Representations of positive projections 1 Introduction - Mathematics ...

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$MATH 5b; Solutions to Homework Set 1 January 2009 1. As K is a ...$

that is a -subalgebra of and let be the corresponding complete Boolean subalgebra of . If is nowhere full with respect to and if is a measure on such that on , then there exists F 0 2 such that (D) = (D \ F 0) for all D 2 . In Section 5 we present a vector-valued version of Maharam's theorem of the following form. Assume that B is an f-subalgebra of the Dedekind complete unital f-algebra A with 1 2 B and that P is a positive order continuous projection of A onto B. If A is nowhere full with respect to B and if 0 < e 2 CA and g 2 B satisfy 0 g P (e), then there exists p 2CA such that p e and P (p) =g. In Theorem 5.7 an application of this result is presented, which is of independent interest. If P is a strictly positive order continuous projection in the unital falgebra A onto the unital f-subalgebra B and if A is nowhere full with respect to B, then it follows from the results in Section 5 that there exist for any 2 [0 1] components p 2 CA with the property that P (p) = 1. The collection of all such components is denoted by S. In Section 6 the structure of this set S and its complete Boolean subalgebras is discussed. The construction of such special Boolean subalgebras is contained in Sections 7 and 8. We would like to point out that these special subalgebras are the essential ingredient in the construction of the measure space (X ) occurring in the representation (1) of the projection P . For more details we have to refer the reader to these sections. In particular the results in Section 8 use Maharam's decomposition of Boolean algebras in homogeneous parts. Finally, in Section 10, the results of the previous sections are combined to obtain the desired Maharam-type representations of order continuous strictly positive projections in Dedekind complete Riesz spaces and f-algebras. 2 Preliminaries In this section we will discuss some results concerning Riesz spaces (vector lattices) which will be used in the sequel. For unexplained terminology and the general theory of Riesz spaces and positive linear operators we refer the reader to the books [1], [9], [13] and [16]. Suppose that L is an Archimedean Riesz space and that K is a Riesz subspace of L. We will say thatK is ( -) regularly embedded in L if for any net (sequence) ff g in K such that f # 0 in K, it follows that f # 0 in L. Furthermore, K will be called a ( -) complete Riesz subspace of L if for any net (sequence) ff g in K and f 2 L such that f " f in L it follows that f 2 K. The latter terminology is inspired by the corresponding notions in the theory of Boolean algebras (see Section 23 in [15]). If L is Dedekind ( -) complete and K is a ( -) complete Riesz subspace of L, then it is easy to see 4
that K itself is Dedekind ( -) complete and K is ( -) regularly embedded in L. If L is an Archimedean Riesz space and 0 w 2 L then we will denote by CL (w) =C (w) the set of all components of w, i.e., C (w) =fp 2 L : p ^ (w ; p) =0g . With respect to the ordering induced by L, the set C (w) is a Boolean algebra. Now we assume that L is Dedekind -complete and that w is a weak order unit in L. Then C (w) is a Boolean -algebra, which is isomorphic to the Boolean algebra P (L) of all band projections in L. The Boolean isomorphism from P (L) onto C (w) isgiven by the mapping P 7! Pw. Consequently, the Boolean algebra C (w) is complete if and only if L is Dedekind complete (see [9], Theorem 30.6). Next we will discuss a construction that will be used frequently in this paper. Let L be a Dedekind -complete Riesz space with a xed weak order unit 0 w 2 L. For f 2 L we will denote by p f : 2 R the left-continuous spectral system of f with respect to w, i.e., p f is the component of w in the band ( w ; f) + dd . The right-continuous spectral system of f with respect to w is denoted by q f : 2 R , i.e., q f is the component of w in the band (f ; w) + d . For the properties of these spectral systems we refer to [9], Chapter 6. De nition 2.1 Given a Boolean -subalgebra E of C (w) we de ne L (E) = f 2 L : p f 2E for all 2 R . We will show that L (E) is a -complete Riesz subspace of L. For this purpose we need a result that goes back to H. Nakano ([14], Theorem 14.4) and which is of interest in its own right. For the convenience of the reader we will indicate the proof of this result. We need the following lemma. Lemma 2.2 For fg 2 L and 2 R the following statements hold (i). p f ^ p g f +g p + f +g (ii). p + ^ ; w ; pf p g . Proof. 5
Page 1 and 2: Representations of positive project
Page 3: for all f 2 bL. We stress the fact
Page 7 and 8: in C (w) and so it follows that f +
Page 9 and 10: Pe (e) =e and Pe (f) =0whenever f 2
Page 11 and 12: (i). L is locally order separable (
Page 13 and 14: this example also shows that the an
Page 15 and 16: 3 Representations Many of the famil
Page 17 and 18: Lemma 3.3 Given a Dedekind -complet
Page 19 and 20: observations that Ker ( ) = bL \ M(
Page 21 and 22: It is readily veri ed that the exis
Page 23 and 24: of 1 by CA = CA (1). Note that CA i
Page 25 and 26: for all c 2 C, bytheaveraging prope
Page 27 and 28: and, using that Qf is 1-periodic, Z
Page 29 and 30: As we have seen in Example 2.12, a
Page 31 and 32: Proposition 4.12 Let A beaDedekind
Page 33 and 34: Then there exists a representation
Page 35 and 36: can adjust the values of the functi
Page 37 and 38: so which implies that u P (p) ; 1 1
Page 39 and 40: Proof. First observe that'jK 2 K s
Page 41 and 42: Proof. Let P : Z(L) ! Z(L) be the c
Page 43 and 44: (ii). If e 1e 2 2F such that e 1 ^
Page 45 and 46: Boolean subalgebra such that CAi =
Page 47 and 48: theorem that B 1 K 1 and 1 (g) = (g
Page 49 and 50: A is a Dedekind complete f-algebra
Page 51 and 52: and (34) now follows by induction o
Page 53 and 54: and since P is strictly positive it
Page 55 and 56:
Lemma 7.7 If 2 D and 0 < 1, then [q
Page 57 and 58:
Since q( ) 2 CB for all 0 1, it is
Page 59 and 60:
(ii). there exists a strictly posit
Page 61 and 62:
y Proposition 4.2 there exists a st
Page 63 and 64:
S (CB [E ). Moreover, there exists
Page 65 and 66:
Proposition 8.6 Assume that A, B an
Page 67 and 68:
Since B is Dedekind complete, it is
Page 69 and 70:
E is a Dedekind complete f-algebra
Page 71 and 72:
Then there exists a - nite measure
Page 73:
[10] Dorothy Maharam, The represent
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