjective.
Proof. (i) -+ (ii) If (ii) is false, there IS E :> 0, a sequence {b,,], of positive
real numbers tending to zero, and a sequence {,f7Jll of functions in the unit
ball of Ha with
:/ l’f,, -% 6,)
together with corresponding elements a,, in K such that
(15, > f,,) ‘.‘, t.
Since K is relatively weakly compact, we can extract a weakly converging
subsequence of {+,jn , by Smulian’s theorem. This subsequence will contradict
Lemma 3.4.
(ii) => (iii) is true by the following duality argument : Consider the set
K” {ffz II” (4, f); .S< 1 for all 4 in K].
By our assumption, EKO contains the unit ball B of H” intersected with
E (f E H- ~ 77 1 K.: St.,
which is weakly (a(H”, Ll/ni)) relatively closed. Rv an application of the bipolar
theorem (e.g., [IO, p. 831):
(I /G)K c (1 /,)A-(“, c F(B” u Eq,

A DUNFORD-PETTIS THEOREM FOR Ll/H=-- 375
where r denotes the weakly closed convex hull. Kow B” is the unit ball of
L1,‘LV (and conversely), and by the weak density of TE in the ball of radius 8
in ZJ’;i\‘:
IP = {b, E Ll/IV / I(+, f) < 1 for allf in E’] (I /6) TB.
Finally, K is contained in the weak closure of EB’ + (c/S) TB, but since B is
relatively weakly compact, this sum is already weakly closed and (iii) is verified
if 111 ~.‘6.
(iii) (i) is proved in [IO, Lemma, p. 2961.
At this point, we arc able to prove that L1/N is weakly sequentially complete:
If :A!111 is a weak Cauchy sequence in Ll/N, Lemma 3.3 applies, such that (ii)
is verified. Hence our sequence is weakly relatively compact, and does converge
in L’ /A\T.
(i) ‘. (iv) An interpolating sequence in K is equivalent to a basis in /‘.
Since in 1’. weak and norm compactness coincide. such a sequence cannot be
weakl! compact.
(iv) .. (i) If a sequence {+,Jn in K does not have a weakly convergent
subsequence, it cannot have a weak Cauchy subsequence (since L’jA- is complete).
By Dar’s complexification [5] of Rosenthal’s result [14], such a sequence
contains an interpolating sequence.
Remark. In the case of the disc algebra it is possible1 to give a direct proof
of (iv) (i), without using the result of Rosenthal and Dor.
The following is an immediate consequence of the weak sequential
complctcness:
~.~OHOIL~Rk7 4. 1. [12, 131. If t/ ze conditions of the theorem in Section 2 hold
and if f f,,j ,, is a sequence in Ll(m) such that for all g in H’, the sequence {Jfng dml,)
converges. Then there exists f in Ll(m) such that fey all g in H’ :
Pwof. Define, for each natural number II, Q, in I,‘,i;Y by:
(6, > R) = .f fi,s dm.
The sequence [4Jli is weakly C’auch!- in Ll/;Y and hence converges to some 4
in I,]/-\‘:
(4% g) --z ;z 1 If,!? dm.
Lifting ~b into f in Ll(m) gives the answer.
’ Second author, preprint

376 CNOI' 4ND DELBAEN
We already know that for a bounded sequence in H”, convergence in L”(m)
(1 ,( p < co) and convergence in Li/K-norm coincide. Moreover. WC have:
COROLLARY 4.2. If the conditions of the theorem in Section 2 hold. then the
following are equivalent for a sequence {g,:,, and g in the unit ball of II’ :
(i) g, converges to g fey the Mackey topology T(H”, Ll/N), i.e., umformly
on weakly relatively compact subsets of LlllT;
(ii) / Tgn - Tg ~ conrierges to 0.
Proof. In fact, (i) 2 (ii) is true without this assumption: since the unit ball
of H1- is weak-star relatively compact, its image in L1/N is weakly relatively
compact, and the implication is trivial.
‘IO prove (ii) dz (i), consider a weakly compact set K in L’/l\:. By the theorem,
we can find, for each plsitive number t a positive number 6 such that if /l belongs
to H x,
imply
Jl ;, ‘-- 2 and ~ T/l i < 6
sup I($, h)] c 2t.
fbt-K
Since for 12 large, 12 72 ,y,i -- g satisfies these inequalities,
on K.
convergence is uniform
COROLLARY 4.3. If the conditions of the theorem in Section 2 hold, then I,‘,iN
has the Dunford-Pettis properti\j.
Proof. ‘Vl’e have to prove [9] that if a sequence in Nx con\-erges lveakly
to zero, then it converges to zero uniformly on weakly compact subsets of
L’/AT. If (h,},, is a sequence converging weakly to zero in H%, we can, with no
loss of generality, assume that this sequence belongs to the unit hall. I{!, (iii)
in Theorem 4.1, elements 4 of a weakly compact subset K of I,’ .\’ can be
uniformly approximated by (images of) elements g in a multiple of the unit
ball of H’. Therefore, for each E positive there is M large and for aI1 A in XI
there is some g in Hr such that:
Since the h,, converge weakly to zero, their Gelfand transforms h,, con\-erge
pointwise to zero on fl, and since

Lebesgue’s theorem implies that
A DLNFORD-PETTIS THEOREILI FOR L1//I' ' 377
tends to zero. \\:e conclude that for all d, in K:
and letting t tend to zero, one finds that lz,, tends to zero uniformly on K.
This corollary implies [9] that if
is a weakly compact mapping of L’lA’ into a separated and quasi-complete
locally convex space E, i.e., transforming bounded sets into weakly relative11
compact sets, then the image under 0 of a weak C’auchy sequence is a Cauch!
sequence in E, and the image under CD of a weak compact set is a compact
set in E.
THEOREM 4.2. If m is dominant the statements (i)-(iv) in Theorem 4.1 nre
equivalent with:
(ii)’ For each positive number t, there is n posit&e number 6 such thnt, for.f itt
the unit bull of A with
we haw
~ Tf II :) 8
(iii’) Fos each positke number E, there is a natural number M such that -fou
each d in K, there is a R in A with
Proof. By the Ahern-Sarason theorem [7, p. 1521, the unit ball of .? is
dense in the unit ball of H” for the norm topology of I,’ (or L’/Y).
REFEREKES
I. P. .IHEIW AND D. SARASON, On some hypodirichlet algebras of analytic functions,
A4m~. /. Mczth. 89 (I 967), 932-94 I.
2. E. A~r.41~, Sur un thCorPmc de Mooney relatif aus fonctions analytiqurs born&s,
Poc(fic /. Math. 49 (1973), 31 l-3 14.

378 CNOP AND DELBAEN
3. I

,omivnr. OF FLW~~IONAI. ANALYSIS 24, 379-396 (1977)
Operators Arising from Representations of
Nilpotent Lie Groups*
LARRY BAGCETT
Department of Mathematics, University qf Colorcrdo, Boulder, Colorado 80302
Conmuniuted by thr byditon
Received November 5, 1974; revised November 25, 1975
‘rhc following two results arc obtained for an irreducible multiplier rcprescnta-
tion T of a connected nilpotent Lie group. First, Tr is a Hilbert-Schmidt
operator if f is square-integrable with compact support. Second, Tf is of trace
class if f has derivatives with sufficiently many moments. An application is made
of the latter result to show that T, can be of trace class even when f is not
continuous.
I. INTRODIJCTION
Suppose G is a locally compact group and that T is an irreducible representa-
tion of G. Let us consider the operators [T,] defined by Jf(,x)T, dx whenever
this integral makes sense. It is a mathematically fascinating question as to how
the operator-theoretic properties of the T,‘s are related to the function-theoretic
properties of thef’s. For which functionsfis the operator Tf a compact operator,
Hilbert-Schmidt, of trace class, a projection, etc . ? Not only is this relationship
a kind of mathematical intrigue, but in fact such knowledge has been useful
in classifying groups, analyzing representations, and computing quantities
important to the theory, e.g., Plancherel measure. Indeed questions of this
sort must be the first to be posed once the connection between representations
of a group and representations of its group algebra has been noted.
If case T is unitary, it is called CCR if T, is a compact operator wheneverf is
integrable. Since the mappingf + Tf is continuous with respect to the L1 norm,
we see that T is CCR whenever T, is compact for every f in a dense subset of
Ll(G), c.g., continuous functions with compact support, differentiable functions,
etc. Connected semisimple Lie groups, connected nilpotent Lie groups, and
motion groups, (semidirect product of a vector group with a compact group),
all have the property that each of their irreducible unitary representations is
* This research WV-IS supported in part by the National Science Foundation under
Grant G.P. 28697.
179
Copyright IC 1977 by .\cadenGc Press, Inc.
.\I1 rights of reproduction in any form reserved. ISSN 0022-1236

380 LARRY BAGGETT
CCR, and this undoubtedly is one of the reasons these groups are so much
better understood than, for instance, discrete groups and solvable groups.
Somewhat more subtle is the question of which functionsf map to Hilbert-
Schmidt operators. In [I] it is shown that this is the case for every L* function
if and only if T is finite dimensional, assuming still that 7’ is irreducible and
unitary. On the other hand, Harish-Chandra [3] has shown that T, is Hilbertt
Schmidt when G is connected and semisimple, T is irreducible and unitary,
and f is square-integrable with compact support. As a special case of the results
in [7] we see that Tf is Hilbert-Schmidt when G is a motion group, Tisirreducible
and unitary, and f is continuous with compact support. It also appears that
Schochetman’s same proof extends to functions which are square-integrable
with compact support. Some time ago the author observed that the same result
is true when T is an irreducible type I multiplier representation of an abelian
group. See Theorem 2.4 below. MTe shall also prove, in Theorem 2.3 below,that
Tf is Hilbert-Schmidt whenever G is a connected nilpotent Lie group, 7’ an
irreducible multiplier representation of G, and f is square-integrable with
compact support. Actually, even if T is a unitary representation, in the inductive
step of the proof given here one is confronted with a multiplier representation
so that it is simplest to state the result at the outset for multiplier representations.
Of course a multiplier representation of a nilpotent group corresponds, in the
usual way, to an ordinary representation of another nilpotent group, so that
this apparently more general statement is really no improvement. In our stud!
of the relation between functions and trace class operators (see Section 4), there
does seem to be an essential difference between the unitary result and the
multiplier result.
Although from the above discussion the square-integrable functions with
compact support seem to form an appropriate class of functions relative to
Hilbert-Schmidt operators, we mention two related facts. Theorem 4.2 of [5]
shows that certain representations of Lie groups having “Large” compact
subgroups map sufficientlv smooth functions to Hilbert-Schmidt operators.
Also, square-integrable representations map every L” function to a Hilbert-
Schmidt operator. See, for example, [6].
hIore delicate yet are the operators of finite trace. The trace norm of a bounded
operator is a bit tricky to compute or even to estimate, and it is not at all con-
tinuous with respect to any of the usual operator topologies. Which functionsJ
map, under a representation T, to operators of finite trace (Trace class operators),
is of great interest since if this class is plentiful then a linear functionalf -+ sp(T,)
exists, and this functional often plays a role similar to that of the character of a
finite group. One of the major results in semisimple theory is that T, is of
trace class whenever T is irreducible and unitary and f is infinitely differentiable
with compact support. The same result is valid for irreducible unitary rc-
presentations of connected nilpotent Lie groups. In fact somewhat more is
true in that case.

REPRESENTATIONS OF NILPOTENT LIE GROUPS 381
1 .l. ‘THEOREM (Kirillov). Let G be a connected nilpotent Lie group and 7’
an irreducible unitary representation of G. Then there exists an element II of the
left emeloping algebra a(G) of G so that the operator Tf is of trace class, and
/ 7; ~ ‘,’ :-; I DF 1~1 , whenever f is su@cient!v smooth so that T,, = T,T, .
(H ere “sufficiently smooth” means that the differential operator I1 can be
passed through the integral sign in ef(x)T3. dx. This is justified, for example,
if Dlfis integrable for sufficiently many elements 1)’ of@(G).)
From this theorem we see that “compact support” is by no means a requirement
in order that T, be of trace class. Rather it is some kind of integrability
condition on the derivatives off that is essential. In Section 4 we shall prove a
result for multiplier representations which is analagous to Theorem 1. I. As a
consequence of that result we shall discover that the “differentiability” condition
in Theorem I. 1 is also by no means necessary. In fact there exist discontinuous
functions which nevertheless map, under an-irreducible unitary representation.
to operators of finite trace. This is a consequence of our theorem (Theorem 4. l),
and the theorem itself appears more restrictive than the one above. M’hat real11
is pro\-cd there is the following:
'I‘HEORE.\I. I,et ‘1’ be an irreducible multiplier representation of n connecferl
nilpotent Lie ,group G. Then there exist a jnite number III,..., IY of elements of
/Z(G) and a corresponding number PI,..., p’ of polynomial functions on G such that
T, is of trace class whenever p”Df is integrable fey each i. Further,
‘1’0 carry out our proof we must require, in addition to the integrabiiity of
certain derivatives off, the existence of certain “Moments” of these derivatives.
It is onl!- after having proved this theorem that we find we can drop the dif-
ferentiability condition in Theorem 1.1 as a necessary assumption.
\Vhether Theorem I .I holds as it stands for multiplier representations is still
unclear. The author has tried hard to discover a counterexample without success.
There arc reasons why one would expect 1 .I not to hold for multiplier re-
presentations. In the first place, since the multiplier might not even be continuous
let alone differentiable, there might not be any differentiable vectors at all and
therefore no operator T, . In the second place, even when the multiplier is
analytic and all the operators T, are densely defined, it is still not true in general
that T,,T, T,, . These two facts are central to Kiriilov’s proof. There is
also snme reason to feel that 1 .l would hold for multiplier representations.
Indeed one should be able to construct a proof by passing up to the group
extension defined by the multiplier, applying 1 .l to that group, and then,
“projecting” back down. The proof given here is based on this idea, but the

382 LARRY BAGGETT
“projecting” back down is not so simple, and the polynomial coefficients seem
inevitably to enter in. The following example may make the complexities of this
situation more apparent.
1.2. EXAMPLE. Let g be an analytic, nonplolynomial, function of a real
variable, and define a function b from R2 into the circle group L by b(q, p) &“Q(~‘)
Define a multiplier 6 on R* x R” by 6((q, , pl), (q2 , pJ) 1 ei(‘~~~~)b(qlpl)b(q2 , p2)/
b(g, -1. 2% , p, + p2). The essentially unique s-representation T of R2 is defined
on L2(R) by (TQ,,)(P), F) =- 41, P) SR ei*” &I -+- m) q(m) dm. Let G, denote the
group extension of R* by L defined by 6, and let T, be the unitary representation
of G, defined by TlcA,Yj XT, . Let a(/\) be a smooth function on I, such that
J-L a(h)X dA _ 1, and for any function f 9n R’ put fr(A, g) = a(A)f(g). ‘l’hcn the
operators T, and TIC, ) are equal.
Now according to ?heorem I. 1, the operator T/ will be of trace class providing
sufficiently many derivatives of fr are integrable over G, . However this inte-
grability of derivatives offi can translate into something quite different about f.
For example, differentiating along the one-parameter subgroup (1, 0, t) in G,
we have:
(44 Vl@T 4, P)(l3 0, w3
== (44 UkWq, P), (0, 41, 4, (P + W3
~~ (44 me (Ir~rr(il)itY(ol-(l~-~f)~~~))
? 97 (P + mP)
(d/dt) [u(Xeit(~(“)-!‘(v))).f(rl, (p + t))](O)
~‘(3 %z(O) - R(Q))f(% P) + 44 4f(% P)*
Presumably higher order derivatives offi would lead to even more complicated
differential operators acting onf. It seems clear that we would have to consider
differential operators with analytic, nonpolynomial, coefficients, and that we
would be forced to assume the integrability of such derivatives off. Therefore
the fact that we can get away with polynomial coefficients is already interesting.
If we integrate out the A, in order to project back down to G, it appears that
the only way we can bound the trace norm of T, , that is the trace norm of TiCi ) ,
1
is with a sum of L’ norms of moments of derivatives off.
The integrabilitv of derivatives of a function, the existence of moments of
those derivatives, and for that matter the very definition of “polynomial function”
on a group, is crucially dependent on the coordinate system being used. In
Section 3 we shall investigate this question in some detail.
It is possible to prove Theorem 4.1 so that the operators Dl,..., IY and the
polynomial functions pr,..., p” are independent of the multiplier, i.e., the same
operators and polynomials work for cohomologous representations. The proof
presented here does not show this, but the other argument, although elementary
in some sense, is long and does not seem to merit the space.

IIEPRESENTATIOKS OF NILPOTENT LIE GHOI:Ps 383
2. HILBERT-SCHMIDT OPERATORS
‘I’he purpose of this section is to prore that Tf is a HilbertMchmidt operator
whenever 7’ is an irreducible unitary representation of a connected nilpotent
Lie group and f is a square-integrable function with compact support. The
proof is b!; induction on the dimension of the group, and in the inductive step
wc arc confronted with a multiplier representation instead of an ordinar\- one
and an apparently unavoidable “unipotent” automorphism. \Ye must state our
theorem then in a somewhat more general-souding form. The technicalities of
the proof involve the structure of nilpotent groups, in particular the constructions
introduced by Kirillov in [4] for the case of a one-dimensional center. \Ye recall
this structure below. ‘l%roughout the section, 7~ will denote the projection of one
group onto a quotient group. The context should make clear which groups arc
imolved.
2.1. Let G be a simply connected nilpotent Lit group of dimension II,
and let C/i denote its Lie algebra. Denote by Z the center of G and h\. Y its
Lit algebra.
2.1.1. There esists a measure-preserving cross-section p of G/Z into G,
and for an\- such cross-section we have
for propcrlv normalized Haar measures. (Indeed p can be taken to he a dif-
fec,morphisk)
2. I .2. Sow let Z be one-dimensional. Then, according to [4], thcrc exist
r
nonzcro elements s, y, and s in 9 such that u” generates L, s does not helong
to the commutator subalgebra Yl of %, and the bracket [s, j-1 of A and 1’ is z.
Let CqO bc the annihilator of y in Cg. Then rg,, is an idcal in CC? of dimension I, I.
It is important to note that whenever elements x, y, z esist satisfying the aho~e
bracket, and for which the annihilator of 1’ is of codimension I, the following
constructions are valid. We write Jr, A-, G, for the closed subgroups of G having
1,ie algebras [y], [xl, and Y,, , respectively. Tl le product I’Z is a subgroup of G
which is contained in the center of G,, , and we denote b\- K the quotient group
G,,,‘17Z. It m-ill be to k- that we appl!- our induct& hypotheses.
2. I .3. Let G bc a simply connected nilpotent Lie group, and let .Y, l..., ,v,~
be a basis of its Lie algebra. Then th e mapping (fl ,..., t,,) + n:‘-, exp(t,.v,) is a
measure-preserving difeomorphism of K” onto G.
2.1.4. Let G bc as in 2.1.3. Let the basis [x1] ha\-e the propcrtics that
YI 2. x’:! .t’, sn ,x, and the span of .x, ,..., .Y,,- 1 is Y,, . Then:
(i) ‘i’he mapping n7-i exp(t,x,) -t ni:i exp(t, &(a,)) is a ciifeo-
morphism whose inverse p is a measure-preserving cross-section of K into G.

384 LARRY HA(;(;W’I
(ii) If we write, as wc can by 2.1.3 and (i) abow, elements I? of G
uniquely as ,y eup(tz) exp(s?;) p(k) exp(yx). or in shorthand ,y I,- .

I

386 LARRY UAGGETI
where ZOO is the automorphism
Therefore
of G/K induced bv W. Of course + is unipotent.
. .
T(,. ((.I
I .i
’ GiN N
.f(np”(Zu~(y))) x( p’(n(n(w’, ?I)) 1)) dn T2, li>,
’ I‘ (f2 zL&y) I..&, N’?~.
G ,’ N
There MY) J”.v f(@‘(mvN x(P’@)) ~TZ x0 ’ ( n ( W. W,‘(F)))). Therefore 7’,, ,),.,
%,.U?) .
Now the support of JA is compact. Indeed it is contained in the set r(C).
Since the dimension of G/2%’ < n, we have from the inductive hypothesis that
there exists a constant c? , depending only on T, and z(C), i.c., depending onI!-
on T and C. such that
where c1 is the supremum, over all y in G/K, of the measure of the set of all 71
in ,V such that n@‘(y) belongs to C. This supremum clearly is bounded 1~~ the
diameter of the compact set r(C) and so depends onI>- on C. This establishes (iii)
of the theorem in this case.
Assume next that no such subgroup % of the center of G, exists. Let y be
a vector in the Lie algebra of G, such that yO =: dn(y) generates a one-parameter
subgroup of the center of G which is pointwise invariant under ZL’. Because we
are in this second case, y does not belong to the center of G, , and there must
exist an Y in the Lie algebra of G, such that [x, y] z a vector which generates
the subgroup R of G, . The annihilator !gO of y is of dimension n, so that m-e may
presume all of the constructions introducted in 2. I. Let sg -= dn(s). Finally let
p”’ be a measure-preserving cross-section of K into G. We make these definitions
in order to employ a different correspondence between functions on G and
functions on Gl . Iffis a function on G, put
i3(tz sy p(k) qx) a(t)f(sy,, p”‘(k) qs,,).

REPRESENTATIONS OF NILPOTENT LIE GROUPS 387
‘I’he operators Tf and Tltf ) are equal. (In this case G itself is simply connected.
and all the integration for$ulas of 2.1.4 hold.)
Let [vi] be an orthonormal basis for L?(R) and let [#.i] be an orthonormal
basis for the space H(S) of S. Define an orthonormal basis [z)~~] for the space
of Tl by zii(y) = q

388 LARRY RAGGETT
where zc,. is the unipotent automorphism of K induced by conjugation bv TX,
and where t(r, k) and s(r, k) are real numbers. Hence
f(q), . p’“(w’(k)) . w(qx,J) eP ds
x ~~1(~,Ir)(k)(s;,,,7(r.))(~j), '~",OU(.S) cfk I 4 drt
where m(,,,,.)(k) L e LL(rJ)eirs(ri’.i.). Recalling again that unipotent automorphisms
are measure-preserving, we have finally that
f(r,n*W)(k) L j-Rf(~y, . p”‘(k) 1 w(qx,)) cisr d.s mc,,,,(zu’-l(k)).
Now the support of f(r,q,fU) lies in the compact set Z-(C) which depends only
on C, and so by the inductive hypotheses there exists a constant c, depending
only on n(C) and S’ and consequently only on C and T, such that

REPRESENTATIONS OF SII.PoTEST LIE C;Ro17PS 389
. a
T(f.d /is .. f (rsqsL”)(k)/2 dk dr dq
-R I J R .r Klf
=-- (. jh lK iR / Jh.i(s?‘,, - p”‘(k) . w(q.yJ) e-“” ds if dv dk dq
. *
3m a’,, . p”‘(k) . zo(qs,,))‘” ds dk dq
.R.K 1 ,I R ‘f(-
-: 27x jR *i; .I, I f(S) ‘,) . p”‘(k) . q d~&,)),2 ds dk dq
by 2. I .3. This completes the proof of the theorem.
Essentially as a corollary to this theorem we have the following:
2.4. 'IhEOREM. Let G be an abelian locally compact group, 8 a t?pe I multipliel
on G x G, T an irreducible &representation of G, and C a compact subset of G.
Then there exists a constant c, depending only on T and C, such that iff is square-
integrable with support in C, then the operator T, is tiilhert-,Schmidt, and
Proof. As usual in this subject, one may immediately reduce to the case
when 6 is “totally skew”; see [2]. Th en G is a direct product G, >, G, where G,
is RPrr and G, is a finite group. Further, 1’ is the outer Kronecker product
T1 x T2 of irreducible multiplier representations T’ of G, and T2 of G, ,
and ‘I” is finite dimensional. Kow iff has support in C’, then T,
where O(G,) is the order of that finite group. Hence 7’, is a finite sum of operators
which are outer products of finite-dimensional operators and Hilbert-Schmidt
operators (by the last theorem), and so T, itself is Hilbert-Schmidt. Q.E.D.
Theorem 2.3 does imply that Tf is Hilbert-Schmidt whenever f is continuous
with compact support, so that, in the terminology of [7], connected nilpotent
Lie groups are H.S.-groups.
3. INTEGRABILITY OF DERIVATIVES
The question of whether a function is differentiable at a point on a manifold
is of course independent of the coordinate system at the point. Whether a
function is integrable on a Lie group is also independent of any coordinate

390 LARRY ISAGGETT
system, the Haar measure being determined by the underlying topological group.
However, if one wishes to investigate the global integrability of a derivative
of a function on a Lie group, then the coordinate system becomes a key factor.
More to the point, it is what we mean by a “Global Derivative” offthat depends
on the coordinate system. Consider the following two examples.
3.1. EXAMPLE. Let G be the group of all triples (t, 9, p) of real numbers
with multiplication defined by (tl , q1 , pl)(t, , qz , pJ _m (tl + t, ~I~ qrpl ,
q1 -1 q2 , p, $ pJ. This is a Lie group, and it is in fact the Heisenberg group.
Its underlying manifold is obviously R”, and one would assume that the first
order derivatives of a function f would be the functions (d/‘dt)f, (djdy)f, and
(d/dp)f. These certainly are the simplest differential operators to apply to f in
order to determine if it is differentiable at a point.
There are, however, some other first order differential operators which are
in fact more intrinsic to the group itself. These are the differential operators
obtained from the left invariant vector fields on G, i.e., by differentiating along
one-parameter subgroups of G. For this group, these operators will be linear
combinations of the following three. Differentiating along (t, 0, 0), we obtain
djdt. Along the subgroup (0, q, 0) we obtain the operator dldq I- pdjdt. And
along (0, 0, p), we obtain didp.
The point is that if we ask whether the first order derivatives of a functionf
are integrable, then we must be sure to specify which first order derivatives
we mean. In this example the integrability of the first order derivatives arising
from the vector fields would imply, in addition to the integrability of certain
ordinary derivatives off, the existence of some kind of “moment” with respect
to the variable p.
3.2. EXAMPLE. The next group is nothing but the simply connected covering
group of the group in Example 1.2. Thus let G be the group of triples (t, 9, p)
of real numbers with multiplication defined by
(t1 9 41 3 Pl)(h Y 42 3 PJ = (5 + f, i- q2Pl 4 P&3,) + p2gk2)
- (PI + P2)&, + 42), 41 + q2 > Pl + PA
where g is an analytic but nonpolynomial function of a real variable. This is
again a Lie group, and it is in fact isomorphic to the previous example. The
underlying manifold is again obviously R”, and again d/dt, d/dq, and didp seem
to be the appropriate first order differential operators. This time, however,
differentiating along the subgroup (0, 0, p) we obtain the operator d/dp J- djdt
[g(O) - g(q)]. Requiring that a function be integrable when acted upon by this
operator is a much more stringent restriction than the existence of a moment.
It should be clear too that other isomorphic copies of the Heisenberg group

REPRESENTATIONS OF NILPOTENT LIE GROUPS 391
would produce first order operators with discontinuous coefficients. The picture
is muddy, and we shall try to clear up at least a part of it.
Throughout the rest of this section we shall be discussing a connected nilpotent
Lie group G. The results stated here may be well known to some experts, but
we present them because the geometric handle obtained is crucial to the proof
of Theorenl 4. I. \Ve give no proofs in this section, the arguments being nontrivial
but routine consequences of nilpotency.
Let G be a connected nilpotent Lie group of dimension X, and let K be a
maximal torus in G. Then it is known that K is a maximum torus and that it is
contained in the center of G.
3.3. 'I'I~EOREM. Let [xi] be a basis for the Lie algebsa !5? of G with the follou.iq
two properties:
(i) .vl ,..., st span the Lie algebra X of K, and A- is the direct product of the
closed we-parumetev subgroups exp(tx,), for 1 S< i C< J.
(ii) [x,] is consistent with the central descending series, i.e., the Lie bracket
of x, zcith the span qf x1 ,..., xi -1 is contained in the span of x1 ,..., .x- 2 .
Then:
(A) Each element g of G can be written uniquely as g -- nyzl exp(t,.xi),
where -.- I, ._ t, -c: l1 for 1 < i < J and ti real for i > J. The mapping which
sends nil, eup(tp~J to the n-tuple (h, ,..., h,, tJrI ,..., t,J, (h, := e7(“tr/zi)), thus
defined is a diffeomorphism of G onto LJ x R”mJ. (L denotes the circle group.)
1Ve denote this global coordinatization of G by pLr,,
(B) If [!(I is another basis of !q which satisfies i and ii, then p)[,,,] [y~[&l
has polynomial component functions and a constant Jacobian determinant.
(c’) Hz=, exp(t?y,) nFS1 exp(tf’xj) -= nz=, exp(p,.x,.), w-here pi,, t,t. -1
f, ’ --I p!,‘, and p, ’ is a polynomial in the variables (t,,.+l ,..,, t, , t,, , 1 ,..., t,,‘).
(D) If f is an integrable function on G, then
where do, represents properly normalized Haar measure on the one-parameter
group rq$f.r,).
3.4. DEFINITION. Any basis [x~] of the Lie algebra Y of G which satisfies
conditions (i) and (ii) of the above theorem is called an exponential coordinate
system for G. A function p on G is a polynomial function if it is a polynomial when
expressed in an exponential coordinate system. By part (B) of the above theorem,
it follows that p is a polynomial in anr other exponential coordinate svstem,
although its degree could change.

392 LARRY BAGGETT
3.5. THEOREM. I)enote by %(G) the left erweloping algebra (!f G. i.c.. the
algebra (associative) generated by the linear difJeerentia1 operators urisin;? fronr the
left invariant cector jields on G. Let [xi] be an exponential coordinate sj*stem jOr C,
and denote by b(G) the algebra of linear d$ferential operators generatrtl II>, the
jk~t order derizatkes did+ with respect to this coordinate system. Then:
(i) L(G) is always abelian, but P(G) is abelian if and only if G is commutatke
(ii) For each element I) in @(G) these exist elements El,..., I:” irl r’(G) and
polynomial functions PI,..., p’ on G such that II x;:, piI?.
(iii) For each element E in CS (G) there exist elements IY,..., IP irl J//(G) and
polynomial functions p’,..., p” on G such that E c;‘_, p’n’.
(iv) If @,(G) is the algebra generated by drfi erential operators of the jkrm
pD for p a polynomial function on G and D an element of o//(G), and if 15 ,,(G) is the
algebra generated bJ- the operators of the fom pi? for p a po~worrrirrl fitru-tion
and E an element of d(G), tlzen -i//,,(G) 6 ,,(G).
This is definitely a theorem about nilpotent groups. \Ve have that c(G)
is generated by the vector fields [si], and K(G) is generated by the operators
d/d.v, . Now (ii) and (iii) follow by induction together with Theorem 3.?* part (C).
Part (iv) follows from (ii) and (iii), and (i) is a fact true in grc‘at generality.
Let us have another look at the examples given in the beginning of this s&on.
In Example 3.1 the evident coordinatization of G by R” is an csponential
coordinatization, i.e., (t, 4,~) (t, 0, O)(O, q, O)(O, 0, p), and the cencrators for
these three one-parameter subgroups form an exponential coordinate system.
From our calculations we see that the differential operators arising from the\-cctor
fields are, with respect to the exponential coordinate system, lincai- diiferrntial
operators with polvnomial coefficients, and we can easily see how to rc’co\-cr
the first order operators d/dt, d/dq, and djdp f rom the vector fields and polynomial
coefficients.
In Example 3.2 the evident coordinatization of G bp R3 is not an c;ponential
coordinatization, for (t, 0, O)(O, q, O)(O, 0, p) (t -- p&(O) - ,g(f)‘i. y, p) and
not (t, q, p) as desired. Our calculations here showed that an element of ‘)/(G),
i.e., a differential operator arising from a left invariant vector field, could not be
expressed in terms of these coordinates merely with polynomial coefficients. The
point seems to be that this “evident” coordinate system is simply the n-ronr one
for this group. Unfortunately the novice may not notice this.
How then shall we define when a functionf has integrable derivati\,cs.

KWKESE~TATIOSS OF NILPOTENT LIE (:I
IT
I
‘(.\.td ,\7:, , and let a(h) be an infinitely differentiable function onl, for which
J,, u(h)h d.\ I. Let [x1 )..., .T,$ , ] be an exponential coordinate system for G,
(see Definition 3.4.) Assume without loss of geneality that x1 spans the Lie
algebra ofL. If pi denotes projection of G, onto G, then the elements [&(x, ,...,
c/T(.Y,, ,)] form
on G, put
an exponential coordinate system for G. ITor an!- function f
Then 7; and 7; (i ) are equal and fi is as smooth as is f.
Let D hc the Llement of the enveloping algebra ?/(G,) of G, guaranteed b!
Theorem I. 1. Let G(G,) denote the algebra of differential operators generated b!
the first order operators d/d.\-, , and choose elements El,..., I?” in d (G1) and polyno-
mial functions PI,..., p” on G, such that I) =- xi= 1 piEi (Theorem 3.5). Now

394 LARRY RAGGETT
assume that f is sufficiently differentiable to carry out the following
computations.
and therefore II T, l’\i, is bounded by a finite linear combination of terms of the
form
where c is a constant, p’ is a polynomial function on G, and E’ is an element of
the algebra Q(G) generated by the differential operators d/d(dn(s,)). \Ve have
shown then that j/ T, /‘S)s is bounded by a finite sum of terms of the form E,,.f l,1 ,
where each E, belongs to G,(G). By Th eorem 3.5, part (iv), each E,, must belong
to q’,(G), and the theorem is completeI!, proved.
R,Iuch more easily derivable directly fr-on1 Theorem I. I is:

REPRESENTATIONS OF NILPOTENT LIE GROI-I'S 395
4.2. COROLLARY. Iff is infinitely dijfeyentiable with compact support, and if 7
is an irreducible multiplier representation of a connected nilpotent Lie group, the?1
T, is of trace class.
4.3. IkAiMPLE . Let II’ now be an irreducible unitary representation of a
connected nilpotent Lie group. Let f be infinitely differentiable with compact
support on G and let b be a Bore1 function of G into the circle group I, for which
b(e) I. Then T,, is of trace class since T,, -- bT, and bT is an irreducible
multiplier representation of G. Hence there are discontinuous functions, for
instance bf, which nevertheless map to trace class operators under the repre-
sentation T.
1T,7e conclude this paper with an example from more classical analysis. It is
largely an example of how slippery the trace is.
Let 6 be the multiplier on R2 x R2 defined by S((q, , pi), (g’&) = ei(u2”1).
The essentially unique irreducible a-representation T of R” is defined onL’(R) by
(T(4,p)(p), v,) = J, ei”“‘dP + ml F(m) dm.
Iffis any integrable function on H”, then the operator Tf is given by a kernel k;
defined by
k;(y~, p) =L jR.f(q, p - m) 2”“ 4.
Of Course these kernels are only defined up to sets of Lebesgue measure zero
in the plane.
Now since every operator of finite trace is the product of two HilberttSchmidt
operators, we know that if f is an integrable function on R’ for which T, is of
trace class, then the kernel k;(nf, p) m: JR k;(m, TZ) K,(n, p) tin, where K, and K,
are Hilbert-Schmidt kernels, i.e., square-integrable over the whole plane.
4gain this last equality is only up to sets of measure zero in the plane. However,
if k; is continuous in both variables, one might expect that the kernels Ki and Kz
could bc chosen so that this equality holds everywhere. Howcler, this is not
the case.
4.4. I'HOP~SITI~N. Suppose f(q, p) ::= g(q) h(p) zcith h(0)

396 I.ARRY HAGGET’
in both variables. Assume, by way of contradiction, that Hilbert-Schmidt
kernels KI and K2 do exist so that K,(wz, p) =L JR KI(m, TZ) Kz(n, p) dn for all or
and p. We would then hare that
which is finite. Hence the function ~77 --f K,(m, m) is integrable. But this is
the function Iz(0) j(nz), and so g would be integrable, which implies that g would
be continuous. Q.E.D.
1. I,. BAGGETT, Hilhert-Schmidt representations of groups, Proc. AWM. il/lnth. Sot. 21
(1966), 502-506.
?. L. BACCETT AIW X. KLEPPNER, Multiplier representations of rlbelian groups, J.
Functional Analysis 14 (I 973), 299-324.
3. HARISH-CIIANDRA, Representations of a semisimple lie group. III, Trtrns. Amer.
IVIuth. Sot. (1954), 234-253.
4. A. KIRILLOV, Unitary representations of nilpotent lie groups, C’spehi AW\/lrrt. ~l’urrk 16
(1962), 57-110 (Russian).
5. E. NELSON AND R. STEINSPHING, Representations of elyptic elements in an enveloping
algebra, Anw. J. AIJath. (1959), 547-560.
6. M. RIEFFEL, Square-integrable representations of Hilhert algebras, J. Func~onal
Analysis 3 (I 969), 265-300.
7. I. SCHOCIIETMAN, Compact and Hilbert-Schmidt induced representations, Duke Altrth.
1. 41 (1974), 89-107.

JOURNAL OF FUNCTIONAL ANALYSIS 24, 397-425 (1977)
Topological Aspects of Algebras of Unbounded Operators
D. ARNAL AND J. P. JCRZAK
Physique-Matf&zatique, Facultd des Sciences Mirande, UniversitC de Dijon,
ZlOOO-Dijon, France
Communicated by the Editors
Received December 16, 1975
The utilization of OF-spaces of A. Grothendieck leads to natural topologies
on *-algebras of unbounded operators. In this way, the origin of pathologies is
clarified, and natural classes of *-algebras with good behavior are introduced.
Examples and particular algebras of countable algebraic dimension are studied
in the second part of the paper.
INTRODUCTION
Infinite-dimensional representations of a Lie group G give rise (by differentia-
tion of unitary representations of G) to algebras of unbounded operators: more
precisely, the enveloping algebra (viewed in the representation) consists of
operators generally unbounded defined on the Girding domain, dense invariant
domain. Since, from the physical point of view, observables have to be chosen
among essentially self-adjoint operators of this algebra [21], it seems reasonable,
for a better analysis of the structure, to define on *-algebras, a topology, which
must be a generalization of the classical situation of @*-algebras. Taking in
account that Van Neumann algehras are @*-algehras dual of some Ranach
space, called the predual, we are lead for our study to introduce &F-spaces
(which arc dual of FrCchct spaces). These aspects accentuate remarks of [13].
The general plan of this paper is to examine topologies associated with
e-algebras of unbounded operators and then to study in details natural classes
of +-algebras (as shown by examples).
Section 1 is devoted to topological considerations: the comparison of the
&topology (topologically satisfying) with the p-topology (algebraically interesting)
is of great importance.
Section 2 is devoted to certain algebras of countable algebraic dimension.
Examples and counterexamples are then investigated, showing the interest
of this topological approach.
We recall briefly the definitions and notations of [ 131. Algebras we will
consider in the following will be algebras of operators, acting in some Hilbert
397

398 ARNAL AND JURZAK
space. We call --algebra, in a Hilbert space 11 an involutive algebra L/Z of operators,
not necessarily bounded, all defined on a domain % dense in lj, with the following
properties:
1” For A t a, the adjoint operator =1” satisfies Dom .AX I) ‘k and
A9CC, A”PC9;
2” Forilr~,,BE,fE~,wehave
(A L- B)f == Af -+- Bf, @B)(f) p= A(Bf).
Moreover, 1 E CY.
Condition d = A* in the algebra Q? means that the operator is symmetric.
An operator d E 0 is called positive (written A ‘> 0 or A E 6P) iff
(Ax, x) 3 0 for all s E 9.
The relation < is clearly an order relation on 4 and a linear map @: CT-+ ~8
from a *-algebra cd (acting in a Hilbert space h) into a *-algebra .B (acting in a
Hilbert space &) is called positive iff A 3 0 in CZ implies @(A) :;- 0 in 3.
In this paper, for an operator A E 67, A denotes the closure of ,4 (the domain
of A being 9).
The p-Topology
1. TOPOLOGICAL CONCEPTS
On each v-algebra @ (acting in some Hilbert space h, with 9 dense domain
invariant under operators of a), we define the p-topology, as follows: Given
A > 0, we introduce, for T E LZ?, the quantity
This defines the normed linear space
9l, = (T E fl; p,,(T) < +a)
with norm 1, iNA = pa / 91, , the canonical injection iA : ‘9, --f GZ and, on a,
the final locally convex topology, relative to the applications iA (for all A E 0P).
We note that Uata+ *A = xaEa+ ‘3, = a; moreover, the relation 0 :> _4 -a B
implies that the injection i,,, : (‘S,,, , 1 ala) ---f (‘3, , 11 liB) is continuous, with
norm smaller than I. Also, if ‘S2,, is strictly included in ‘3, , i,,B(%n,) cannot be a
dense subspace of (%, , I, / n), because the open ball B + w, with w {TE%, ;
p,(T) < 3) does not intersect 5X7, . The p-topology is well defined by choosing

TOPOLOGI('AI. ASPECTS 399
maps i,, (j t J directed set, AJi iS. 0) such that LJ! : U,,tJ ‘Jl,, [3], and is separated
since, for every 7’ ;’ 0 E G?, there exists .I” t 9 such that (TX, .Y) rf 0 [lo, 13. 1381.
For a continuous linear map .f : CT --, C, we put, for -4 f Q
i$fi!#,,, supif( YE 0 with p,(T) S. I].
PuoposI’rro~ I. I. Let LX U,t, 91,.,, (I directed. --fi 1%: 0) he a i -algeha, md
.f : tt F C be 0 lineor fem.
.f is positive if ad only if f A, ,,, .f (Ai) for all i E I.
Proof. \\‘e denote by GT, the real linear space of symmetric elements of LT.
Then, clearly, 0’ = IY~ @i C& , and from the continuity of T+ 1’^, the
above direct sum is a topological direct sum. Hence a linear form is continuous
if and only if its restriction to C& is continuous.
\4’e remark that, for a hermitian linear form g on 67 (this means that x(7’“)
g(T), or equivalently x(T) E R for symmetric 7’) the following equalitv is true
g ,(,,,i, mu: sup{g( T); T t CZQ with f,,,(T) :.< I 1.
Because, for every E :> 0, there exists T E 67, with p,,(T) 0. Hence
i g(h(T* + T)), = 3 /g(T) + g(T*)l = g(T) ;;> 1 g !IA,,, - E
and, from fA(g( T + T”)) S’ I follows our equality. ‘The proposition is therefore
a consequence of [I 21.
DEFINITION 1. I. A +-algebra CZ is said to be countably dominated, if? there
exists an increasing sequence of subspaces ‘San (Art > 0. n E N) such that
One can check that algebras of countable algebraic dimension are countably
dominated.
DEFINITION 1.2. A +-algebra CT is said to be p-closed, iff there exists a
decomposition LT m- (JiE, ‘LX,, (with A, > 0, I directed) such that, for ever);
i E 1, ‘SAi is a Banach space.
In this paper, we restrict ourselves essentially to countably dominated v-
algebras. Endowed with the p-topology, they are bornological quasi-barrelc-I

400 ARNAL AND JURZAK
SF-spaces [lo]. We refer to [ 131 f or complements. For a p-closed +-algebra CT,
the bilinear map
is continuous (since 02 is a barreled 9.F-space; and see [IO, p. 1681); moreover,
for every A E 6V , %,, is a Banach space. Indeed, if (Tl,)ptN is a Cauchy sequence
in WA , II MI
I((T,, - T,)x, x>I T-. c(p, q)(Ax, 4
Hence. for a certain i E I
((T, - T&, x), < Me(p, q) p&x, x) (for some M < S-m).
By completeness of the Banach space $XA, , there exists an operator T: Y -F 3’
such that:
I((T,, - T)x, x)1 e:: l (p)(A~x, x) with F-2 ~(9) 0.
Since, for every x E 22, (TX, X) : lim,,, (T,x, x), by letting p --f nj in the
first majoration, we deduce that (sI, , 1~ ii.4 ) is complete. We recall that, for
x E ~2, y E h, the linear form on LZ?
T -+ (Tx, y)
is denoted w~,~ . If y E 9, the linear form w~,~ is continuous on LY; this follows
from polarization equality and from [I 31. If y E h, the linear form w~,~ is a
simple limit of continuous linear form wz I, (yFL E 9), and is not necessarily
continuous (except, for example, if G? is p-clo”,ed).
PROPOSITION 1.2. Let (72 be a countably dominated *-algebra, and S C 67 a
subset of rY. The following assertions are equivalent:
(i) S is bounded for the p-topology.
(ii) There exists A 3 0 such that
l(Tx, x): < (Ax, X) fey all .2: E 2 and all T E S.
Proof. The implication (ii) 3 (i) is immediate. For the converse, we first
note that, for A, > 0, A, > 0,
%A,
+- ‘xAgc flA1+A2 .
By Proposition 5 of [IO, p. 1711, th ere exists an operator A .> 0 such that
s c c,

where
TOPOLOGICAL ASPECTS 401
1,’ standing for the closure of the set 5, relative to the p-topology. Hence,
for x E Y,
UC Gvw~ TN (r ==- (Ax, x))
(B(0, Y) being the closed ball of the complex plane, of radius I), and by [ 131,
which achieves the proof.
The X-topology
Let CT be a v-algebra. We introduce, for A E QZ, the normed space 2R,4
with norm
and on 67 the final locally convex topology relative to the canonical injection j, :
!I.N, + 0’. We note that the spaces 912, constitute a directed set, since
As previously, this topology (called the h-topology) is well defined by choosing
operators iii E GT such that QT = Uit, 9.RA, Let us note that the X-topology
is invariant under isomorphisms (i.e., h,(T) = h,(,,(@(T))). More precisely, if
CD: CT? ---z .?I is a homomorphism (positive multiplicative map), then
implies
and by [I 31
that is
‘1 T*x 11 : c(n) 11 Ax :I for all x E 2, lim ~(72) 0,
n *I
(@(T,*T,)x, x) < 2(n) (@(A*&. x),

402 ARNAL AN;D JUKZAK
Also, from the equivalence of assertions (a) and (h) (proof left to the reader)
(a) CZ can be written IT : = u,Eh !IJl,i (‘+li E CT),
(h) W can be written /I UIEN !I&+, (B, 1 0, B, t (;%),
it is not ambiguous for a +-algebra (7 to be countable- dominated.
DEFINITION I .3. A “-algebra ci’ is said to be h-closed, iff there exists a
decomposition 67 (JtE, 9Ji,d6 such that, for ever!. i t f, 9J1, is a Banach space.
It follows that, for every .I E L7, !13{,., is a Hanach space. ’
THEOREM I .I. I,et FZ he a p-closed t-algehru, countabl,’ dominated. Then,
(2 is h-closed. und the h-topology a

TOPOLOGICAL ASPECTS 403
PROPOSITION 1.3. Let 6!? = &,91)31,41 be a *-algebra, with domain 9:. c/ is
f7-dosed if and only if 5’ =:- nii, Dom(A,).
Proof. By hypothesis, for T E 67, there exists M < j-m, i E I such that
TX I

404 ARNAL AND JURZAK
closed domain B. For a closable operator S de$ned on 9, such that X9 C 8, we
canJind an integer n E N (resp. m E N) such that
Proof. It follows from hypothesis that 53 endowed with its natural topology
is a FrCchet space. It is now easy to check that the graph of the map S: 9 - 9
is closed. The closed graph ensures that S is continuous, which implies, for
some n fz N,
Then, by routine majorations, we deduce
l(Sx, x)1 -5 M(&x, x) forsome mEN,andallxE9.
PROPOSITION 1.5. Let G! be a countably dominated *-algebra, with closed
domain 9, and 99 a jr-algebra with the same domain 9 such that
Then, every positive linear form f on CJ? can be extended to a linear form fU positive
on 8.
Proof. By Lemma 1.1, GZ is a cofinal subset of %‘, and the proposition
is a consequence of Theorem 1.6.4 of [12, p. 241. In particular, if Z(9) denotes
the set of operators (generally unbounded) acting in the Hilbert space lj, which
satisfy
DomA =g A% c 2,
DomA*
A*g?cC,
we can choose in the last proposition g = 9(g), or ti = 02” (W being calculated
in the algebra P(9)).
PROPOSITION 1.6. Let OY be a countably dominated *-algebra, with closed
domain 9, and S be a subset of QZ. Then, the following statements are equivalent:
(i) S is bounded for the h-topology.
(ii) There exists A E Ol, such that, for every T E S
11 Txlj ,(;iAxl~ for all XE9.
(iii) S is bounded fov G!? endowed with topology dejined by seminorms
T+l’ TX‘ XE9.

TOPOLOGICAL ASPECTS 405
(iv) S is bounded for 6?! endowed with topology defined by seminorms
T+ j(Tx, y)’ .x: E 9, y E l).
Proof. It suffices to establish the implication (iv) * (ii) By the Banach-
Steinhaus theorem, it follows from (iv) that, for every x E 9
As in Lemma 1.1, we can apply the closed graph theorem to operators T of S,
viewed as a linear map from 9 endowed with its natural topology into the
Hilbert space 11. The preceding estimate shows that S is a simply bounded
subset of dip,(9, 6). Thus, the equicontinuity principle implies that the set
S C 5??(9, lj) is equicontinuous, which gives (ii).
COROLLARY 1 .I. Let G! be a countably dominated x-algebra, with closed domain
9, endowed with the X-topology. Then, assertions (i) and (ii) are equivalent:
(i) For every S E Ot, the map T---f ST is continuous.
(ii) The bilinear map (S, T) ---f ST is continuous.
Let us mention that the map T---f TS is always X-continuous, because the
relation 11 TX 11 < 1~ Ax ‘1 (for all x ES) implies ‘/ TSx 1; -5 ~1 ,4Xx Jo.
Proof. By virtue of [lo, p. 1621, we establish (ii) =- (i) by showing that
(S, T)- ST is hypocontinuous. We put n(S)T :- ST. Let S be a bounded
subset of 6Y; by the preceding result, one can find A E QJ such that for every T E S,
1~ TX / < li Ax for all . 2” E 9 .
The equicontinuity of maps n(T) (T E S) amounts to show that maps r(T):
YVJJ1, -+ ed are equicontinuous, for every BEG!. Since n(A) is continuous, the
unit ball of the normed space (9JIa 1 X,) is sent into a unit ball of some !I&
(C E a), which means that
// Rx ]j ,( I] Bx /I for all x E 9 implies 1: ARx 11 < /I Cx jj.
But I] TRx /I < 1: ARx 11 for all x E 9’. Thus, the unit ball of $9X, is sent by
operators r(T), T E S, into a unit ball of some 9Nc . This achieves our proof.
One can show that under the hypotheses of Corollary 1.1, the completion of
the normed spaces 93, (B E a) can be identified with operators S such that
Sg C 9. We thus obtain a bigger space @, which is an algebra (with natural
multiplication).
THEOREM 1.2. Let E? be a countably dominated Y-algebra, with closed domain 9.
The following statements are equivalent:

406 AKNAL ANLI JUKZAK
(i) The A-topolq,qv is identical to the p-topolog>,.
(ii) The bounded s&sets of the X-topology coincide with the hounded subsets
of the p-topology.
(iii) The linear forms OJ,~, ,, (x t 9, -L’ E I)) are p-continuous.
(iv) FOB e7*ery .-I E (7 , there exists B E cl such that 91R C !llls and the
injection i: ‘Jl,, + !LN8 is continuous.
(v) The bilinear map (S, ‘f) t ST is p-continuous.
hoof. For simplicity of notations, we denote by (U, h) resp. ((I, p)) the
space 67 endowed with the h-topology (resp. the p-topology). The equivalence
of (i) and (ii) follows from the fact that ((7, h) and (n, p) are bornological spaces.
Clearly, (i) implies (iii). Sow-, if S is a bounded subset of (G!, p), and if (iii)
holds, then the injection i: (Cl, p) -+ (U, ,Y) (.Y topology on 0 defined 1~)
seminorms 7’ + (7:x, ~1) .x t 9, T t I)) is continuous; hence, S is bounded for -7,
and by Proposition 1.2, S is bounded for (U, X). This gives (iii) -:- (ii).
The equivalence of (iv) and (ii) is obvious, and the equivalence of (i) and (v)
follows from C’orollary I. I and from p-continuity of T -+ T*.
Of course, if /r consists of bounded operators, ((r, h) is equal to (/Z, p),
and these topologies agree with the norm topology.
DEFINITION 1 S. Let (r be a .-algebra, with dense domain 5’. A sequence
kLN of elements of ‘/ is called o-conrcrgcnt. it?, for every ‘4 t (7,
If U lJEJW,, ) then a sequence (,v,) of elements of 9 is a-con\crgcnt
if and only if (1) holds foi- operators -4, (J’ t 1). Moreover, if 5’ is (i’-closed,
then for every closable operator S such that SY C Y, the relation (1) holds,
with =3 S: this is a consequence of Lemma I. I, and this shows that this
definition is not related to the algebraic structure of f7.
Rut this concept leads to a natural definition of the ampliation (r St> li K of
a .-algebra r%. In fact, let R he a (separable) Hilbert space. We identif!, the
Hilbert space 1) i$~ R to ‘IT’,, ,i I),, (with I),, 1~ for every n E N) and WY define.
for 7’ E r%, the operator 7’ ,‘; I R to bc
and

TOPOLOGICAL ASPECTS 407
ever!- o-convergent sequence can be viewed as an element of 1) $j, K (take
A ~~ Id in relation (I)) and under this identification, the set D of o-convergent
sequences appears as the intersection of the domains of the closure of operators
T @ I defined on the algebraic sum Q&,, 9,, (9 being M-closed). Therefore,
D is invariant under operators T g I, I3 is ((r I; E 11, I finite.
(ii) The follozuing statements are equivalent.
(ii. 1) g, is a-weakly continuous,
(ii.2) p i.s a-stvongl~* continuous,
(ii.31 q = ILEN WI,.?,‘ > zcith ,Y, C 9, ~1, t 11, xii”i J*, 1 -: t-x, und
(,x,)iih heiy a n-conzqeyent sequence.

408 ARNAL AND JURZAK
Proof. We first prove part (i). Since the set of linear forms of the type
v = C&l wxi,?l, (I finite, xi E 9, yi E I?) is a vector subspace F of @%*, algebraic
dual of 02, the weak topology of 0? is the topology a(& F), and therefore (i.3) is
equivalent to (i.1). Of course, (i.3) implies (i.2). Conversely, let v be a strongly
continuous linear form. Then, for some finite family (x~)~, I, we consider the ampliation map
Then, putting 0 -= 9 0 Q-r, P == (x,)l$iX,n , we get
1 0(T @ I)! S; i’(T @ 1)1 i’
which gives our result.
Part (ii) follows from part (i), because, if (xi)iGN is a u-convergent sequence,
then, for every T E Q?
with R = LC2(N), 2 = (x&~ , 9 (Y~),~~ .
COROLLARY 1.2. Ez?ery weakly (resp. o-weakly) continuous linear form on CY
has a weakly (resp. u-weakly) continuous extension to Z(9).
This is a consequence of a remark following Definition I .5.

TOPOLOGICAL ASPECTS 409
I,et a = (JZEN 9q be a #-algebra, with closed domain %:. We endow fl
with its X-topology, and denote 02’ the strong dual of (a, X). For f E a’, n E N,
we put
:iflln,A,, ~= sup{/ f(T)]; T E ‘JJIA,& with X&T) : 11.
By [lo], the strong topology of a is the coarsest for which the transposed
mappings ti.dr of fl’ into the strong duals (!N>L , 11 iA,Ac) are continuous.
We denote by
(a) -Ip, the space of weakly continuous (equivalently strongly continuous)
linear forms on a. Of course, 9- C a’,
(b) 9.. the adherence of F, in fl’ (endowed with its strong topology).
THEOREM 1.4. 1” Every a-weaklqt continuous linear form on r% belongs to 9.. .
2” Conversely, if the algebra 67 satisfies condition (C), then every element
of Z.. is c-weakly continuous.
Condition (C) is:
There exists an operator A E LZ, which is the restriction to 9 of the
inverse of a compact operator in I), such that
Before plunging into the proof of this theorem, let us note that if we differentiate
a strongly continuous unitary irreducible representation x of a Lie group G
(with Lie algebra g), the enveloping algebra Q? generated by dz(g) admits the
domain 9 of differentiable vectors as closed domain and
Moreover, (C) is fulfilled if G is C.C.R. [17].
The proof rests on
(A Laplacien: [16]).
LEMMA 1.2. (adaptation of Lemma 2 of [23]). The vector space of operators
of Y(g) with finite dimensional range, is dense in Y(g) endowed with the X-
topology.
A similar proof of Lemma 3 of [4, p. 361, and Lemma 1.2 establish
LEMMA 1.3. Let (x~)~

410 ARNAL AND JURZAK
coincide. Then, for ecel:\, T E Y(P)
Proof of the theorem. We first treat part I . Let cp := x,tN w,,.,.,, be a CJweakly
continuous linear form on 0’ ((x~)(~~ is a o-convergent sequence, and
xi”=, 11 y, ,/a < -I- co). Let yll xi:.li, w,,.~.,,, . Then ?Ii F Y- , and q~,< tends to q
in 67’ because, for every .4 E a%,
since the relation
implies
/ f
~IJ~~,~~(T) / - - i 1; A.Yj ~1 ‘IFi 11 C i i (;’ ,hi ii2 + j’i ~i2).
i=n+1 i==ntl i-=W~l
For part 2”, it suffices to consider the case U : Y(9) because of Corollary I .2
and the relation
sup(~f(T)i; T E G! such that 11 TX ~ T;- j =I.v ;,)
s< ’ sup(‘f (T)! ; T E Y’(3) such that 11 TX 11 < 11 Ax j,
forfeQ?.
We can assume with no loss of generality that Ax > I/ x 11 (for all m t 9).
Thus
0 _ p 5. I, dns ,’ .

The polar decomposition of the map
g, (y, J’,) Llhsj -- “F X,(.y, e;) d”e,’
with F,’ t 9, C, E 11, Xi -,z 0, the system of (e,),, ;. ,, (resp. (dbe,‘)lci~C,) being
orthonormal.
B!: injectivity of Ah, and Lemma I .3, we deduce that
Indeed, the relation ,! TX !( --I I A”.Y , (T E Y’(9)) for all .Y E 9 implies
i-l
411
(2)

412 ARNAL AND JURZAK
is a-convergent, since using Lemma 1. I, for every n
converges because of relation
obtained from
Now
achieves the proof (see Definition I .6(iv)).
2. SPECIAL *-ALGEBRAS AND EXAMPLES
In this section, we examine particular *-algebras. Examples will be discussed
and listed after a few abstract results. We introduce for convenience the following
definition:
DEFINITION 2.1. A p-closed +-algebra of countable algebraic dimension,
with closed domain, is called hyperfinite.
Let us recall that algebras of countable algebraic dimension are countably
dominated.
PROPOSITION 2.1. Let O? be a *-algebra of countable algebraic dimension.
Consider the following statements.
(i) GY is hyperjnite.
(ii) For every A E UP, $, is finite dimensional.
(iii) The p-topology of G? is the finest locally convex topology on GY.
(iv) GZ is X-closed.
(v) For every A E GZ, 9jIa is jnite dimensional.
(vi) The X-topology of 02 is the jnest locally convex topology on LX.
Then, one has the implications
i e ii 0 iii > iv G v 0 vi.
Of course, if CT is hyperfinite, then h = p.

TOPOLOGICAL ASPECTS 413
Proof. It is an easy consequence of the Baire theorem that a Banach space
cannot be of countable algebraic dimension unless it is a finite-dimensional
space. Therefore i * ii (resp. iv * v).
Clearly, ii * iii (resp. v 2 vi), since the p-topology (resp. the h-topology) is
the inductive limit of finite dimensional spaces. If iii (resp. iv) is true, then the
bounded subset of 0Z are finite-dimensional: From Proposition 1.2 (resp.
Proposition 1.6) this implies (i) (resp. iv).
COROLLARY 2.1. (existence of a predual). Let 67 he a hyperfinite +-algebra.
Then, there exists a Frechet space Cl, , ,whose strong dual is Ii(. JIloveovev, if d * is
a secondFrechet space with strong dual G?, then f?!* is topological& isomorphic to (r, .
Pyoof. The first assertion follows from reflexivity of U; it suffices to take
fl* = CT’, endowed with strong topology (the strong topology agrees with
u(C?!‘, ~7) which is metrizable since c;% is of countable algebraic dimension).
If d, is a second predual for CY, then d, is reflexive ([IO, p. 961). Since the
topology of n; is the topology of uniform convergence on equicontinuous
subsets of l;il, it necessarily agrees with the strong topology of (il’. This achieves
the proof.
It is clear from Proposition 2.1 that a hyperfinite algebra GY is a nuclear space,
and that (S, T) --f ST is continuous; hence, if f is a linear form on fZ, the map
T E 0 +-.f ( T*T)lJ2 is continuous.
An application of Theorem 1 of [1 I] leads to
COROLLARY 2.2. Let 62 be a hyperjnite --algebra, and f a state on 6Y. Then,
there is a standard measure space Z, a weakly measurable map 6 ---z fc from Z into
the set of extremal states of ~7, and a positive measure EL on Z (with p(Z) = I)
such that
COROLLARY 2.3. Let a be a hyperjinite h-algebra; and cilk the convex cone of
positive elements qf 0. Then
1” for ecevy T E al, there exists S E Cl-~ such that
O

414 ARNAL AND JURZAK
Pyoof. We recall that an element T of 0P belongs to some extremal ray
of GY+ if and only if the relation 0 < S < T implies S = AT, for some h 3 0.
Let @ = LN %, , for certain iz, > 0. Then @~I = unerm (aA n CZ+) =-
lJneN ‘%2,. Each set ‘Jzfi, being a closed convex pointed cone of a finite dTmensiona1
space admits a compact (convex) base. The correspondence between extreme
points (of the convex compact base) and extreme rays, the Krein-Milman
theorem, the finite dimension of the base imply assertions 1” (note that an
extreme ray of %2, , is an extreme ray of 07, and conversely). Thus, the set of
extreme rays of GP is the union,
Assertions 2” and 3” are therefore
for n E N, of the set of extreme
clear.
rays of ‘%:
n
.
COROLLARY 2.4. If Q! is a hyperfinite *-algebra, and .@ a a-subalgebra qf
GZ, then L% is hyperjinite.
PYOO~. Obvious (note, however, that the domain 9 of GZ is not ,%?-closed:
by definition, the domain of g is obtained by the processes of [19]).
We now recall a few facts about infinite tensor products. First, let (gJieN be a
sequence of vector spaces, each 5Zi being dense in some Hilbert space hi . We
denote, for every iE N, by ai an element of si, with norm equal to I. The subspace
of the Hilbert space lj :A @tai) h, ( incomplete tensor product), generated by
elements @ xi , with xi E Bi and xi = ai for almost i, is dense in h, and will be
noted simply (j& gi . We consider now a sequence (QZi)iGN of x-algebras, with
closed domains Bi (dense in the Hilbert spaces hi). For a family (T& of
operators, with Ti E G& , and T, = Id hi f or almost all i, one can define the
operator &J T, to be the operator whose action on the element (ZJi xi of 6Ji g:i
is @ji T,xi .
The algebra 02 generated by such operators will be denoted (& 6& , @J gi).
Since we are mostly concerned with algebras with closed domains, by applying
the method of [19] Lemma 2.6, we can extend the domain oi g< to a new
domain 9 (denoted 2 = @,A G#J such that G@ becomes Qkclosed. By the
infinite tensor product of *-algebras 6Yi, we mean from now on the couple
(a, B), noted simply @$!, G& .
PROPOSITION 2.2. 1” 1f (&)isi+ is a finite set of hyper$nite i--algebras,
then @yzl G$ is a hyperjinite *-algebra.
2” If (O&N is a sequence of hyperjinite *-algebras, such that ‘%(I ; L&J _ @
Id for almost all n, then the infinite tensor product @r==, UZ, is a hyperfinite *-algebra.
In the above formulation, %(l; &J d enotes the set of bounded operators
ofC&.
Proof. For assertion I”, it suffices to treat the case of two hyperfinite *-

TOPOLOGICAL ASPECTS 415
algebras a and %?. Let Ai (resp. BJ be operators of a-’ (resp. 37”) such that
G? = uy=, gAi (resp. d =m u:r 91B,). We prove that
For this, it suffices to consider elements of the form S @ T (S E C?, T E .%).
From the relation
we have to treat elements of the form S @ 1. But, for some i,
which etablishes our assertion.
Now let CAEA S, @ TA (A finite set) be an element of 91Aio1+,~!8~. Then for
every vector of the form T &? #, we have
for some A4 > 0.
Thus, for every vector p), the operator CA (S,rp, 9)T, belongs to sBi (one can
assume A, > 1 and Bi > 1 for all i E N). By polarization, operators of the form
‘&CA (S,F~ , yj)Th (for a finite set of vectors ~~ , yj) belong to +XJt . This last
space being finite-dimensional, Cnf(SA) T,, E gBi for every (continuous) linear
formfE 02’. Choosing the S, linearly independent, and using the Hahn-Banach
theorem, we conclude T, E ‘iRBi , for every X E A. In the same way, by symmetry,
S, belongs to SAL . Thus
91 A,@l+l@A, c SAIOl + %@B, *
This achieves part 1’.
For assertion 2”, it suffices to imitate the preceding proof.
Weyl Algebras
We denote by .Y the space of Ca rapidly decreasing functions of the real
line. Y is dense in the Hilbert space lj = L2(R; dx) (dx Lebesgue measure),
and invariant under the operators of the complex algebra 6Z, generated by p, q,
with
Pf =&f,
qf = xf, fE.Y.

416 ARNAL AND JURZAK
Moreover, Y is G&closed and, by [16], (“% = Un,o%(lm.d)n with A = p2 - q2.
We refer to [6] for a detailled algebraic study of G!?.
PROPOSITION 2.3. Cl is hyperfinite.
Since
Proof. Let V be the linear algebraic space, spanned by Hermite functions
yk(x) ;= (711/22kf3!)-1/2 (-1)‘; ex”i2
pv, = 2-1’z(k1’2~7+l - (k + 1)1’2~A+1),
qvk = 2-1’2(k1’3~g-l + (k + I)1’2Fr+l),
V is stable under operators of G?. Moreover, 1~’ being dense in .Yj endowed
with its Frechet topology (defined by seminorms qz + li(l - .Ap, 11, n E N) the
operators of C7? are completely determined by their restrictions to V (being
continuous for this Frechet topology). We introduce
P == 2-1yp + q) Q = 2- l,‘(q - p)
and note that c;! is graded by G? == zjPi=,. C Pip, Y EZ [6]. Let u - xr u, be
an element of %(1-A)7L . Then
with vl, = 0 iff k > 0. Thus, ug belongs to !I$_,,, since
where &‘i < + co. If Y + 0, the consideration of
and
leads to the estimate
((u - 4 9k + %+r > Fh -c %i-r)
(h - %) ‘Pk + &k+, ) v7c + +k+?)
@h?k 7 %+r)l < M2[((1 - A)% vk ) P)k) + (tl - dn qk+r ) %+r)l
= J%(k) (k + I)“,

TOPOLOGICAL ASPECTS 417
where M, < +co, and M,(k) positive bounded function of k. Thus, putting
we conclude that
Then, necessarily 1 < (2n - ~)/2, showing that ‘%c,-,), is finite-dimensional.
COROLLARY 2.5. Let GYf,L be the complex associative algebra generated by
operators pi and pi of L2(R”, dx), dejned by
Pif = (W%)f> qi(f) = xif, 1 < i < m,
f belonging to Y(Rm), the space of C” rapidly decreasing functions in m variables.
Then, G&, is hype@nite.
Proof. The proof is an immediate consequence of Proposition 2.2, 1”. A
similar corollary holds for an infinite tensor product.
We can now turn to a more complete description of Corollary 2.3. For
simplicity we denote I the set
I = (U E CZ?; u = hp + pq + v with u symmetric, h, p, v E C}.
PROPOSITION
u~I,andk>O,
2.4. 02 can be written 6l = &>. ut, ‘%z,2k . Moveover, for every
Proof. Let u be an element of I: u being symmetric, there exists a unitarily
implementable automorphism @ of @ such that :
@(ip) = u PI.
Then we just have to prove that dim !QD)2k = 1 Vk 3 0. Let v = Ciej hijqjp”,
hij # 0, be an element of ~(is)zrc . For all unitarily implementable automorphisms
16 of GY such that t/(p) = p, we have:
Now, the space generated by the 4(q), 4 a such automorphism, is infinite-
dimensional since the degree of 4(q) can be arbitrarily large. So if there exists
580/24/4-9

418 ARNAL AND JLJRZAK
an index j > 0 in zj =- C h,,qjp” , %(iD)2k is infinite-dimensional. By Proposition
2.1 this is impossible. Thus we have:
But after a Fourier transform, this relation gives
Therefore, choosing the function f with support in [n, n - 1], next in
[l/n --:-- 1, l/a], we conclude that i z-m 2k and %(is)zk is one-dimensional. Finally,
let 9 be the space &a,, lLCI $J1,,91C . For any integer n, (ip)” belongs to .W since
(ip)*” and (z’p + 1)“” belong to ~9 for all k. For the same reason, q” and z’, mu
(;p $ hq)” (A E R) a 1 so b e 1 ong to d. But the uA for n $ I different values of A,
generates a vector space which contains all the monomials
of the system:
q~p~~-i, the determinant
being a van der Monde determinant, which proves the proposition.
Cnitayv Representations of Lie Groups
Let G be a real Lie group with Lie algebra 9, and 1.: a continuous unitary
representation of G in a separable Hilbert space 6. We denote by 9 the space
of differentiable vectors of U, by rlc. the differential of Z,;; dL’ is a representation
of 9, and operators dC(X) (X E Q) are clearly defined on Y. LVe also introduce ZI
the complex enveloping algebra of the complexification qa: of n, and f’Y the
complex .-algebra Ltr’(ll) (6’ ac t. s on the &closed domain 5’).
PROPOSITION 2.5. If G is nilpotenf and L’ irreducible, U? is hdvper@itr.
Proof. Indeed, CT is the algebra studied in Corollary 2.5.
PROPOSITION 2.6. Let 1: be an iweducible unitary representation of the group
G =~ SL(2, Iw). Then, the algebra (r is hyperfinite.
Proof. Indeed, CT is an algebra 3,, studied in [7]; with notations of this
paper, we get a basis of (?I of the form

‘I’OPOLOCICAL ASPECTS 419
RIoreorer. we can find in 11 an orthonormal basis ye,, satisfying
with k CZ or N, according to the case, q a constant and A,- , cx,, , /j,, numbers
of order k when k i goes to infinity. Now it suRices to adjust the proof of
Proposition 2.3.
Of course, for irreducible representations of compact groups, CT is hypertinite.
Some insight in factorial representations can be obtained b>
LF:>Lrni.a 2. I. Let 1,’ br a ,factorial representation of a group G of tjppP I. Then,
there exists u unique unitary irreducible representation L’,, of G such that 11 tll.(U)
is isomorphic to dC;,(ll).
Proof. Let 1~ -= J;’ h(f) dp([), CT ~ sd C’(t) dp([) a decomposition of ( into
a direct integral of irreducible representations of G. By [5], the representation
r’(t) is almost everywhere equivalent to an irreducible unitary representation
7 of G. Thus, we can write [4, p. 165, I931
P,, being the space of differentiable \-ectors for C’,, , we get by differentiation
nr 111 ‘,) :, I on P,, 1 2’; I,‘(*\~, d/L) (algebraic tensor product).
RIoreover. .P ,, :+Z,‘(.Y, dp) is dense in 9 for the topology of C/ since this space
is invariant under 1 -(G) [I 81. ‘Therefox, the map 7’ f 7’ ‘t I of t/l -(,(!I) onto
r/l.(U) is an isomorphism, bicontinuous by [13]. Q.E.D.
It is worth noting that for nonfactorial representations, the structure of
// dC:(II) can be complicated: for instance, certain unitar! rcprcsentations
of abelian groups leads to algebras of functions, showing the great 1-xietv
of situations. Nevertheless.
I’KOPOSITION 2.7. I,et G be the Heisenbey gvozlp (dejned it1 [22]), anti % ~/, 0
a central element ?f q. A4ssume that one of the following h~~potheses is trw
(i) til .(Z) is an unbounded operator,
(ii) the spectrum of dl:(Z) is jinite, and dijjJerent from [Oj.
Then, c/ /s l~~pe~jinite.

420 ARNAL AND JURZAK
Proof. Let {X, ,..., X,,, , Y, ,..., Y,,, , Z} b e a b asis of g satisfying the relations:
[Xi , Y,] = zijz [Xi ) Xj] = [I’; , Yj] = [Xi p Z] z [Yj y Z] E 0.
We decompose U in U, @ Lra where Ui is a direct integral of infinite dimensional
irreducible representations of G and U, is a direct integral of characters of G.
If (i) is true, we can write:
U, = I;I!’ c’, dp(X),
where (1, is factorial, X is the spectrum of --i dU,(Z) and TV is a positive Radon
measure on R. Moreover, Ii, can be expressed as U, = oA @ 1 with c, irreducible
infinite-dimensional of G (Lemma 2.1): CA acts inL2(R”) and we have:
d&(X

By the same method:
TOPOLOGICAL ASPECTS 421
(1 - A)‘” _ [I @ 1 - I(&,’ @ 1 + c (Lvj)Z @X21
z z I
Then almost everywhere:
And so 111 + 1 J / < 2nm following the extimations of Proposition 2, and
this corollary. Now if q belongs to $J” V, jj v 11 = 1 for all functions f of A’-,
we have almost everywhere:
i 2 ilc+k (j$ y 9’j’ (+a ~1” (~lJl+kf(3, f(4) 1 G Mb, TVV) f(4, f(4),
where M(n, F) is finite and P(A) is a polynomial of degree 2n. If h goes to infinity,
we obtain: i J i + k < 2n, and %(l~d)n is finite-dimensional.
xow assume (ii) is true, and denote {A, ,..,, A,} the spectrum of dU(Z). It is
easy to see that 02 is linearly spanned by the monomial XIYJZ’i with 0 :< h < r,
I and J being elements of W. We can decompose U in:
where C:,,, is the factorial representation of G considered in the first part of the
proof. If u ~~ Ca,,J,,XIYJZ” belongs to ‘%~r-~)~ , then
l(dUA(4!h $11 < Jw - 4” 94 $4
hence i 1; + 1 J 1 < 2nm.
Topology of 9 and Structure of ~2
for *E 6 V,
U.E.D.
The aim of this Section is to show that there is no link between nuclearity
of 9 and hyperfiniteness of 02, even for enveloping algebras (viewed in some
representation). In fact, this is not too surprising since if 0! is written a =
uis, “nAi , then the topology of $3 admits seminorms x--f jl Aix /I, and the set
of Ai , i E 1, does not necessarily reflect the algebraic structure of QL
LEMMA 2.2. Let E, be the two-dimensional Euclidean group, and C’ the
Tepresentation of E, dejined in L2(T, dx) (T torus, dx Lebesgue measure) by
(U(m, /?)f)(t) = eirRecde-“)f(t - a)

422 ARNAL AKD JURZAK
o( beivzg real (0 :< ac -‘; 27-r), /3 comp1e.y bein.g a parametrization of E, [22]. Then
1 L7 == dU(U) is not h-vperjkte,
2 the space Y of d#erentiable oectom qf t ix nuclear.
Proof. 9 is the space of C’ functions on the torus [22] since !J is threc-
dimensional with basis (S, Y-r , I’.,) satisfying
(dU(,Y)f)(t) =- --- $;J.
(dC;(I,)f)(t) = ir cos tf(t),
(dCi(l*z)f)(t) = iv sin tf(t).
The space Y, endowed with seminorms f ~+ ‘(d”/dt’ f I)” f! is nuclear
(1 - 0) -1 being a nuclear operator on L2(T). H owever. the algebra CT’ is not
hyperfinite, since the algebra C[I’r] of polynomials in Yr is included in 91~,-,, .
LEMIA 2.3. Let G be the aflne group of the real line, and L. the representation
of G &fined in L2(R, dx) bv
(W% P)f l(t)
eiRt”,l/4f (alq
CL -, 0 and /‘3 real being a parametrization of G.
Then
I (;I = dU(Il) is hyperjnite,
t E R,
2‘ the space 53 of differentiable sectors of L is not nuclear.
P~of G admits two irreducible infinite-dimensional unitary representations
V and I- (we refer to [22] for details and notations), U1 being defined on
L”(R. ds) by
(LTL(% P)P>W ei~~rcp(x -- Log a), x E R,
and C’ is unitarily equivalent to C’ 0 C; 1 : indeed, CT splits in I. Lz,U.rt (-.’
u rz1 5 .oJ and these two components are unitarily equivalent, since the unitar!
operator I’: L2(-- cu, 0) --+ L2(0, -103) (V(f))(t) f (-t) is an interwining
operator. Also, l,; !&Q,,,) is equivalent to 1’~’ because I&‘: L”(0, L “c) p-C’@)
((Wf)(x) .f (e2z) ez/2112)) is . an interwining operator (for C mL’(,,, , j and F’).
Now, 9 is two-dimensional, with basis (X, I’) satisfying
[X Yl = y,
dU(X) -~= ;t(djdt) A- :,
dl;(l’) = it?.

TOPOLOGICAL ASPECTS 423
Since f belongs to 9 if and only if the restrictions off to (- co, 0) and (0, + a)
are in 9, 9 is isomorphic to the topological direct sum % @ 9 -~, where 9 is
the space of differentiable vectors for L’ 1. This sum is not nuclear because the
Laplacian A of G is such that nr-’ (I ~ 0)~~ is not compact, G being not liminar
[17, 221.
Finally, the space 9’ of CJ rapidly decreasing functions of the real line is
dense in 9 [1] and so 11 is isomorphic to a subalgebra of the l\‘eyl algebra
considered in the first part. Thus II is hyperfinite.
A Situation with X f p
Let 1) be a separable infinite-dimensional Hilbert space, (e,, ; n E N) a complete
orthonormal system of 1~. 1Ve note B the space algebraically spanned by the e,,
and 0 the -‘Te n -: 0 if 11 --< k and LT*LTe,, = nenWk if k < n,

424 ARNAL AND JURZAK
If h = p, the *-operation is continuous for the h-topology and ST* is bounded.
But for any M > 0, any positive integer n and p, we can find k such that
11 TVe, I/ ~-- ~l(k + n)e,+, 11 -=- k + n > Mn” = MI1 Tf’e, 11,
which proves that the element TV of GY* is not in ‘!I&, . Following Proposition
16 we conclude that 3?* is not bounded. Q.E.D.
ACKNOWLEDGMENTS
The authors would like to thank Professor M. Flato for helpful advices and suggestions
of many improvements; and Daniel Sternheimer for his kind interest in this work.
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TOPOLOGICAL ASPECTS 425
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JOL’RNAI. OF I~I:N(‘TIDN.AI. ANALYSIS 24, 426 (I 977)
&?.\lS, Ii. .A., 241
XRIRIEN, \y. o., 258
ARKAI., III., 397
B.A(~~ETT, LARW, 379
BAMBERGEH, AIAIN, 148
BEAUZARIY, BERNARD, 107
BERTH~ER, A. M., 258
Crlor, ILIAN-I)IXN, 156
CNOP, I., 364
CONNES, ALAIN, 336
CRONE, L., 21 I
DAVIDSON, KENNETH K., 291
DELBAEN, F., 364
DUBINSKY, ED, 21 I
EFFROS, EDWARD G., I56
EMBRY, MARY R., 268
GEORG, KURT, 140
GOOTMAN, ELLIOT C., 223
Copyright pi, 1977 by Academic I’rcss, inc.
All rights of reproductiml in .m) f,rrm rcscrved.
Author Index for Volume 24
YIAR(Y s, ~IosI1B, 303
~hH.l’EI.I.l, ik~.ARIO, 140
%~.411REY, &RNAHD, i 07
MIXL, V~wott J., 303
Rol3rwo~, I~RFK IV., 280
ROIHNSO~, \V. B., 91 I
Rossr, I-Ir-GO, I I
TAYLOR, JOSEPH I,., I I
TOLIMIFHI, RI(.HARI~, 353