Direct Integrals and Decompositions

CHAPTER 14 DIRECT INTEGRALS AND DECOMPOSITIONS In Section 2.6, Direct sums, we studied direct sums of Hilbert spaces. In Chapter 5 (following Corollary 5.5.7), we considered direct sums of von Neumann algebras. The present chapter deals with a useful generalization of the concept of “direct sum” as it applies to Hilbert-space constructs. In this generalization the “discrete” index set X of the sum is replaced by a (suitably restricted) measure space ( X , p). In the simplest case, with one-dimensional component Hilbert spaces of complex numbers, the generalization amounts to passing from l,(X) to L,(X, p). In the case of direct sums, we assign Hilbert-space constructs, for example, operators, to each point of X and “add” them. In the theory of direct integrals, we assign such constructs to each point of the measure space ( X , p) and “integrate” them. For the case of direct sums, we may have to impose a convergence condition (especially when X is infinite). For the case of direct integrals, we must impose both measurability restrictions (on the assignment of constructs to points) and convergence (that is, integrability) restrictions. To avoid the possible pitfalls inherent in the consideration of measure spaces of a very general nature, we shall assume, throughout this chapter, that our measure space ( X , p ) consists of a locally compact a-compact space X (that is, X is the countable union of compact sets) and p is a positive Bore1 measure on X (taking finite values on compact sets). At the same time, many of the measure-theoretic arguments we give will involve eliminating collections of subsets of X of measure 0 (“p-null sets,” or simply, “null sets,” when the context makes clear what is intended). Of course, these collections must be countable for such an argument to be effective. The possibility of keeping these collections countable relies on an assumption of separability of the Hilbert spaces that enter our discussion. This assumption applies throughout the chapter. At a certain stage (following Theorem 14.1.21), we shall want to assume that our measure space can be given a metric in which it is complete and separable. The reader who finds this assumption reassuring is urged to consider it in force throughout the chapter. There is no serious loss of generality if we think of X as the unit interval plus at most a countable number of atoms and p as Lebesgue measure on the unit interval. 998

14.1. DIRECT INTEGRALS

999

The chapter is divided into three sections. The first, and longest section, describes Hilbert spaces that are direct integrals and develops their theory. In particular, the operators and von Neumann algebras that are decomposable relative to such a direct integral are studied. Section 14.2 deals with the possibility of decomposing a given Hilbert space as a direct integral of Hilbert spaces relative to a given abelian von Neumann algebra on it. Section 14.3 is an appendix composed of those less standard measure-theoretic results needed in the earlier sections of this chapter. 14.1. Direct integrals In this section, we define direct-integral decompositions of Hilbert spaces, operators that are decomposable and diagonalizable relative to such a decomposition, and von Neumann algebras that are decomposable relative to such a decomposition. We study the basic properties of these constructs. If we follow this development in the familiar special case of direct sums of Hilbert spaces (the case of direct integral decompositions over discrete measure spaces), the point of view we adopt is that each vector of the direct sum is a function on the index set to the various Hilbert spaces (subspaces) that make up the direct sum. To guarantee that we have the full direct sum rather than a proper subspace, we make the technical assumption embodied in Definition 14.1.l(ii). The diagonalizable operators are those that are scalars on eiich of the spaces; and the decomposable operators are those that transform t’he subspaces of the direct sum into themselves (see Definition 14.1.6). While it is relatively easy to show that the bound of a decomposable operator is the supremum of the bounds of its various components, the corresponding result (Proposition 14.1.9) for direct integrals requires some more effort and care. As one might suspect from the case of direct sums, the families of decomposable operators and diagonalizable operators form von Neumann algebras with the latter the center of the former (Theorem 14.1.10). Direct-integral decompositions of representations of C*-algebras and states appear (Definition 14.1.12) in a manner analogous to their direct-sum decompositions. Defining direct integrals of von Neumann algebras requires a more circumspect approach than is needed for their direct sums. The countability demands of the measure-theoretic situation require us to operate from some countable “staging area.” A norm-separable C*-subalgebra and the components of its identity representation are used for this. (See Definition 14.1.14.) Fine points of normality of components of normal states and the nature of the components of projections with special properties (for example, abelian, finite, etc.) take on greater significance in the context of direct integrals (see

1000

14. DIRECT INTEGRALS AND DECOMPOSITIONS

Lemmas 14.1.19 and 14.1.20) and allow us to identify the types of the components of the von Neumann algebra. (See Theorem 14.1.21.) The type 111 situation presents some special problems that have been avoided to that point. To illustrate these difficulties, note that if we form the direct sum, 10 Sa, of Hilbert spaces X, (a E A) and have assigned to each index a some collection Y , of bounded operators on Xa,there is no problem in selecting and forming the direct sum operator, XO T, an operator T, from each 9, (provided (11 T, 1) : a E A} is bounded). In the case of direct integrals, where the index family A must be replaced by the measure space (X, p), we have the added requirement that the selection must be made in a “measurable manner.” The techniques of Borel structures and analytic sets used in establishing the measurable selection principle needed for this appear in the appendix (Section 14.3). The results that draw on this principle appear at the end of this section-notably, the result that the components of the commutant are the commutants of the components (Proposition 14.1.24) and the proof that the components of a type I11 von Neumann algebra are of type 111. 14.1.1. DEFINITION.If X is a o-compact locally compact (Borel measure) space, p is the completion of a Borel measure on X, and (X,,}is a family of separable Hilbert spaces indexed by the points p of X, we say that a over (X, p ) (we write: separable Hilbert space 2 is the direct integral of {S,,} 2 = Jx@ X,, d p ( p ) ) when, to each x in 2, there corresponds a function p -+ x ( p ) on X such that x ( p ) ~ X , for , each p and

(i) p -+ ( x ( p ) , y ( p ) ) is p-integrable, when x, ye&, and (x, y) = J x ( x ( P x Y(P)> 4 4 P ) (ii) if u,E Sp for all p in X and p -, ( u p , y ( p ) ) is integrable for each y in X, then there is a u in S such that u@) = u p for almost every p. We say that Jx@ 2,, d p ( p ) and p -,x(p) are the (direct integral) decompositions of 2 and x, respectively. 14.1.2. REMARK.From (ii) of the preceding definition, with x and y in X there is a z in 2 such that ax@) y ( p ) = z(p) for almost every p. Since

+

it follows that z = ax + y. That is, the function corresponding for all u in S, to ax y agrees with p ---t ax@) + y ( p ) almost everywhere. It follows that if x(p) = y@) almost everywhere, then x = y; for then (x - y)@) = 0 almost everywhere and, from (i) of Definition 14.1.1, (Ix - y1I2 = 0.

+

1001

14.1. DIRECT INTEGRALS

It follows, as well, from (i) and (ii) that the span of { x ( p ) : x E 2 }is X pfor almost all p. I[n the lemma that follows, we prove an expanded form of this fact that will be useful to us. 14.1.3. LEMMA. If {x,} is a set spanning 2, then Ye: = X pfor almost every p, where 2; is the closed subspace of X pspanned b y {x,(p)}. Proof: If X , = { p : p E X , X ; # iVP}and up is a unit vector in X p0%: or 0 as P E X , or p $ X , , then 0 = ( u p , x,(p)) for all p. With y in %#, let { y , } be a sequence of finite linear combinations of elements in { x , ) such that Ily - ynll +O. If y j = blx,, ... bnxan, then yj(p) = b l x O l ( p ) ... b,x,,(p) except for p in a null set N j . Thus 0 = (:up,y j ( p ) ) for p in X\Nj. Since

+ +

IIY - YnI12 =

I

I~Y(P)

+ +

-

y,(p)1l2 d p ( p )

+

0,

some subsequence { Ily(p) - ynk(p)ll}tends to 0 except for p in a null set N o . For p not i n the null set uy==o N j , then, ( u , , y ( p ) ) = 0. In particular, p -+ ( u p , y ( p ) ) is integrable for each y in 2.From Definition 14.1.1(ii), there is a u in 2 such that up = u ( p ) almost everywhere. But

0 = (UP?U ( P ) >

= (UP? up>

almost everywhere. As upis a unit vector when p is in X , , X , is a null set. 14.1.4. EXAMPLES. (a) The space L,(X, p) is itself the direct integral of one-dimensional Hilbert spaces { Cp> (each identified with the complex numbers). To see this, select from each equivalence class of functions in L,(X, p) a representative f: Then (i) of Definition 14.1.1 is a consequence of the definition of L,(X, p). For (ii) of that definition, we note that iff is a complex-valued function on X such that f.g E L , ( X , p ) for each g in L,(X, p ) , then f~&(A:, p). (Compare Exercise 1.9.30.) (b) The (discrete) direct sum of a countable family of Hilbert spaces {Hn} may be viewed as the direct integral of {Yen}over the space of natural numbers provided with the measure that assigns to each subset the number of elements it contains. Each element of the direct sum is a function n -+ x(n) with domain N, where x ( n ) ~ # ~If. y is another element with corresponding function n -, y(n), then m

(x, Y> =

1 ( x ( 4 , m>,

n= 1

by definition of the inner product on the direct sum. But the sum in this last equality is the integral relative to the (“counting”) measure on N just described; and (i) of Definition 14.1.1 is fulfilled.

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14. DIRECT INTEGRALS AND DECOMPOSITIONS

To verify (ii) of that definition, suppose U , E ~for each n in N; and each y in the direct sum. Let f be a suppose that c . " = l ( ( u n y, ( n ) ) ( < co forfunction in 12(N),and let y(n) be (Iu,(I- ' f ( n ) u , if u, # 0 and 0 if u, = 0. Then ~ ~ = l ( uyn( n. ) ) = ~."=ll(u,l(. f ( n ) < co. It follows that n -+ ((u,((is in Z2(N). (See Exercise 1.9.30.) Thus u is in the direct sum, where u(n) = u,. 14.1.5. REMARK. If 2 is the direct integral of {X,,} over ( X , p), it may occur that the spaces Z, have varying dimensions (finite as well as countably infinite under our separability assumption). We note that the set X , of points p in X a t which 2,has dimension n is measurable. To see this, let { x i } be an orthonormal basis for 2.Let r , , r 2 , . .. be an enumeration of the (complex) rationals, where rl 7 I . Withj,, . . . ,j,, kl, ... ,k,, and m positive integers, let X j . k , , ,be { p : I(rjlxk,(p) ... + rjnxkm(p)/1< m - l } (where j and k denote the ordered n-tuples ( j , , . . . ,j,) and (k,, . . . ,k,,), respectively, some j , = 1, and { k l , . . . , k,) are distinct). From Lemma 14.1.3, with the exception of points p in a null set X , , {x,(p)) generates 2,,. For p not in X,,H,, has dimension less X j , k , mThus . the set of points at which than n precisely when p lies in H,, has dimension less than n is measurable; and each X, is measurable. 1

+

nk,,uj

If 2 is the direct integral of { H p }over { X , p}, an 14.1.6. DEFINITION. operator T in B ( X ) is said to be decomposable when there is a function p -+ T ( p ) on X such that T(p)E B ( ~ and, , ) for each x in 2, T(p)x(p) = ( T x ) ( p ) for almost every p . If, in addition, T(p) = f ( p ) Z , , where I, is the identity operator on Yi",, we say that T is diagonalizable. 14.1.7. REMARK. If p + T(p) and p + T'@) are decompositions of T, then T(p) = T'(p) almost everywhere. For this, let { x j } be a denumerable set spanning 2. From Lemma 14.1.3, there is a null set N o such that { x j ( p ) } spans Hpfor p in X\N,. At the same time, T(p)xj(p) = (Txj)(P) = T ' b ) x j ( P ) ,

except for p in a null set N j . It follows that the (bounded) operators T ( p )and T ( p ) coincide on X\N, where N = Nj. Conversely, if T and S are decomposable and T(p)= S(p) almost everywhere, then T = S ; for, then,

u?=,

r

for all x and y in &

r

1003

14.1. DIRECT INTEGRALS

Iff is a bounded measurable function on X , then p + ( f (p)x(p), y ( p ) ) is integrable for all x and y in 2.From Definition 14.1.1(ii), there is a z in 2 such that f (p:rx(p)= z(p) almost everywhere. Defining M x to be z, we have that M , is a diagonalizable operator with decomposition p +f(p)Z,. In particular, if j r is the characteristic function of some measurable set X,, then M , is a projection--the diagonalizable projection corresponding to X,. If H is a diagonalizable positive operator, it will follow from Proposition 14.1.9 that H has the fonn M , with f measurable and essentially bounded. From this we can conclude the same for each diagonalizable operator. H

,

14.1.8. PROPOSITION. If 2 is the direct integral of { X p ouer } ( X , p) and T,, T, are decomposable operators in a(%), then aT, T,, TIT,, TT, and I are decomposable and the following relations hold for almost every p :

+

(i) (aT1 + T2)(P)= aT,(p) + T2(p); ( 4 (Tl~-z)(P) = Tl(P)T,(PL (iii) T:(p) = Tl(p)*; (iv) I ( p ) = I , . Moreover, (v) if 71(p) I T2(p)almost everywhere then Tl I T2.

Proof: For (i) note that, given x in 2,and defining (aTl + T,)(p) to be aTl(p) T2(p),we have

+

(aT1 +

?;)(P)X(P>

+ T,(P)X(P)= (aT,x)(p)+ (T,X)(P)

= aT,(p)x(p) = (aT1x

+ T2X)(P)= ((aT1 + T,)X)(P)

for almost every p , from Definition 14.1.6 and Remark 14.1.2. Thus aT, is decomposable with decomposition p -+ aTl(p) + T,(p). Similarly, defining ( Tl T,)(p) to be TI(p)T,(p),we have

+ T,

almost everywhere, for each x in 2. Thus TIT, is decomposable with decomposition p + Tl(p)T,(p). Defining F*(p) to be T(p)*,we have (7'*(P)X(PXY(P)) = < X ( P X T(P)Y(P))= (X(P),( T Y ) ( P ) )

almost everywhere; and p + (x(p), (Ty )(p))is integrable. From Definition 14.1.1.(ii), there is a z in 2 such that T*(p)x(p):= z(p) almost everywhere. Since (T * x - z, Y > = ( x , TY) - (2, Y )

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14. DIRECT INTEGRALS AND DECOMPOSITIONS

for each y in #, T*x - z = 0. Thus (T*x)(p)= z ( p ) = T(p)*x(p) almost everywhere, and T* is decomposable with decomposition p + T(p)*. Defining I ( p ) to be I , , we have I ( P ) X ( P ) = I&)

= x(P) = (W(P),

so that I is decomposable with decomposition p If T,(p) I T,(p) almost everywhere and X E

+I,. ~ ,

n

so that T,

s

T,.

The converse to ( u ) of Proposition 14.1.8 is valid and allows us to show that p + IIT(p)ll is essentially bounded with essential bound 11 TI1 for a decomposable operator 7: 14.1.9. PROPOSITION. If% is the direct integral of { S pouer } ( X , p ) and A , , A , are decomposable, self-adjoint operators on # such that A , 5 A , , then A , ( p ) < A , @ ) almost everywhere. If T is decomposable, then p -+ 11 T(p)ll is an essentially bounded measurable function with essential bound 11 T 11. Proof. From Proposition 14.1.8(i),A , - A , is a positive, decomposable operator with decomposition A , @ ) - A,(p). Thus it will suffice to show that, if 0 IH and H is decompqble, then 0 5 H ( p ) almost everywhere. Choosing a dense denumerable subset of fl and forming finite linear combinations of its elements with rational coefficients, we construct a dense denumerable subset { x i } of X t h a t is a linear space over the rationals. From Lemma 14.1.3, (x,@)} spans 2, for p not in some null set N o . For each finite rational-linear combination, rlxl + ... r,,x,,, there is an xi equal to it; and, from Remark 14.1.2, there is a null set outside of which rlxl(p) + ... r,,x,,(p)= x,(p). If N,, N , , ... are these null sets (corresponding to an enumeration of the rational-linear combinations), then { x,@)} is a rational-linear space spanning ZP for p not in N j , a null set N . With p not in N , then, {xi@)} is dense in

+

+

uTz0

Z P .

If 0 5 H, then 0 2 ( H x j , xi> = fX(H(p)x,(p),x,(p)> d p W . Suppose H(p)xj(p), x x p ) ) < a < 0 for p in some subset X , of X of finite positive

14.1. DIRECT INTEGRALS

1005

measure. With f the characteristic function of X o , p -+ (f (p)xJ(p),y ( p ) ) is integrable for each y in 2;so that (just as in Remark l4.1.7), for some z in 2,z ( p ) = f ( p ) x i ( p )almost everywhere. In this case

contradicting the assumption that 0 I H . Therefore 0 s (H(p)xj(p),xJ(p)) except for p in a null set M i . If M = Mi and p $ N u M , then 0 I (H(p)xi(,a), xJ(p)) with {xJ(p)} a dense subset of 2,.It follows that 0 IH ( p ) for p not in N u M . With T decomposable, T* and T*T are decomposable with decompositions T*(p) and T*(p)T(p), respectively, from Proposition 14.1.8. Since 11 T(p)l12= I/ T*(p)T(p)ll;to show that p -P 11 T(p)II is measurable and essentially bounded with essential bound 11 TI[,it will suffice to deal with a positive Now 0 I H I IIH(II so that, from what we decomposable operator H on S. have just proved, 0 H ( p ) I IIHI(I, almost everywhere. Conversely, from Proposition 14.1.8, if 0 I N ( p ) I a l p almost everywhere, then 0 I H Ia l and JIHllI a. It follows that the essential bound of p -+ \lN(p)l[ is llN\\. To establish that p + IIW(p)ll is measurable, we make use of { x j } and N , introduced in the first paragraph of this proof. If s (>O) is rational then the set X , of points p not in N where H ( p ) I s l , is

uT=,

W

0{ P : IsIIxj(P)I12,P + N } .

j= 1

Now IIH(p)II lies in an open interval if and only if there are rationals I and s in that interval such that H ( p ) $ r l , and H ( p ) Isl,; so that the set of such p in X\N is a countable union of the sets X,\X,. Thus p + liH(p)/(is measurable.

(x,

14.1.10. -rHEOREM. If x is the direct integral of {x,}over p), the set R of decomposable operators is a von Neumann algebra with abeiian commutant 9 coinciding with the family %? of diagonalizable operators. Proof: From Proposition 14.1.8,9?is a self-adjoint algebra of operators on X containing I . It remains to show that 9 is strong-operator closed. Let A be a n operator of norm 1 in the strong-operator closure of 9, and let (xi} be a denumerable dense subset of .#. Using the Kaplansky density theorem,

1006

14. DIRECT INTEGRALS AND DECOMPOSITIONS

there is a sequence { T,} of operators in the unit ball of W such that T,xj + A x j for all j. Then for all j,

II T,xj - Axj II ’ =

II T,,(P)xj(P) - (Axj)(p)II ~ A P )0. +

l x

There is a subsequence { Tn1} of { T,,} such that 11 T,,(p)x,(p) - (Ax,)(p)JI tends to 0 almost everywhere. Again, there is a subsequence { Tn2}of {Tn1}such that I) q 2 ( p ) x 2 ( p )- (Ax,)(p)ll tends to 0 almost everywhere. With { T,,) the “diagonal” (that is TI,, T 2 , , . . . ) of these subsequences, we have that 11 T,,@)x,(p) - (Axj)(p)I(+,O almost everywhere. Using Lemma 14.1.3 and Proposition 14.1.9, there is a null set N such that (x,{p)} spans X p , ~lT,,,@)l~ 5 1 for all n, and T,,(p)x,(p) +,(Axj)(p) for all j, when peX\N. It follows that, for p in X\N, there is an operator A @ ) in the unit ball of 9Y(Xp) such that A(p)xj(p)= ( A x j ) ( p )for all j. With x in S, let ( x j . ) be a sequence chosen from { x j } tending to x. Using the L,-subsequence argument of the preceding paragraph, we can choose a subsequence {xY} of {xi’}such that x&) -+ x(p) and (Axj..)@)+ (Ax)(p)for p not in some null set M. Then for p not in N u M , A(p)x(p) = (Ax)@).Thus A is decomposable with decomposition p + A@), W is strong-operator closed, and W is a von Neumann algebra. With % in place of 9, this same argument shows that % is a von Neumann algebra. If A is diagonalizable with decomposition f(p)Z, and T is decomposable, then A T and T A are decomposable with decompositions f(p)f,T(p) and T(p)f(p)Z,, respectively, from Proposition 14.1.8(ii). Since AT and T A have ‘ . the same decompositions, A T = T A (from Remark 14.1.7), and A E ~ We show that W = W , and since, as just noted, % is a von Neumann algebra, 9’ = 5%‘‘ = %. As W G W and 92 and W are von Neumann algebras, in order to show that W = W, it will suffice to show that each projection E in v‘ is in 9. For this, let {ui} and { u j } be orthonormal bases for E ( S ) and (I - E ) ( S ) , respectively; and let { x j } be an enumeration of the set of finite rational-linear combinations of elements in { u j , u j } . As in the first paragraph of the proof of Proposition 14.1.9, there is a null set N such that, if rlxl + ... + r,x, = x j , then r , x , ( p ) ... + r,x,(p) = xJ(p), for rationals r , , . . . ,r,, and {xJ(p)} is , p 4 N. For p not in N, let E(p) be the projection with range dense in X P when spanned by {u,(p)}. If u is a finite rational-linear combination of elements in { u j } and p 4 N, ( E u ) ( p ) = u(p) = E(p)u@). Let u be a finite rational-linear combination of { u j } . Suppose, for the moment, that we know (uJ@), u ( p ) ) = 0 if p 4 M for some null set M. Then 0 = E(p)u(p) = (Eu)(p) for p not in N u M. Hence, if p $ N u M, ( E x i ) ( p ) = E(p)x,(p). With x in X, there is a sequence ( x r } of elements in { x i } tending to x. As in the preceding paragraph’of this proof, there is a null set N o such that ( E x ) ( p ) = E(p)x(p) if p 4 N o u N u M. Thus E E W .

+

14.1. DIRECT INTEGRALS

1007

It remains to prove that ( u ( p ) , u ( p ) ) = 0 almost everywhere when Eu = u and Eu = 0. A.t this point, we use the assumption that E commutes with %. Let P be the diagonalizable projection corresponding to the measurable subset X , of X . (See Remark 14.1.7.) Then 0 = (Pu, Eu)

=

(EPu, U )

=

(PEu, U )

Since this hdlds for each measurable subset X , of X , ( ~ ( p )u(p)) , everywhere. w

= 0 almost

Note that the first two paragraphs of the preceding proof establish that if { T,} is a (bounded) sequence of decomposable operators converging to A in the strong-operator topology, then A is decomposable and some subsequence { T,,} of {T,} i:s such that { T,.(p)} converges to {A(p)} almost everywhere. If { T,} is monotone, then { T , ( p ) } is monotone for almost all p , from Proposition 14.1.9, and the sequence {T',(p)), itself, is strong-operator convergent to { A @ ) } almost everywhere. In particular, if { E n } is an orthogonal family of projections with sum E, then E,(p) = E ( p ) almost everywhere. 14.1.11. EXAMPLES. (a) With reference to Example 14.1.4(a),the algebra of decomposable operators on L,(X, p) (considered as a direct integral of one-dimensional spaces) coincides with the algebra of diagonalizable operators. It is the (maximal abelian) multiplication algebra of L,(X, p ) see Example 5.1A). (b) In case H is the discrete direct sum of a countable family {#,} of Hilbert spaces (see Example 14.1.4(b)) each decomposable operator T is the direct sum (see Section 2.6, Direct sums) of a family (T,} of operators T, on 3, (so that T ( x , } = {T,,xnJand jjT[j = sup(/jT,(j}).In case T is diagonalizable, each T, is a scalar. H In the definition that.follows, we refer to representations of general C*algebras rather than norm-separable algebras simply because norm-separability plays no role in the definition. In practice, however, we shall have to assume that our algebra is norm-separable in order to prove the results that interest us.

If H is the direct integral of Hilbert spaces { H p } 14.1.12. DEFINITION. over ( X , p), a representation cp of a C*-algebra CU on H is said to be decomposable over ( X , p) when there is representation cpp of 2l on Hp such that q ( A ) is 'decomposable for each A in CU and q ( A ) ( p ) = cp,(A) almost

1008

14. DIRECT INTEGRALS AND DECOMPOSITIONS

everywhere. If cp(A) is diagonalizable as well, for each A in W,we say that cp is diagonalizable. The mapping p + q P is said to be a decomposition (or diagonalization) of cp. A state p of W is said to be decomposable with decomposition p --t p p when pp is a positive linear functional on Iu for each p, such that p,(A) = 0 when cp,(A) = 0, p + p,(A) is integrable for each A in W, and P(A) = j x P , ( 4 dP(P).

As in Remark 14.1.7, which applies to decompositions of operators, one can show that if p + cpa and p + c p b are decompositions of cp. a representation of the norm-separable C*-algebra \LI, then cp, = cpb almost everywhere (see Exercise 14.4.2); and that if cp and cp‘ are decomposable representations of W whose decompositions are equal almost everywhere, then cp = cp’ (see Exercise 14.4.1). The condition that p,(A) = 0 when cp,(A) = 0 guarantees that there is a positive linear functional pb of cp,(W) such that p’,(cp,(A)) = p,(A) for all A in W. In application, it will suffice to have p p defined on the complement of a null set N ; for with p in N , we can let pp be q cpp, where rj is an arbitary state of W,. The resulting mapping p + p,, defined for all p in X , will satisfy the conditions of Definition 14.1.12. 0

14.1.13. THEOREM.If % is a direct integral of Hilbert spaces (X,} over ( X , p ) and cp is a representation of the norm-separable C*-algebra M in the algebra of decomposable operators, then there is a null set N and a representation cpp for each p in X\N such that p + c p p is the decomposition of cp. If p is a state of M and x o is a vector in X such that p ( A ) = (cp(A)x,, x o ) for each A in Iu, then p is decomposable with decomposition p + p,, where p,(A) = (cp,(A)XO(P),XO(P)). Proof: Let 210 be the self-adjoint algebra over the rationals consisting of finite rational-linear combinations of finite products from a self-adjoint denumerable generating set for W. Let Z,be a dense denumerable rationallinear space in %. With A , , A , in Iuo and rl, 1, rationals, (r,cp(A,)

+ rzcp(A,))(p) = J-Icp(A,)(P)+ r2cp(A2)(P)r

(cp(A,)4442))(P) = cp(Ad(P)cp(A,)@),

and cp(Al)*(P) = rp(Al)(P)*

for almost every p , from Proposition 14.1.8. There is a countable union N o of null sets such that these relations hold on W o for all p in X\N,, that is p + cp(A)(p)is a representation cpp” of W, in W ( Z , , )for p in X\N,. Using Proposition 14.1.9 in this same way, we can locate a null set N , such that - I , I cp(A)(p)I I , for each self-adjoint A in the unit ball of W,

1009

14.1. DIRECT INTEGRALS

and all p in .Y\N,. Thus cp; is bounded on 910 and extends (uniquely) to a representation 'p, of 9l on X p ,for p not in N o u N,. To see that cp,(A) = cp(A)(p)almost everywhere, let { A , } be a sequence in a, such that (\A,- All .+ 0. Then if p $ N o u N , , IIcp,(A,) - cp,(A)II + 0. Employing X0 together with the L,-subsequence, "diagonal" argument of (the first paragraph of) the proof of Theorem 14.1.10, since we have that I(cp(A,)- cp(A)II -+ 0, there is a null set N , such that {cp,(A,)} (= {cp(A,)(p)}) is strong-operator convergent to cp(A)(p) if p $ N , u N , u N , . Thus if p $ N o u N , IJ N , , then cp,(A) = cp(A)(p), cp is decomposable, and p + cpp is its decomposition. If we define p,(A) for p in X\(N, u N , ) to be (cp,(A)x,(p), x o ( p ) ) ,then, with A and N , as above, (cp(A)(P)XO(P),XO(P)) = (cp,(A)X,(P), X d P ) )

for p in X\(No

u N , u N , ) ; and

p ( A ) = (rp(A)xo, xo>

=

b

=

I p , ( A ) dP(P)-

(cp(A)@bOCP),X,(P)) dP(P)

Thus p is decomposable with decomposition p

.+

pp.

14.1.14. DEFINITION.If X is the direct integrai of Hilbert spaces (.Ye,} over ( X , p), a von Neumann algebra W on X is said to be decomposable with decomposition p + 92, when W contains a norm-separable strong-operatordense C*-subalgebra 2l for which the identity representation I is decomposable and such that I,(%) is strong-operator dense in W p almost everywhere. H The lemrna that follows establishes that the decomposition p + W,of W is independent of the C*-subalgebra 9l giving rise to the decomposition. Before proceeding to that lemma, however, we remark on some consequences of the preceding definition. Since I is decomposable, 2I consists of decomposable operators (from Definition 14.1.12). Thus each operator A in 92 is decomposable, from 'Theorem 14.1.10. At the same time, the argument of the first paragraph clf that theorem assures us that A(p) E B p for almost all p . 14.1.15. LEMMA. I f X is the direct integral of Hilbert spaces {Yi",] over ( X , p), 2l and 9?l are norm-separable C*-subalgebras of the algebra of decomposable operators, and 2l- = B - , then '$I =;9 9; almost everywhere, where 21p and g p are the images in 9?l(ZP) of the decomposition of the identity representations of 2I and a.

1010

14. DIRECT INTEGRALS A N D DECOMPOSITIONS

ProoJ: From Theorem 14.1.13, there is a null set N o such that A -+ A(p) and B + B(p) are representations of Q and l B on X p ,when p $ N o . Let 910 and Bobe (norm-)dense denumerable subsets of % and a;and let Ho be a , %- = B - , dense denumerable rational-linear space in X. If B E B ~since there is a sequence { A , ) in the ball of radius ((BIIin Nosuch that A,x -+ Bx for each x in Xo.Again, using an L, - subsequence “diagonal” argument (as in the proof of Theorem 14.1.10), there is a subsequence { A , , } of { A , } and a null set N such that A,,(p)x(p)-+ B(p)x(p)for all x in X, when p 4 N . Let N , be the (countable) union of null sets formed by applying this process to each B in go.Then B(p)€%Z, for all B in go;hence, since such B(p) form a norm-dense subset of B p ,BpP 5 a;,when p # N o u N , . Similarly, there are null sets M o and M , such that %; G B; when p # M o u MI. Thus 9I; = B; almost everywhere.

Once we establish that von Neumann algebras on separable Hilbert spaces have strong-operator-dense C*-subalgebras that are norm separable, combining Theorem 14.1.13 with 14.1.15, we have the following theorem. 14.1.16. THEOREM.I f X is the direct integral of Hilbert spaces { X p } over ( X , p) and W is a von Neumann subalgebra of the algebra of decomposable operators, then 9!? is decomposable with unique decomposition p + 9,. Although it could be proved by introducing an appropriate metric and quoting some elementary results from the topology of separable metric spaces, we make use of operator-theoretic techniques instead to prove the following lemma. 14.1.17. LEMMA. Each von Neumann algebra 92 acting on a separable Hilbert space X contains a strong-operator-dense norm-separable C*-subalgebra. Proof: With (xj} a dense denumerable subset of 2, the vector ( ~ ~ X ~ (~~ ~ ~ ~- x~~ X [ ~~) -, ~isx separating ,,...) for (z 0 @...)(a), where z is the identity representation of W . Thus we may assume that W acting on S has a separating vector; so that each normal linear functional on W has the form O,,~(W.Choose Ajk in (W), so that mxj,xk(Ajk)2 llmxj,xklW(1 - t;and let % be the (norm-separable) C*-algebra generated by { A j k } . If IIo,,,,I% II = 0, Il0~,~l9ll = 1 and llxj - xII, Ilxk - ylI are small, then llmx,ylB- m x j , x k l ~is ll small. But

101 1

14.1. DIRECT INTEGRALS

Hence each normal linear functional annihilating rU annihilates B; and %-=B. a

14.1.18. PROPOSITION. If&? is the direct integral of Hilbert spaces I*,} over ( X , p } , $@ and Y are decomposable von Neumann algebras on X,each containing the algebra %? of diagonalizable operators, with decompositions p + B,, p -,Y,, and A is a decomposable operator on y14 then A E Y if and onZy if A ( p ) c i Y , almost everywhere. If 9,= 9,almost everywhere, then 9= Y . Proqf: Suppose, first, that A 2 0, A ( p )E 9, almost everywhere, and A is not in 9'.The Hahn-Banach theorem ("separation" form) applied to a(&?) with its (locally convex) weak-operator topology, provides us with a hermitian normal linear functional p on a(%)annihilating Y but not A . From Theorem 7.4.7, there are positive normal functionals p + and p - such that p = p t - p - . Since p l Y = 0, p t IY = p - 1 9 . From Theorem 7.1.12, there are countable families { x . } , { y , } of vectors in &? with llxnllZ and C IJyn(12 finite such that p + = wXnand p - = C my,. If H is a positive operator in 9,

1

c

En"=

In"=

where f : p -, (H(p)x,(p), x , ( P ) ) and 9 : P -, 1 (H(ply,(p), yn(p)>are understood its the L,-limits of the finite partial sums (which form Cauchy sequences of positive, integrable functions on X ) . At the same time, these sums converge almost everywhere (since a subsequence of each of these sequences does and each sequence is monotone increasing). As Y contains %?, we can replace H by its product with the diagonalizable projection corresponding to a measurable subset X , of X (see Remark 14.1.7). In this case, we have jx,f(P)dp@) = Jx,g(p) dp(p). Thus f = g almost everywhere. Since the null set involved in this last equality varies with H , we cannot assert that w(,)I Y , = (En"=wynCp,) I Y , at this point. However, since 9' is decomposablle, it contains a norm-separable C*-subalgebra 'i!lsuch that B -,B(p) is a representation of 2I on X, with range %, strong-operator dense in 9, for alniost all p . If { H j } is the denumerable set of positive operators in % obtained by expressing each operator in a (norm-)dense denumerable

(En"=

1012

14. DIRECT INTEGRALS AND DECOMPOSITIONS

subset of 'u as a linear combination of (four) positive operators, then, except for p in some null set N , m

OD

C1 < H j ( p M p ) ,xn(P)) = 1 n= 1

n=

(c

(c

for all j. Since the functionals on B(X,) involved in this last equality are oxn(,JIY, = o,,(pJl.4pp for p not in N . Now A(p)E Y , almost normal, everywhere, by assumption, whence we have that Z,"=,(A(p)x,(p), xn(p))= (A(p)y,(p), y,(p)) almost everywhere. Integrating over X , this yields p + ( A ) = w,,(A) = @,,,(A) = p - ( A ) ; so that p ( A ) = 0-contradicting the choice of p . Thus A E Y .The last assertion of the statement follows from this. If A is a self-adjoint decomposable operator such that A(p)E Y, almost everywhere, we have that A = IIAlll - (IIAllZ - A ) and that A(p) = J)AJ)Z,- (l)AllZp- A(p)) almost everywhere. Hence, from the preceding argument, JIAJIZ- A E Y . Thus A E Y .

1

1

14.1.19. LEMMA. Zf &' is the direct integral of Hilbert spaces (XP) over ( X , p), 9 is a decomposable von Neumann algebra on X, and o is a normal state of 9, then there is a mapping, p -,up,where o,is a positive normal linear functional on B p ,and o ( A ) = jxwp(A(p))dp(p)for each A in 9. If W contains the algebra W of diagonah'zable operators and W I E Z E is faithful or tracial,for some projection E in 9, then opJ E(p)W,E(p) is, accordingly,faithful or tracial almost everywhere. ProoJ: From Theorem 7.1.12, there is a countable set of vectors (y,] in

H such that o ( A ) = 1(Ay,, y,) for A in W and

1 IIY,IlZ

n=

1

c."=

c m

00

1=

=

n= 1 I X

(YAP), Y A P ) ) dP(P).

It follows that (yn(p), y,(p)) is finite almost everywhere; so that A, + ,
p =) =

[(f 2 X n=l

I=1

)

(A(p)yn(p),Y n b ) ) dP(P)

jx(A(p)Yn(P), Y n b ) > d p b ) =

W

1 = o ( ~ )

n= 1

14.1. DIRECT INTEGRALS

1013

for each positive A in 9. By expressing an arbitrary operator in 92 as a linear we have the same equality, now, combination of four positive operators in 9, for each A in 9, If $2 c 9., then R c W; so that 22' is a decomposable von Neumann algebra. (See Theorem 14.1.10.) Let { A ; } be a denumerable strong-operatordense subset of 92'.Let { E , A,} be a norm-dense subset of a strong-operatordense norm-separable C*-subalgebra CU of 9.If w l E 9 E is faithful, then (ALEy,} spans E ( X ) , for w annihilates the projection (in E g E ) on the orthogonal complement of this span in E ( S ) . Now &(p) commutes with CUP for almost every p ; so that { A ; @ ) } and 9,commute almost everywhere. At the same time, {Ah(p)E(p)y,(p)} spans E ( p ) ( X , ) almost everywhere. To see this, choose a denumerable set (xk} spanning ( I - E ) ( X ) ; so that {xk, A k E y , } spans S.From Lemma 14.1.3, {xk(p),Ak(p)E(p)y,(p)} spans 2, almost everywhere. Since Ex, = 0 for all k , E(p)xk(p) = 0 for all k, almost everywhere. Thus { Ak(p)E(p)y,(p)} spans E ( p ) ( S p ) almost everywhere. Hence there is a null set N such that {R,E(p)y,(p), n = 1, 2, ...I spans E(p)(X,,} when p $ N . If H , is a positive operator in E ( p ) B , E ( p ) such that wP(HP)= 0 for some p not in N , then 0 = H,y,(p) = H,E(p)y,(p) for all n, so that HPB)6E:(p)y,(p)= 0 and 0 = H , E ( p ) = H , . Thus o,lE(p)B,E(p) is faithful when p $ N . With w I E 9 E tracial and P the diagonalizable projection corresponding to a measurable subset X , of X (see Remark 14.1.7) we have

= op('(P)A,(P)E(p)An(P)E(p)) almost Thus op(E(P)An(P)'(p)Am(P)'(P)) everywhere--first for the given n and m, then for all n and m. Since apis a normal state and E(p)YIU,E(p) is strong-operator dense in E(p)W,E(p), o,I E(p)B',Ejp) is tracial almost everywhere.

The mapping p -,a,,of the preceding theorem, gives rise to a mapping p -,o',, where o;= wPoI , and 1 is the identity representation of 9 on X. Although decompositions of states are defined for norm-separable C*algebras (Definition 14.1.12), the mapping p --f o',is, in effect, a decomposition of the normal state o into positive normal linear functionals o;.

1014

14. DIRECT INTEGRALS AND DECOMPOSITIONS

14.1.20. LEMMA. l f ,X is the direct integral of Hilbert spaces {X,}over ( X , p), W is a decomposable von Neumann algebra on X, and E is a projection in 9, then the following assertions hold almost everywhere:

(i) E(p) is a projection in 9,; (ii) if E is in the center of 9, then E(p) is in the center of ap; (iii) if E F in 9, then E(p) F ( p ) in 9,; (iv) if E is abelian in W,then E(p) is abelian in W p ; (v) CdP) = CE(p,; (vi) if E is properly injinite in B? and C , = I , then E(p) is properly injinite in B?,; (vii) if E is finite in 9 and 9 contains the diagonalizable operators, then E(p) is jinite in 9,.

-

-

~ E(p) = E(p)* almost Proox (i) Since E2 = E = E*, we have E ( P ) = everywhere, from Proposition 14.1.8;so that E(p) is a projection in B?, almost everywhere. then E(p)T(p)= T(p)E(p)almost every(ii) If ET = T E with T i n 9, where, again, from Proposition 14.1.8. Allowing T to take on a denumerable dense set of values in a strong-operator-dense norm-separable C*-subalgebra '2l of 9,we have that E(p) commutes with 21p almost everywhere. Thus E(p) is in the center of 9,almost everywhere, if E is in the center of 9. (iii) If there is a partial isometry V in W such that V * V = E and V V * = F , then V*(p)V(p) = E(p) and V(p)V*(p)= F(p) almost everywhere, from Proposition 14.1.8. Thus E(p) F ( p ) in 9,almost everywhere. (iv) If E is abelian in 9 then EAEBE = EBEAE for each A and B in 9. Thus E(P)A(P)E(P)B(P)E(P)= E(P)B(P)E(P)A(P)E(P) almost everywhere. Letting A and B take on a denumerable (norm-)dense set of values in the we conclude strong-operator-dense norm-separable C*-subalgebra 2I of 9, that E(p)21pE(p)is abelian almost everywhere. Hence E(p)WpE(p) is abelian almost everywhere; and E(p) is an abelian projection in g Palmost everywhere. (v) From Proposition 5.5.2, the range of C , is [ B E ( X ) ] .With { x j >a denumerable set spanning E ( 2 ) and {A,) a denumerable strong-operatordense subset of 9, {Anxi)spans C,(X). Using Lemma 4.1.3 and arguing as in the second paragraph of the proof of Lemma 14.1.19, we have that {xj(p)} spans E(p)(X,) and {A,(p)xj(p)}spans C,(p)(SP), almost everywhere. If 2I is the norm-separable C*-subalgebra of 9 generated by { A n } , then {A&)} generates 21p and 2I; = 9, almost everywhere. Thus (A,(p)xj(p)} spans C E ( p ) ( Xalmost p) everywhere; and C,(p) = C,(,, almost everywhere. (vi) If E is properly infinite in 9 and C , = I , there is a subprojection F of E such that E F E - F i n 9, from Lemma 6.3.3. Thus E(p) F(p) E(p) - F(p) in W,, almost everywhere, (from (iii)) and either E(p) = 0 or E(p)

-

- -

- -

14.1. DIRECT INTEGRALS

1015

is properly infinite, almost everywhere. But C , = I , so that I , = C,(p) = CE(p) almost everywhere (from (v)). Thus E ( p ) is properly infinite almost everywhere. (vii) If E is finite in W,then EWE admits a faithful normal tracial state. Composing this state with the mapping A -+ E A E yields a normal state w of 9 that is both tracial and faithful on EWE. If R contains the diagonalizable operators, Lemma 14.1.19 applies, and w,lE(p)W,E(p) is a faithful tracial state for almost all p . Thus E(p) is finite in W,for almost all p . In the next theorem, we identify the types of the components in the decomposition of von Neumann algebras of various types. For unity of statement, we have included the algebras of type 111, although the proof that we give that the components are of type 111, in this case, requires a special technique whose development we postpone to the discussion after the theorem. For the proof of the following theorem, we shall make use of some simple observations concerning decompositions of subspaces. If 2 is the direct integral of (.ui",} over ( X , p), W is a decomposable von Neumann algebra on 2' with decomposition p -+ 9 p ,and E is a decomposable projection, then E ( 2 ) has a direct integral decomposition and ERE is decomposable relative to it with decomposition p -+ E(p)B',E(p). To see this, note that E E W , where V is the algebra of diagonalizable operators on #; so that E has a central carrier relative to 59' in %. This central carrier is the diagonalizable projection corresponding to a measurable subset X , of X ; and E ( 2 ) is the direct integral of { E ( p ) ( I x p ) }over ( X , , p). (Although X , is a measurable subset of X and not itself locally compact, our results apply without change to this situation; arid we can speak of direct integral decompositions over ( X , , p). Indeed this direct integral is essentially the same as that over ( X , p o ) where po( Y ) = p ( Y n X , ) for a measurable subset Y of X . ) If Tis decomposable on Ix, then E 7 E is decomposable on E ( 2 ' ) (with decomposition p-' E(p)T(p)E(p)).If So is decomposable on E ( 2 ) , then S on 2,agreeing with So on E ( 2 ) and 0 on ( I - E ) ( 2 ) is decomposable (with decomposition S(p) = So@)for p in X , and S(p) = 0 for p in X\X,) and E S E = S . Thus EWE is the algehra of decomposable operators on E ( 2 ) and %'E is the algebra of diagonalizable operators. Moreover, the decomposition of EWE is p E ( P ) ~ ~ J ?If( %' ~ )s. W,then %E c E 9 E . -+

14.1.21. THEOREM.If 2 is the direct integral of Hilbert spaces (2,) over ( X , p ) and 9 is a decomposable von Neumann algebra containing the diagonalizable operators, then 9 is of type I,, II,, II,, or 111 if and only if, correspondifilgly,a, is of type I,,, II,, II,, or I11 almost everywhere.

1016

14. DIRECT INTEGRALS AND DECOMPOSITIONS

Proof. If W is of type I, (n possibly infinite), there are n orthogonal equivalent abelian projections E l , ..., En in 9 with sum I. From Lemma 14.1.20 and Proposition 14.1.8, E,{p) is abelian in W,,CEj(,)= I , , E k p ) E,(p), and E,(p) = I, almost everywhere. It follows that .9, is of type I, almost everywhere. If 9 is of type II,, then from Lemma 14.1.20(vii), I, is finite in 9,almost everywhere. An infinite orthogonal family of projections in 9,each with central carrier I, gives rise to an infinite orthogonal family each with central carrier I, in W,almost everywhere. Thus a, is of type 11, almost everywhere. If 9 is of type II,, then from Lemma 14.1.20(vi), I , is properly infinite in 9,almost everywhere. If E is a finite projection in 9 with central carrier I , E has an infinite orthogonal family of subprojections each with central carrier I. Thus E(p) is finite in 9, and admits an infinite, orthogonal family of subprojections with central carrier I, almost everywhere. Thus W , is of type 11, almost everywhere. Assuming, as we shall at this point (see the discussion following the proof of Proposition 14.1.24), that 3, is of type 111 almost everywhere when 92 is of type 111, there is no difficulty in showing that the types of 9, determine that of 9. If 9,is of type either I,, or II,, or II,, or 111, almost everywhere and W has a central projection Q such that W Q is not of the corresponding type, then, from the discussion preceding this theorem (WQ), is Q(P)a,Q(P). From (ii) of Lemma 14.1.20, Q(p) is in the center of W, for almost all p ; so that Q(p)g,Q(p) = 9,Q(p) almost everywhere. Thus W Q acting on Q(Z) is a decomposable algebra of one type, containing the diagonalizable operators, whose components W,Q(p) are of a different type almost everywhere. But this contradicts our assumption in the type IT1 case and what we have established in the other cases. Thus W has the same type as its components. W

,

-

We have deferred the proof that a type I11 von Neumann algebra containing the diagonalizable operators has type I11 components in its decomposition until after the proof of Proposition 14.1.24. For its proof, we use a special type of argument that must wait until some preliminary results have been developed. The argument involves a measurable “selection” or “cross-section” principle. This principle entails an excursion into an area we have not encountered thus far and requires methods for its proof foreign to those we have been using. For this reason, we have placed the discussion of this principle in an appendix to this chapter (Section 14.3). The measurable-selection principle makes it possible for us to use a very natural (and powerful) strategy of proof in decomposition theory (that we have, nevertheless, avoided until now). This strategy is best described with an is the decomposition of 2 over (X, p). If W is a illustration. Suppose (2,) decomposable von Neumann algebra containing the algebra of diagonaliz-

14.1. DIRECT INTEGRALS

1017

able operators, the same is true of 92’.Is (B’), equal to (a,)’almost everywhere? The affirmative answer to this question is the substance of Proposition 14.1.24.Let us consider how we might prove it. If { A j }and ( A ; } are denumerable families of operators, strong-operator dense in 9 and w’, respectively, then, except for p in a null set N , all A&) commute with all A>@).Thus (w’),E (a,)’for p not in N . (This argument has appeared in the =’ (B’),,let A ; be 0; otherwise choose A; in proof of Lemma 14.1.19.)If (9,) (.%!,)’\(w’),.Suppose that we can make our choice of A’, in (B,)’\(a’), in a measurable manner-specifically, so that A; = A’@) almost everywhere for some decomposable A’. Then A’(p) commutes with all A,@) almost everywhere; so that A’ commutes with all A j and A ’ E ~ ’ .But, then, A’(p) = A ; E (B’)almost , everywhere; and (92,)’= (B’)almost , everywhere. The question of the possibility of making a “measurable selection” of A; in (B,,)’\(w’),masks an additional problem. Is the set at which (9,) =’ (w’),, a measurable set? The application of the measurable-selection principle entails establishing that this set is measurable. The topics presented in Section 14.3 supply the techniques for proving this measurability as well as the measurable-selection principle. What is needed is the fact that an “analytic subset” of X (the continuous image of a complete separable metric space) is p-measurable. The selection principle amounts to the fact that, if we of a complete associate with each p in a subset X , of X a non-null subset 9, separable metric (csm or Polish) space (one with a denumerable base admitting a complete metric)-usually the unit ball in B(X)-and the set Y of pairs ( p , A ) with p in X , A in 9,is an analytic set, then there is a measurable mapping, p -+ A,, from a measurable subset of X containing X , into the csm space such that A , E Y , . (The mapping “selects” A , from the non-null sets 9,in a measurable manner.) The precise details and proofs appear in Section 14.3. In order to apply these measurability and measurable-selection techniques to our direct-integral problems, we must transfer our decomposition {.Ye,} over ( X , p ) to a single Hilbert space X in a measurable manner and develop the notion of measurable mappings from X into X and A?(X).We carry out this program in the following few paragraphs. The lemma that follows prepares us for the transfer to a single space X. 14.1.22. LEMMA. I f X is the direct integral of Hilbert spaces { X p over ) ( X , p), then the algebra 9of decomposable operators is of type I,, if and only if X, is n-dimensional almost everywhere. If B is of type I,, and { E j :j = 1,. . . ,n} is an orthogonal family of abelian projections with central carriers I and sum I , then, with xi a generating vector for E j under w’(= U), xJ@) # 0 and {xJ{p)) is an orthogonal family of vectors that span X, for almost all p .

1018

14. DIRECT INTEGRALS AND DECOMPOSITIONS

Proof: If .9is of type I, and the family { E j :j = 1 , . . . ,n) is as in the statement of this lemma, then (EJ(p)} is an orthogonal family of abelian projections with central carriers I , and sum I , for almost all p . In particular, Zp has dimension not less than n for almost all p . We show that the dimension of 2, does not exceed n, for almost all p . by establishing the last assertion of this lemma. With j distinct from k, let X , be the set of p such that (xJ(p),x k ( p ) ) # 0. If Q is the diagonalizable projection corresponding to a measurable subset Y of X,, then

0 = (QEjxj,

Ekxk)

=

(Qxj,

xk)


= J-Y

Since this holds for every measurable subset Y of X , , (xJ(p), x k ( p ) ) = 0 almost everywhere; and X , is a null set. Thus {x,(p)} is an orthogonal family for almost all p . Let { C j } be a denumerable strong-operator-dense subset of %. Since x f generates Ek under V, { C j x k:j = 1, 2,. . .} spans &.(a?). As { E k } has sum I . ( c j x k : j , k = 1,2,. ..} spans X. Thus (CJ(p)x,(p):j , k = 1,2,. . .>spans X pfor almost all p , from Lemma 14.1.3. Except for p in some null set CJ(p)is a scalar multiple of I , for all j (since C j is diagonalizable). Thus {xJ(p)}spans X, for almost all p ; and X, has dimension n. Since x j generates E j and C E j= I , the range, [W%xj] (= [Wxj]), of CE,is Y?' (see Proposition 5.5.2). Thus x is separating for V (see Proposition 5.5.11 in conjunction with Theorem 14.1.10). It follows (from Lemma 14.1.19, for example) that x i ( p ) # 0 almost everywhere. Suppose, now, that Z phas dimension n for almost all p . Since al = V and % is abelian, W is of type I (see Theorem 9.1.3) and there are central projections P , such that %'P, is of type I , or P , = 0 (see Theorem 6.5.2). Assume P , # 0 for some m.Since P , E V, P , is the diagonalizable projection corresponding to some measurable subset X , of X of positive measure (see Remark 14.1.7). In this case, P , ( Z ) is the direct integral of {X,} over ( X o ,p), V P , is the algebra of diagonalizable operators, and %'P, is the algebra of decomposable operators, relative to this decomposition of P , ( Z ) . Since 9 P m is of type I,, from what we have just proved, Xpis rn dimensional for almost all p in X,. By assumption, rn = n. Thus P , = 0 unless rn = n ; and W is of type I,. With the notation of Lemma 14.1.22, we carry out the transfer of the decomposition {Yt",} to a single Hilbert space X of dimension n. Let { y j } be an orthonormal basis for X, and let N be a null set X such that xJ@) # 0 for all j and (xJ(p)}is an orthogonal family of vectors spanning X, for each p in

1019

14.1. DIRECT INTEGRALS

X\N. There is a unique unitary transformation Upof 2,onto X, for p not in N , such that U,xj(p) = Ilxj(p)IIyj for all j . With x in 2,

and \

n

If y is a vector in X, then ( U , x ( p ) , y ) = , ( y j , y ) a j ( p ) for each p not in N . Each function p + (yj, y)a,(p) is measurable, so that p + ( U , x ( p ) , y ) is the (finite) limit almost everywhere of measurable functions. Thus p + ( V , x ( p ) , y ) is measurable. We say that the mapping p + V , x ( p ) of X into X is weakly mecuurable in this case (that is, when the functions p + (U,x(p), y ) are measurable for all y in X ) . At the same time, p -+ U,A(p)U;' is a weakly ) each decomposable A on 3 (that is, measurable mapping of X into @ ( X for p + U , A ( p ) U ; l y is weakly measurable for each y in X ) . To see this, note that it suffices, for weak-measurability of a mapping p + z(p) of X into X, to verify that p + (z(p), y j ) is measurable for each y j in the orthonormal basis ( y j } ;for, as above, p + (z(p), y) is, then, the (finite) limit almost everywhere of measurable functions on X . In the same way, it suffices, for weakmeasurability of a mapping p + B , of X into 9 ( X ) , to verify weakmeasurability of the functions p + B,yj for each j , and, hence, of (B,y,, y , ) for each pair j , k. In the present case, ( U p A @ ) U , l Y j , yk)

= (Ilxj(P)ll

'

Ilxk(p)ll)-l(A(p)xj(p),x k ( P ) ) ;

so that p + l/,A(p)U; is weakly measurable. It will be useful for us to note, in passing, that p-+B,* is weakly measurable if p -+ B, is (clear, from the definition) and that p + TpSpis weakly measurable if each of p + T, and p + S, is. For this, we observe that we can recast the criterion for weak-measurability of p + B,. Let B , y j be b,(p)y,. Then p + ( B , y j , yk) ( = b&)) is measurable; so that p -+ B, (and, by the same argument, p + z(p)) is weakly measurable if and only if each of the functions b, is measurable (the functions a&), where z ( p ) = ,a,(p)y,., in the case of p -+ z(p)). It follows that p -+ z ( p ) and p + B , are strongly measurable (that is, measurable as mappings into X in its metric and B ( X ) in its strong-operator topology) if and only if they are weakly measurable. We say, henceforth, that such mappings are measurable. Note, too, that p + (u(p), u ( p ) ) is measurable if p + u(p) and p + v ( p )are; for it can

1020

14. DIRECT INTEGRALS AND DECOMPOSITIONS

be factored as p -+ (u(p),v(p)) + (u(p), v ( p ) ) , a measurable mapping of X into X x X (since X is separable) followed by a continuous mapping of X x X into C. Returning to the mappings p -+ T, and p S,, we see that if = r m = 1 sjm(P)Ym and T z Y k = r m = 1 t,*,(p)ym,then ---f

rm=

Since p -+ sjm(p)t,*,o is a measurable function (finite almost everywhere) p + S , Tpis measurable. In the case of a general direct-integral decomposition {%,} of 31” over ( X , p), the set X , of points p such that % , has dimension n is measurable (see Remark 14.1.5) and corresponds to a diagonalizable projection P, such that 9 P n is of type I , (if X , is not a null set) and is the algebra of decomposable operators arising from the decomposition of P,(%) as ( X p }over ( X , , p). Associated with this decomposition is the family of unitary transformations U,,, p in X,, of %, onto a fixed Hilbert space .X, of dimension n. If we form X, the direct sum of X,, n = 1, 2,. . . , K O (where X,is absent if X , is a null set), then the total family { U,, :n = I, 2,. ..,K O ;p E X > transfers the constructs of this general decomposition of % onto the one space X in a “measurable” manner. We summarize this discussion in the lemma that follows. 14.1.23. LEMMA. If 31“ is the direct integral of the Hilbert spaces {2,,} over ( X , p) and X, is the set of points p in X at which 2,has dimension n, then, fi X , is not a null set for p, the diagonalizable projection P, corresponding to X , is the maximal central projection in 8, the algebra of decomposable operators, such that WP, is of type I,. In this case, there is a family { U , , : P E X,} such that U,, is a unitary transformation of 8,onto a jixed Hilbert space X, of dimension n. I f X is the direct sum of those X,such that X , is not a p-null set and U p is U,, when P E X , , , then { U p } is a family of unitary transformations such that U p maps 2, into N,p + U,x(p) is measurable for each x in H,and p + U,A(p)U,* is measurable ,for each A in 9.

We illustrate the actual technical use of the measurable-selection principle by giving the full argument for the result on decomposition of commutants whose proof was sketched prior to Lemma 14.1.22. For the remainder of this chapter, the assumption that X is metrizable as a csm space is in force. 14.1.24. PROPOSITION. If 3E” is the direct integral of {31”,) over ( X , p) and

92 is a decomposable von Neumann algebra on X containing the algebra %?of

1021

14.1. DIRECT INTEGRALS

diagonalizabk operators, then w’ is decomposable and (B’),= (B,)‘ almost everywhere. Proof: Let { A j } and { A ; } be denumerable strong-operator-dense subsets respectively. Let { y j } be a denumerable of the unit ball in each of 9 and 9, dense subset of 2.To employ the techniques of Section 14.3, we introduce the Hilbert space X of Lemma 14.1.23 and the family of unitary transformations described there. We use the unit ball Bl of B ( X ) provided with its strong-operator topology and the translationally invariant metric, d(S, T ) = c’=12--ll(S -- T)ejll, where { e j > is an orthonormal basis for X, which is (uniformly) compatible with this topology, as our csm space. Since %? E B, 3’ is decomposable. Let F , be the orthogonal projection of X onto X,. We consider the set of pairs ( p , A ) in X x B, satisfying the following conditions: (i) AU,A,U,* = U , A j ( p ) U , * A , j = 1, 2,... , EX,, and F,AF, = A ; (ii) there are positive integers m and h such that, for eachj, there is a positive integer k not exceeding h for which (*)

ll(A - L‘pA~(P)U,*)Upyk(p)ll 2 llm,

PEX.,

and

FJF,

= A.

Except for p in a Borel p-null set N o in X , a pair ( p , A ) satisfies (i) if and only if EX, and A E U,(W,)’U,* and satisfies (ii) if and only if F , A F , = A , P E X,, and A $ UP(9Y),U;. We may also assume that (X\N,) n X , are disjoint for Since p -+ U,Aj(p)U,*,p -+ U,Aj(p)U,*, different n and have union X\N,. and p + U p y i p )are measurable for allj; there is a Borel p-null set N , such that, restricted to X\N,, these mappings into Bl and X are Borel for a l l j (see Lemma 14.3.1). Let X , be X\(N, u Nl). It follows that the mappings (p, A ) + U,A j(p)U,*A and ( p , A ) -+ AU,A,(p)U,*, from X , x Bl, with its (product) topological Borel structure, to Bl, are Borel mappings. Indeed, they can be factored as the composition of the strong-operator continuous mappings (U,Aj(p)U,*,A ) .+ U,A,(p)U,*A and (U,Aj(pW,*, A ) AU,Aj(p)U,* of B1x B l into Bl with the mapping ( p , A ) + (U,Aj(p)U,*,A ) from X , x BI into Bl x Bl; and this last mapping is Borel relative to the (product) topological Borel structures since the strong-operator topology on Bl has a countable base by virtue of the separability of X. The subset Y j ,of ( X , n X,) x (F,991F,) where these mappings agree, for a given j, is a Borel (=9;), the subset of X , x B , fulfilling condition (i). set, as is ny=lYjn ( = 9’) is the Borel subset of points ( p , A ) in X , x Bl such Again, uF= 9’; that A E Up(Bp)’Up*. Similarly, the subset 9 j k m n of ( X , nX,) x (f’,BIF,) satisfying (*), for givenj, k, m, and n, is a Borel set, since ( p , A ) -+ (U,Axp)U,*, A , U,yk(p)) iS a Borel mapping of X , x Bl into Bl x B l x X and the mapping +

1022

14. DIRECT INTEGRALS AND DECOMPOSITIONS

6 nu m

h

h.n,m=l j = 1 k = l

Yjkmn

is the subset Y “ of pairs ( p , A) in X , x @, such that, for some n, FnAF, = A, p E X n and A 4 Up(.@’),U,*.Thus Y‘ n Y” ( = Y )is the subset of pairs (p, A) in X , x B , such that A E (U,(W,)’Up*)\(Up(B’)pUp* and is Bore1 subset of x x 39,. From Theorem 14.3.5, X x Bl is a csm space and Y is an analytic subset of it. The image X, of Y under the projection mapping X x B1onto its X-coordinate is (an analytic, hence, measurable subset of X and is) precisely =) U,(W,)’U,*\U,(W’),U,* # 0. the set of points p in X , for which (9, From Theorem 14.3.6 there is a measurable mapping p + U,A’,Uf from X, into B1such that U,A’,U,*EF,. Then A; (= UfU,A‘, UfU,)E(~,)’\(B’), for p in X , . Defining A’, to be 0 for p in X\Xl, with x and y in H,we have

so that p

-+

=

(UpA’,Up*U,x(P), U,Y(P)>;

(A’,x(p), y(p)) is measurable. Since l
U,Y(P))l 5 IIx(P)II. IIY(P)IL

p + (Abx(p), y b ) ) is integrable. It follows that A>@) = (A’x)@) almost everywhere, for some A’x in 2,from Definition 14.1.1. Moreover, A‘ is linear, since each A‘, is linear; and r

r

so that IlA’II I 1. It follows that A’ is a decomposable operator on 2 such that A’(~)E(W,>’\(B’), for all p in X,. But, in this case, A’(p)Aj(p)= A,(p)A’(p) almost everywhere for allj. (Recall that A’@) = 0 for p not in X , ) . Thus A’A, = AjA‘ for all j, and A’€%”. From the discussion following E for almost all p . Hence X is a pnull set, and Definition 14.1.14, A ’ ( ~ ) (g), (9,= )’ (W’), almost everywhere.

,

We complete the argument of Theorem 14.1.21 by showing that if a decomposable von Neumann algebra W of type I11 contains the algebra V of diagonalizable operators on a Hilbert space 2,the direct integral of {X,}

1023

14.1. DIRECT INTEGRALS

over ( X , p), then W, is of type 111 almost everywhere. Since % G W and W G %' (9is decomposable), '%5 is contained in the center of B. Since 92 is of

type 111, the algebra %' of decomposable operators on X must be of type I,; for a non-zero, central projection P in W lies in %? (the center of V'). Hence P lies in the center of R. Thus P is infinite in 9, and, therefore, in %'. Employing the transfer process from {X,,)to a Hilbert space X (Lemma 14.1.23), since %?' is of type I,, we can use a single infinite-dimensional X rather than a direct sum. Let { U p: p in X } be the family of unitary transformations U p of X, onto X with the properties noted in Lemma 14.1.23.Let .gobe the unit ball in the set of self-adjoint operators in g ( X ) provided with its strong-operator topology (in which it is a complete separable metric space). Since W is of type 111 and &f is separable, there is a unit separating (and generating) vector y o for R (see Proposition 9.1.6). It follows (from Lemma 14.1.19,for example) that yo(p) is separating for W,,except for p in a null set N o . Let { A j ) .and { A ; } be denumerable strong-operator-dense subsets of the intersection of the unit ball in g ( X )with the self-adjoint operators in W and w',respectively. Then { A j ( p ) }and {A>(p)}generate BPand, from Proposition 14.1.24, RP, respectively, except for p in a null set N , . Let { y j } be a denumerable dense rational-linear subspace of X, so that (y,(p)> is such a except for p in some null set N , . We consider the pairs ( p , E ) subspace of 8,, in X x Bo satisfying the following three conditions: (i) E I= E 2 # 0. (ii) EU,,AJ(p)U; = UqA>(p)U;E,j = 1, 2,. . . . (iii) For each positive integer n, there is a positive integer rn such that

I(

up

-

j(P>u,*E

(

up

< II EU,Y,(P)lI n

upYm(P),

up

U,*Eu p Y r n ( P ) ,

U,*EUpYrn(P)> U p A j(P)

u;

vpYrn(P)>

I

for j , k = l , 2,....

If p in X\N, where N = N o u N , u N , , and E in gosatisfy (i), (ii), and (iii), then E is a projection, from (i) (since B0 consists of self-adjoint operators), E E U,W,U;:, from (ii) (since {A>(p))generates aP), and EU,W,U,*E is not properly infinite, from (iii). To see this last, note that, from (iii), the vector state of EU,B,,U;E associated with EU,y,(p) approximates a trace to within l/n since (EU,A,(p)U,*E} is strong-operator dense in the set of self-adjoint elements in the unit ball of EUpBpUZE.Passing to a limit (of some subnet of these states in the weak* compact unit ball of the dual of EU,W,UBE), we see that EUpWpU,*Epossesses a tracial state p and E is not properly infinite.

1024

14. DIRECT INTEGRALS AND DECOMPOSITIONS

Conversely, if E is a non-zero projection in U,W, U,* that is not properly infinite, there is a normal tracial state of EUP9,U,*E (compose the centervalued trace of Theorem 8.2.8 on a finite summand of EU,%UU,*E with a vector state, for this), which must be a vector state, from Theorem 7.2.3, since EU,y,(p) is separating for EU,B,,U,*E, when p # N o , If we choose {y,(p)}, a sequence of vectors in {y,(p)) such that {U,y,(p)} tends to a vector corresponding to that vector state (this is possible when p $ N 2 ) , we see that p (in X\N) and E satisfy (i), (ii), and (iii). Thus (i), (ii), and (iii) determine precisely the set Yo of points p in X\N at which Bp contains a non-zero projection that is not properly infinite, that is, at which 9,is not of type such that p and E satisfy (i), (ii). 111-equivalently, at which the set of E in go, and (iii), is non-empty. To apply the measurable-selection principle, we shall locate a Borel (hence, analytic) subset Y of X x W,consisting of pairs ( p , E ) satisfying (i), (ii), and (iii), such that the image of Y under the X-coordinate projection is an analytic set X , differing from Yo by a p-null set. Suppose, for There is, then, a measurable mapping the moment, that we have found 9. p + U,E,,U,* from X I into 9,such that p and U,E,U,* satisfy (i), (ii), and (iii). Defining E , to be 0 for p in X\X,, we have, precisely as at the end of the proof of Proposition 14.1.24, that E , = E(p) almost everywhere for some decomposable operator E on H.Since E(p) is a projection in 92, almost everywhere, E is a projection in 9 (from Proposition 14.1.18, by forming the von Neumann algebra generated by 9 and E). Since 9 is of type 111, either E = 0 or E is properly infinite. If E = 0, E(p) = 0 almost everywhere, X, is, is of type 111 almost everywhere. If E is properly therefore, a p-null set, and 9, infinite, there is a projection F in 9 such that F < E and E F E - F , from which, F ( p ) IE ( p ) and E(p) F(p) E(p) - F(p) (see Proposition 14.1.9 and Lemma 14.1.20). In this case, E(p) is either 0 or properly infinite almost everywhere. In either event, X I is a p-null set, and 9,is of type I11 almost everywhere. It remains to locate a Borel subset Y of X x gowith the properties noted. Using Lemma 14.3.1 let X, be a Borel subset of X such that the measurable mappings p + U,Aj(p)U,*, p + U,AJ(p)U,*,and p + U,y,(p) are Borel mappings on X, for all j and such that X\X, is a p-null set containing N . Since (p, E ) + E and ( p , E ) + E 2 are continuous mappings of X, x go into go,the points at which they agree form a Borel subset Yoof X, x go; and Yb\(Xo x (0))is a Borel subset Y oof X, x go.The pairs in Y oare those at which (i) is satisfied. Just as in the proof of Proposition 14.1.24, the in X, x W,satisfying (ii) is a Borel set. Again, the inequality set of pairs 9, of (iii) determines, for each j , k, n, and rn, a Borel subset y j k n m , and

- -

n u ii m

m

n = l m=1 j.k=l

Yjknrn

- -

14.2. DECOMPOSITIONS RELATIVE T O ABELIAN ALGEBRAS

1025

is the Bore1 set Y2 of points of X , x @I, satisfying (iii). Let Y be Y , n 9, n 9,. Then Y is precisely the set of points in X , x satisfying (i), (ii), and (iii). The image X I of Y under the X-coordinate projection is an analytic set differing from Yo by a p-null set. Bibliography:

[21, 54, 671

14.2. Decompositions relative to abelian algebras In Section 14.1, we developed the basic theory of Hilbert spaces that are direct integrals (as characterized in Definition 14.1.1) and studied the resulting decompositions of decomposable operators (Definition 14.1.6) and decomposable von Neumann algebras (Definition 14.1.14). In the present section, we discuss the possibility of recognizing a given separable Hilbert space as a direct integral and a given von Neumann algebra on it as decomposable. More precisely, we shall begin by asking when X is the direct integral of spaces { X p }in such a way that a given abelian von Neumann algebra d on it appears as the algebra of diagonalizable operators. We shall see that this is always the case (Theorem 14.2.1) and note the details of the construction expressing 2 as such a direct integral. Theorem 14.1.10 tells us, then, that d'is the (von Neumann) algebra of all decomposable operators on X relative to this direct integral decomposition; and Theorem 14.1.16 tells us that a von Neumann algebra is decomposable if and only if it is a subalgebra In this perpsective, given an abelian von Neumann algebra d on X of d'. and a von Nmmann subalgebra 9 of d' (equivalently, 9containing d), we may ask ourselves about the effect of a special relation between d and 93 on the components g Pof the decomposition of 9. For example, if .c9is the center can anything specific be of W,or d is a maximal abelian subalgebra of 9, said about W p ?We shall see (Theorems 14.2.2 and 14.2.4) that d is the center of W if and only if 9,is a factor almost everywhere, and d is maximal abelian in W'if and only if W pis 9#(Xp)almost everywhere. The prohlem of expressing 2 as a direct integral of Hilbert spaces is largely one of identification. The serious work was done in Section 9.3 when we described the spatial action of type I von Neumann algebras and in Section 9.4 when we described the maximal abelian algebras on separable Hilbert spaces. To illustrate this, we begin by considering the simplest instance. Suppose d is a maximal abelian subalgebra of @I(X).From Theorem 9.4.1, d is unitarily equivalent to exactly one of the multiplication d j (1 < j 5 No), or d,0 d , (1 5 k INo). If we denote algebras d,, by ( X , p ) the appropriate measure space associated with d ([0, 13 with Lebesgue measure in the case of .dc,and S j in the case of dj-see the

1026

14. DIRECT INTEGRALS AND DECOMPOSITIONS

discussion preceding Theorem 9.4.1 for this notation), then there is a unitary transformation of 2 onto L , ( X , p ) that carries d onto the multiplication algebra of L,(X, p). Thus each vector in 2 corresponds to a function f in L,(X, p). We have noted in Examples 14.1.4(a) and 14.1.1 l(a) that L,(X, p ) is a direct integral of one-dimensional Hilbert spaces and that the algebra of diagonalizable operators is the multiplication algebra of L , ( X , p). Thus we have our desired (unitary equivalence of 3“ with a) direct integral decomposition of Z in which d is (unitarily equivalent to) the algebra of diagonalizable operators. The next level of complexity occurs when d’is of type I , . In this case, 0 I J , where Theorem 9.3.2 tells us that d is unitarily equivalent to 1 0 (do d ois a maximal abelian algebra acting on a (separable) Hilbert space X (see the discussion following Theorem 6.6.1 for this notation). In other words, there is a unitary transformation of 2 onto & 0 X that carries .d onto the set of operators ( A 0 A : A E d o }If. we view X as L,(X, p) and d oas the multiplication algebra on it (so that d is isomorphic to d o ,though not unitarily equivalent to it), an element of 2 is transformed onto a pair (f, g) of functions in L,(X, p). To each p in X,there corresponds (f(p), g(p)), a vector of a two-dimensional Hilbert space Z p .If an operator A in d o corresponds to multiplication by h, then, since A 0 A transforms (f; g) onto (hf;hg) which has component ( h ( p ) f ( p ) , h(p)g(p)) in Sp,A 0 A on X 0 X has component h(p)l, on X p .It is readily verified that X 0 X is the direct } that .do0 I , (which corresponds to d)is the algebra of integral of { Z pand diagonalizable operators relative to this decomposition. If d‘is of type I , , with n finite, then .dis unitarily equivalent to d o0 I , , and the preceding discussion is altered only in replacing pairs by n-tuples. In a formal sense, the same is true when d’ is of type I,. In this case, the Hilbert space to which 2 is unitarily equivalent consists of sequences {A> of functions fj in L , ( X , p ) such that ~ ’ L l l l f j l lis~ finite. It follows that Cj”= 1 fj(p)12 is finite for almost all p . If we start with an orthonormal basis {x,) in X and the corresponding sequences { fnj> in X 0 Y 0 . . ., except for p in a p-null set, we associate with each x, the vector {fn1(p),fn2(p), . . .} in 1, ( = X p ) Again, . i@ is (unitarily equivalent to) the direct integral of { Z p and } .dis (unitarily equivalent to) the algebra of diagonalizable operators relative to this direct integral decomposition. In the most general situation, d’is the direct sum 0 d ‘ P , , n = 1, 2,. . . , KO,where P , is a projection in d and d ’ P , is of type I , or P , = 0. In this case (with P, # 0), we apply the preceding considerations to the abelian von Neurnann algebra d P , acting on P , ( Z ) to construct the measure space ( X , , p,) such that P , , ( Z ) is unitarily equivalent to the n-fold direct sum X, of L , ( X , , p , ) with itself and such that d P , is carried onto the algebra of diagonalizable operators on X, relative to the direct integral decomposition

14.1. DECOMPOSITIONS RELATIVE TO ABELIAN ALGEBRAS

1027

{A?,} of X,, over (X,,, p,,)we described. Let ( X , p) be the direct sum of these constitutes a direct integral measure spaces. Then the total family {H,} X,, ( = X ) , and the direct sum of the unitary transdecomposition of formations of P,,(H) onto X,, is a unitary transformation of A? onto X that carries d onto the algebra of diagonalizable operators relative to this decomposition. We summarize this discussion in the theorem that follows.

c@

14.2.1 THEOREM.If d is an abelian von Neumann algebra on the separable Hilbert space A? there is a (locally compact complete separable metric) measure space ( X , p) such that Y? is (unitarily equivalent t o ) the direct integral of Hilbert spaces {&,} over ( X , p) and d is (unitarily equivalent to) the algebra of diagonalizable operators relative to this decomposition. With Theorem 14.2.1 in mind, we may speak of the (direct integral) decomposition of a (separable) Hilbert space X relative to an abelian von Neumann algebra d on H,as well as the decomposition of a von Neumann subalgebra .% of d' relative to .#. We study, now, the effect of special assumptions about the relation between W and d on the components 9,. 14.2.2. THEOREM.If d is an abelian von Neumann subalgebra of the center W of a von Neumann algebra B on a separable Hilbert space A? and {A?,} is the direct integral decomposition of 2 relative to d,then V, is the center of W,almost everywhere. I n particular, 9,is a ,factor almost everywhere if and only (f sd = V.

w' E W c d'. As V G w',%? E W c W E d', Proof: Since .dc V E 9, and each of .%, B?',V, and W' is decomposable (relative to d). Let and 212 be norm-separable strong-operator-dense C*-subalgebras of W and W', respectively:,and let be the C*-algebra they generate. Then % is a normseparable strong-operator-dense C*-subalgebra of the von Neumann algebra V' generated by W and 92'. It follows that W,, (B?'),,and (W), are the (a2),, and a,, respectively. From Propostrong-operator closures of (al),, =, (a,)'and (V), = (%?,)' almost everywhere; so that sition 14.1.214, (9) (al),,and (a2),generate the commutant of the center of W p almost and (a2),, generate a,,(V,)' is the commutant of the everywhere. Since (a,), and V, is the center of B,, almost everywhere. center of W!,; If V = d ,then V, is the algebra of scalars, since d is the algebra of diagonalizable operators; and W,is a factor almost everywhere. Conversely, if V, is the algebra of scalars (that is, if 9,is a factor) almost everywhere, then V, = d,almost everywhere and V = d,from Proposition 14.1.18. Combining Theorems 14.1.21 and 14.2.2, we have the following corollary.

1028

14. DIRECT INTEGRALS AND DECOMPOSITIONS

14.2.3. COROLLARY. I f 9 is a von Neumann algebra of type I,, 11,,II,, or I11 acting on a separable Hilbert space 2,the components W pof 9 in its direct integral decomposition relative to its center are, almost everywhere, factors of type I,, II,, II,, or 111, respectiueiy. The decomposition of a von Neumann algebra W relative to its center is referred to as “the central decomposition of 9? (into factors).” To what extent is this central decomposition of 92 into factors unique? In the most primitive case, when the center V of R contains two minimal projections Q 1 and Q2 with sum I, the measure space can be taken to consist of two points p 1 and p z , each with positive measure, and BPIis (unitarily equivalent to) WQ1 acting on Q1(X) while W,,is (unitarily equivalent to) W Q , acting on Q 2 ( XIn ) . this , where HpI= Q1(X‘) and case, 2 is the direct integral of { X P ,SPz}, XD2 = QAW.

If {Z,,} is another direct integral decomposition of 2 over ( X , p ) relative to which W is a decomposable algebra containing the diagonalizable operators and W,is a factor almost everywhere, Theorem 14.2.2 tells us that W is the algebra of diagonalizable operators. We know that W is * isomorphic t o the multiplication algebra of L 2 ( X ,p); so that, measure-theoretic minutiae aside, X consists of two points q1 and q2 to each of which p assigns positive measure. With Q;, Q; the corresponding minimal projections in W, W,,and W,,are unitarily equivalent to 9Ql and 9 Q ; , respectively. Of course the decomposition of W into factors is “unique” in this case. We have, simply, t o discover which of Q1 or Q 2 the projections Q; and QL are and “match” the corresponding factors. The essence of the uniqueness is the “converse” part of Theorem 14.2.2 stating that each decomposition of an algebra W containing the diagonalizable operators into factors is such that the center of W coincides with the algebra of diagonalizable operators. In the case of more general directintegral decompositions into factors, the basic ingredients of the preceding discussion still apply. The center W of W is the algebra of diagonalizable operators and is * isomorphic to the multiplication algebra d ,of the measure space ( X I ,p , ) of the decomposition. If d 2is the multiplication algebra of ( X 2 , p 2 ) and 8 ‘ is the algebra of diagonalizable operators in a decomposition of 92 over ( X 2 ,p,), the isomorphisms of W with d ,and d, provide us with an isomorphism rp of . d , onto d,. Since .d, and d,are maximal abelian algebras. Theorem 9.3.1 applies and assures us that cp is implemented by a unitary transformation. Following the pattern of the argument in the case of the two-point space, we should hope to map X , onto X , in a manner that implements cp. It is too much to expect that we can find a one-to-one mapping of X , onto X , that preserves measurable sets and measure 0 sets; for X , and X , may not even have the same cardinality. For example, nothing prevents us from adding a third point to the two-point

14..2. DECOMPOSITIONS RELATIVE TO ABELIAN ALGEBRAS

1029

space and assigning it 0 measure, in the framework of the general direct integral theory. But we may hope to exclude Borel subsets of measure 0 from each of X , and X , and map the remaining portions of the space onto one another by i i one-to-one mapping such that both the mapping and its inverse preserve measurable sets and measure 0 sets. That there is such a mapping (in the case of our restricted measure spaces) is the substance of a theorem due to von Neuma.nn [63], whose proof is a measure-theoretic construction. (See Theorem 14..3.4.)Suppose that Y, and Y2 are Borel subsets of X , and X , such that pl(Xl\,Yl) = p2(X2\Y2) = 0, and q is a one-to-one mapping of Yl onto Y, that, together with q-', preserves measurable sets and measure 0 sets and for which tp(M,-l) = M,, where f, =f2 0 v] almost everywhere. Let p, q denote the measure on X , (equivalent to p,) that assigns the measure p,(q(Y)) to a measurable subset Y of Yl. There is a measurable function finite and positive almost everywhere on X , , such that for each p, v]-integrable function g on X , , 0

0

&l(P)

I x m ( m ) 2

= IXl

d P ) dP2

O

?(PI.

Let { x j } be a dense denumerable rational-linear subspace of S so that { x j ( p ) } and { x j ( q ) ) are such subspaces of and Afq for almost all p in X , and almost all q in X,. Let U p x X p )bef(p)xAq(p)). We note that U p extends to a unitary transformation of YPp onto &',,(p). Let 2 , and Z , be measurable subsets of Y, and Y, such that q ( Z , ) = Z,. From our assumption that q implements cp and that cp is engendered by the isomorphisms of % with d , and d,; 2 , and Z , correspond to the same diagonalizable projection P in V . Since (xj(v](p)),x,(v](p))) is 1.1, q-integrable on X , , 0

r

r

As this hollds for all measurable subsets Z , of Y,, p -+ (Upx,(p), Up&)) is pl-integrable over X , and ( U P x J p ) ,u p x k ( p ) ) = ( x x p ) , xk(p)) for all j and k except for a p,-null set. It follows, since { x j ( p ) } and {x,(q)} are dense rational-linear subspaces of X Pand Xqalmost everywhere, that U pextends to a unitary transformation of A?p onto X,,(,,),for almost all p .

1030

14. DIRECT INTEGRALS AND DECOMPOSITIONS

If A is a decomposable operator on X relative to ( X , , p l ) then A commutes with %'; so that A is decomposable relative to (Xz, pz). Let { A , } be a denumerable self-adjoint strong-operator-dense subalgebra of 92 over the rationals; and let be its norm closure (a norm-separable C*-subalgebra of a).If (xi'} is a sequence of vectors in { x i } tending to AkXj, an &-subsequence for argument tells us that, for some subsequence, {xj..(p)}tends to A&.(p)xJ
and W is decomposable. Let { B j } be a Proof: Since d zR; W c d', denumerable strong-operator-dense self-adjoint subalgebra (over the rationals) of .%, and let { A j } be such a subalgebra of &'. Let al and 212 be their norm closures (the C*-algebras they generate). If is the C*-algebra then is the norm closure of the set of finite sums generated by 211 and '$I2, of B,A, (an algebra over the rationals, since B j A , = A$,.), and 2I is strong-operator dense in the von Neumann algebra Y generated by .% and d.Thus apand 9,are the strong-operator closures of (al),and aP, respectively. But a, (al),,and 9,= .Y, almost everywhere, since (BjA,)(p) = B,(p)a,(p)Z, = a,(p)Bj(p), where ak(p) is a complex number. (Recall that A , is a diagonalizable operator and that W is not assumed to contain d ;so that the conclusion 9,= Y , does not imply that W = Y nor does it conflict with Proposition 14.1.18.) If .%, = @ ( X palmost ) everywhere, then Y, = B(XP)almost everywhere and Y contains d,the algebra of diagonalizable operators. On the other so that fd'),= (d,)', from Proposition 14.1.24, and hand d'has center ,d, d, is the algebra of scalars, since d is the aigFbra of diagonalizable operators. Thus (S$")p = a(%,) = Y , almost everywhere. Applying ProposiSince an operator commutes with the tion 14.1.18, d'= Y ; and d is 9'. (von Neumann) algebra generated by W and d if and only if it commutes with both 92 and d,w'n d'= 9' = d.Thus d is maximal abelian in 9' when 9,= B(Xp)almost everywhere.

14.3. APPENDIX-BOREL

MAPPINGS AND ANALYTIC SETS

1031

Suppose, now, that d is maximal abelian in B?'; so that 9" = d . Then := (9')p = (9,almost )' everywhere, from Proposition 14.1.24; and g P= Y;, = 93(Xp)almost everywhere. W

d , = {aZ,}

I f cp is a representation of a C*-algebra 2l on a 14.2.5. COROLLARY. separable Hilbert space X, the decomposition of cp relative to an abelian von Neumann suhalgebra d of cp(2t)' has components that are irreducible almost everywhere ij' and only if d is maximal abelian in cp(2l)'. Bibliography: [21, 54, 671

14.3. Appendix-Bore1 mappings and analytic sets A Borel structure on a set X is a family W of subsets (the Borel sets) containing (3,the complement of each set in B, and the union of each countable subfamily of 93. Since the intersection of Borel structures on X is a Borel structure and the set of all subsets of X is a Borel structure on X , each family 9 of subsets of X is contained in a smallest Borel structure, the Borel structure generated by F.If X is a topological space the Borel structure generated by the open sets (equivalently, the closed sets) is called the topological Horel structure. For our purposes, we may restrict our attention to topological spaces and their topological Borel structures, although some of the results that follow are valid for the more general Borel structures. If @ is a Borel structure on X and p is a (positive) measure defined on the then the family of sets of the form X , u N o , where X , sets of 9, and N o is a subset of a p-null set, is a Borel structure on X . Defining p ( X o u N o ) as p(X,), ji is a measure defined on the sets of B and each subset of a p-null set is in g . We say that a measure space ( X , ji) with this property (each subset of a null set is measurabk) is complete. In the present case ( X , p) is the completion of ( X , p). If X is a topological space and W is its topological Borel structure, we refer to either ( X , p ) or ( X , ji) as a Borel measure space. We say that a Borel measure space ( X , p ) is regular when (as agreed in Remark 1.7.6) p ( C ) < co for each compact set C and each measurable subset X , differs from both the intersection of a descending sequence { O n } of open sets containing it and the union of an ascending sequence { C,} of closed sets contained in it by p-null sets. In this case, if some p ( 0 , ) < 00 when p ( X , ) < m, then lim p(0,) = p ( X , ) = lim p(C,). Either the condition on open sets or the condition o n closed sets implies the other. For example, knowing the condition on open sets and given a measurable set X , , let (0,) be a descending sequence of open sets containing X\X, such that = 0. ~ u t 0,)\(x\x,) = xo\(u(x\O,)); Xw,

a

p((nm(w,))

cn

1032

14. DIRECT INTEGRALS AND DECOMPOSITIONS

(= C,) is a closed subset of X, (since X\X, E On), and {C,} is ascending since (0,)is descending. If Bl is a Borel structure on X I and B, is a Borel structure on X,, a mappingfof X, into X, is said to be a Borel mapping whenf-'(X;)EB, for each X 2 in a,.If {X,} is a subfamily of W,generating the Borel structure a, andfisamappingofX, intoX,such thatf -'(Xb)~W1foreach b,thenfisa Borel mapping. To see this, note that, since f - preserves countable unions and complements and since B l is a Borel structure, the family F of those sets X i in 93, such that f -'(X;)EB~ is a Borel structure. By assumption { X , ) s F.Since {x,) generates B,, 9 = B,, and f is a Borel mapping. If (X, p ) is a complete Borel measure space, a mappingf of X into a space Y with a Borel structure is said to be measurable when f-'(Y,) is pmeasurable for each Borel subset Yo of Y: Since p is complete,f-'(Yo) may not be a (topological) Borel set in X (and f may not be a Borel mapping). In the lemma that follows, we note the possibility of finding a Borel p-null set on the complement of which f is a Borel mapping when the Borel structure in Y is countably generated.

'

14.3.1. LEMMA. If (X, p) is a complete Borel measure space and f is a measurable mapping of X into space Y with a countably generated Borel structure, then there is a Borel p-null set N of X such that f ((X\N) is a Boref mapping.

Proof: Let { q} be a denumerable family of Borel subsets of Y that generates the Borel structure. Thenf- '( 5) is a measurable subset of X and, hence, is the union of a Borel subset Xi of X and a subset of a Borel p-null set N , . Let N be N,. Then

uT=

f-'(

y) n (X\N)

= X, n (X\N),

and X j n (X\N) is a Borel subset of X\N. Thus f I(X\N) is a Borel mapping. H

14.3.2. LEMMA. lf (X,p) is a regular Borel measure space, Y is a completely regular space such that C(Y) is countably generated, and q is a mapping of X into Y that is Borel and one-to-one on a compact subset X, then there is a Borel subset Xoof X such that q is a Borel isomorphism on Xoand p(X',X,) = 0. Proof: Let Fobe the (denumerable) subalgebra over the rationals is generated by a countable generating family of functions in C( Y )(so that go norm, dense in the algebra C(Y) of continuous bounded functions on Y ) . Since q is a Borel mapping on X,f 0 q is a Borel function on X for each f i n C( Y). Let f i , f 2 , . .. be an enumeration of the functions in 9,. By Lush's

1.1.3. APPENDIX-BOREL

theorem, given a positive

E,

MAPPINGS A N D ANALYTIC SETS

1033

there is a closed subset Xl of X such that

p ( X \ X l ) < ~ / 2and f l o q is continuous on T , .Again, there is a closed subset X, of X , such that f 2 q is continuous on X, and p(X1\X2)< ~ / 4 . Inductively, there is a closed subset X, of 37,- such that f, q is continuous on X, and p(X,- l\X,) < ~12".Let XEbe X,. Then p(X\Xe) < E and Xeis a closed set on which all f, q are continuous. Since {f,}determines the topology of I:q is a continuous mapping of X, into Y: As q is one-to-one on Xe,Y is Hausdorff, and Xeis compact, q is a homeomorphism on Te. It follows that q(XJ is compact and that the image of each Borel set in X, under q is a Borel set in q(XJ and hence in Y: Let X, be U:=lXl,n. Then X, is a Borel subset of X , X, G X, and p(X\X,) == 0. If X , is a Borel subset of X,, then X , n XI,,,is a Borel subset q ( X , n XIIn). Thus q(X,) is a Borel set of X,,, for each n ; and q(X,) = in Y and q is a Borel isomorphism on X,. 0

n=;

0

0

uF=,

I f ( X , p) is a o-compact regular Borel measure 14.3.3. PROPOSITION. space, Y is ~i subset of X of jinite measure, and q is a one-to-one Borel mapping of 9 'into a completely regular space Y for which C( Y ) is countably generated, then there is a Borel subset X , of Y such that p(Y\X,) = 0, q(X,) is a Borel subset of and q is a Borel isomorphism of X , onto q(X,). Proof: Since X is regular, there is an ascending sequence { C,} of closed = 0. As X is o-compact, there is an subsets of 9 such that p(Y\u:=lC,) X:. ascending sequence { X : }of compact subsets of X such that X = Then Xk n C, ( = X,) is compact, { X"} is an ascending sequence of compact and p ( Y \ U ~ = l X , , ) = 0 (for, with {Xh} and {C,} ascending, subsets of 9, X i ) I? C,) = X,). From Lemma 14.3.2, each X, contains a Borel subset B, such that p(X,\B,) = 0, q(B,) is a Borel set, and q is a Borel isomorphism of B, onto q(I3,). By taking u . " = l B n as X , , our result follows.

u;=

(u;=

(u;=,

u;= ,

x,

14.3.4. -rHEOREM. If X I and are a-compact separable metric spaces, pl and p, are complete regular Borel measures on X , and X , , and cp is an isomorphism of L , ( X , , pl)onto L , ( X , , p,), then there are Borel subsets X i , X i of X , , X, and a Borel isomorphism q of Xl onto X ; that identijies pl-null sets with pL,-nullsets such that pL1(X1\X;)= p2(X2\X;) = 0 and f i x ; = cp(f)~ilforeachfinLm(X,,p,). Proof: Throughout this proof, the subscript j takes the values 1 and 2. Since X j is separable and p j is regular, L,(X,, pj) is a separable Hilbert space. Its multiplication algebra d-j admits a separating (and generating) unit vector x j , b,y Corollary 5.5.17. In fact, since the state ox, @ is normal, where @(Mf) = Mq(,-); by Theorem 7.2.3, it is a vector state of -01",, and we may 0

1034

14. DIRECT INTEGRALS AND DECOMPOSITIONS

choose x1 so that w,,1J1 = w,, +. If we assign the value ( M f J x j 7 x j ) to a measurable subset Y jof X j , where& is the characteristic function of Y j ,the resulting set function p> is a positive regular Borel measure on X j , equivalent to p j , with the following properties: 0

(i) p i ( X j ) = 1; (ii)

lx,

f(PJ

U l @ d= I x 2 d f ) ( P 2 ) 4 4 P J

f o r f in L d X l , Pl).

If we can establish the assertion of this theorem for the measure spaces (XI,pi) and ( X 2 , pi), then it follows for (Xl, pl) and ( X , , p z ) . We assume, henceforth, that p j satisfies (i) and (ii). Let B j be the denumerable strong-operator-dense subalgebra over the rationals of d: generated by the characteristic functions of the open sets in a denumerable basis for the topology of X i . Let and g2be the algebras over and gz],respectively. Let the rationals generated by {Bl,+-'(.%I,)) LBj and Vj be the algebras of equivalence classes of functions in L,(Xj, p j ) corresponding to giand gj.Choosing a representativefi for each (real) class in W j , we can find a pj-null set N j such that j , is a Borel function on Xj\Nj, &(Xj\Nj) E [ - 11 &I/ io, I l f J ,I, and the algebraic operations in g j correspond to pointwise operations on Xj\Nj. We may treat g j (and gj),now, as algebras of functions. It will be convenient to assume, as we may, that N j contains the complement of the support of p j . (See Remark 3.4.13.)It follows that each point p j in Xj\Nj corresponds to a bounded multiplicative linear functional p p , (hence, from Theorem 3.4.7, pure state) on the norm closure of g j .Since B2 is an abelian C*-algebra, there is a * isomorphism of g2onto C( Y ) , where Y is the compact Hausdorff space of pure states of g2 with its weak* topology (see Theorem 4.4.3).Let 5, be q2 (GIGl); and let $ j be the corresponding isomorphism of g j , the algebra of functions in L , ( X j , p j ) associated with g j . Then p p J qJrl is a multiplicative linear functional on C( Y ) ,and corresponds to a point t j ( p j )of I:by Corollary 3.4.2. By construction $ J < f j ) ( t j ( p j ) )= f j ( p j ) for fj in g j and p j in Xj\Nj. The mapping @ j ( & ) -+ Jxjfj(pj) d p j ( p j ) ( & E g j ) defines a state of C ( Y ) and, hence, a regular Borel measure vj on Y such that vJ(Y) = 1. Since $2(q3(f1)) = t+bl(f,) forfl in g1; from (ii), the two states of C( Y ) determining the measures v1 and v2 are identical. Thus v1 = v2 ( = v). By construction r r

{+(a,),

aj,

q2

0

0

for& in g j .Since g j is norm dense in G j ,the functions $,
14..3. APPENDIX-BOREL

MAPPINGS AND ANALYTIC SETS

1035

since the (representing) functions in LBj separate the points of Xj\Nj. From Proposition 14.3.3, there is a (cr-compact) Borel set X y contained in Xj\Nj such that p j ( X ; ) = 1, t J ( X ; ) (= 5) is a Borel set in E: and T j is a Borel isomorphism of X ; onto 5. Having just noted that p j and v induce the same integration when functions in $Bjare “transported” by Il/j and that tJrl “induces” t,hj on g i ,a Yo))for each dense subset of Q j , we should be led to feel that v(Y,) = pLi(tJrl( measurable subset Yo of Y-equivalently, that

for each characteristic function g j of a measurable subset Yo of Y. We prove (1) in the argument that follows, noting in the next computation that it holds when g j is t,hj(f,) and fj is the selected representative of an equivalence class in qj (cQj). For such an f,,

Let Fjbe the class of all bounded complex-valued Borel functions g j on Y for which (1) holds. Then Fj is linear, norm closed, and contains a norm-dense subset of C( Y ) . Thus F j contains C( Y ) . From the monotone convergence theorem, if a bounded function g j is the pointwise limit, everywhere on E: of an increasing sequence of (real-valued) functions in Fj, then g j E F j , We show, next, that F j contains the characteristic function of each open smet in Y. For this note that, since Y has a countable generating family of continuous functions, it is homeomorphic to a subset of the product of it countable number of unit intervals. Hence Y admits a metric (compatible with its topology). If 0 is an open subset of Y and f , ( y ) is min{n dist(y, Y\0), I}, then {fn} is an increasing sequence of functions in C( Y ) tending pointwise to the characteristic function of 0; and this function is in Fj. Let Y jbe the class of all Borel subsets of Y whose characteristic functions satisfy (1). From the results of the preceding paragraph, .CPj contains all the

1036

14. DIRECT INTEGRALS AND DECOMPOSITIONS

open sets in Y. Since Fj is linear and contains 1, the complement of a set in Y j lies in Y j .Moreover, the countable union of sets in .Yj is in Y j (from the “monotone closure” property of Yi).Hence Y jcontains all the Borel sets in Y Applying (1) to the characteristic function g j of 5, we have v( 5) = pJ(Xy) = 1. Hence v(Y, n Y2) = 1 ; and applying (1) to the characteristic function of Y, n Y,, we have p,CX[i)= 1, where X[i is the Borel subset I ( Yl n Y,) of Xy. For p in X’,, let q ( p ) be < ; ‘ ( ( , ( p ) ) . Then q is a one-to-one, measure preserving, Borel isomorphism of X‘, onto X i , and pJ(Xj\X>) = 0. If EX', and ,f €S1,then (cp(f)f l ) ( P ) = (cp(f)O 5; l)(tI(P)) = ($2(cp(f)N(Sl(P)) = (ICll(f))(Sl(P)) = f(P).

rJ:

But the mapping M , -+ M , , , (g E L , ( X , , p 2 ) ) is a * isomorphism $ of ..I”, onto J1. Since and ij’ agree on the self-adjoint strong-operator-dense subalgebra @, they agree on &. Thus q is a Borel isomorphism of XI onto X ; such that fix’,= q(f)oq when f e L , ( X , , pl), and pj(Xj\XJ) = 0.

+-’

A topological space X is said to be a csm (complete separable metrizable) space if it has a countable base for its open sets and admits a metric in which it is complete. The set consisting of a point chosen in each set of a countable base is dense in X so that, with its metric, X is a complete separable metric space. Let d be a metric on X in which it is complete. Defining d’(p, q ) to be d(p, q)[1 + d(p, q)]-’, we have that d is a metric on X , d(p,q ) I min(d(p, q), l), and d(p, q ) I 2d’(p, q ) when d(p,q ) I $ (since d(p, q ) = d ( p , q)[l - d ( p , q)]-’). It follows that d and d determine the same topology on X and that X is complete relative to d’. There is no loss of generality in assuming, therefore, that our csm space X is equipped with a metric bounded by 1 (that is, d(p, 4) I I for each pair p, q in X ) . If ( X , , d,,), n = 1, 2,. .. are csm spaces (d, a metric bounded by l), their topological product X is a csm space. To see this, let d((p,), (q,,)) be ~ 2 - ” d , ( p , , q , ) . Then d is a metric on X , compatible with the product topology, relative to which X is complete. Choose a point p , in each X , ; and let { p n j : j = I , 2, ...} be a denumerable dense subset of X,. Then {(plj,, p Z j z , ... ,pnjn,pn+ pn+,, . . .)} is a denumerable dense subset ofX. If the spaces X , are disjoint and X, is their (topological) sum, then, defining d,(p, q) to be 1 when p E X,, q E X,, and n # m,and do@,, 4.) to be d,(p,, 4). when p,, q. are in X,, do is a metric on X , compatible with its sum topology relative to which X , is complete. Thus X is a csm space. Of course each closed subset of a csm space X is a csm space. The same is true of an open subset G. With p . q in 0, let d(p,q ) be

J(inf{d(p,p’) : p ’ ~ X \ 0 } ) - ’ - (inf{d(q, 4’): qfEX\0})-’I

+ d(p, q).

14.3. APPENDIX-BOREL

MAPPINGS A N D ANALYTIC SETS

1037

Then d' is a rnetric on 0 compatible with its (induced) topology relative to which it is complete. To see that 0 is complete relative to d',note that if d ' ( p , , p , ) + O . then d ( p , , p , ) - + O ; so that there is a p in X such that d(p,, p ) + 0. If p ~ X \ 0 , [inffd(p,, p') : p ' ~ X \ @ } ] - l + co,contradicting the 0 (by compatibility) fact that d ' ( p n , p m ) + O as n, m + 00. Thus ~ € and d'(p,, p ) -+0. Hence (0,d') is a csm space. It follows from the preceding discussion that the space Z, of positive integers (and 0) is complete relative to the metric d, where d(m, n) = Im - nl(1 + Im - nl)-' and that S,the countable product of 7 , with itself (that is, the space of sequences of positive integers) is complete relative to d', where m

d'({mj}, { n j } ) =

C 2-jlmj - n j l ( l + Imj

-

njl)-',

j= 1

and d' is compatible with the product topology on S.Thus (S, d') is a csm space. We prove that each (non-empty) csm space ( X , d ) is the continuous image of S.The closed balls with radius $E and centers a dense denumerable subset of X form a denumerable covering of X by non-null closed sets with diameter not exceeding E. Let X , , X , , . . . be such a covering for diameter Let X n l ,X n 2 , .. . be such a covering of X , for diameter Let Xnml,X,,,,,.. . be such a covering of X,, for diameter $. Continuing in this way, we define with diameter not exceeding 2-' such that closed non-null sets Xn,nz...nk X,,l...,,k E X,l...,,k- for each finite sequence ( n , , . .. , nk) of positive integers. With ( n , , n,, ...) in S,choose a point Pk in X , ,...nk for each k. Then { p k } is Cauchy convergent in X and tends to a limit f(n,, n,, .. .) in X . If {p;} is an alternative choice of defining sequence, d(pk, p ; ) I 2-', since both pk and p ; lie in Xn1. ..nk. Thus ( p i } tends to f ( n , , ... ,rzk); andfis a mapping of S into X . If (n,,n , , . . .) and (m1,rn,,. . .) are suitably close in S,then n , = m,, . . . ,nk = mk for some large k. Hence d ( , f ( n , ,n2, ...),f( m,, m2,...)) < 2-' (since both f ( n , , n 2 , . . .) and f ( m , , rn,, . . .) lie in X,,,.. .,,k); and f is continuous. If P E X, p lies in some .Y,, (since X , , X , , .. . is a covering of X ) and p is in some X,,,, (since X n l l ,X n l , , ... is a covering of X,,). Continuing in this way, we construct a sequence i n l , n,, ...} such that f ( n , , n,,. ..) = p. Thus f is a continuous mapping of S onto X . A subset X , of a csm space X is said to be analytic if it is the continuous image of a csm space. Of course each csm space is an analytic subset of itself. Composing mappings, we see that the image of an analytic set in a csm space under a continuous mapping is an analytic set. If X , is a csm space, n = 1, 2, ... and f , is a continuous mapping of X , into the csm space X (so that f,(X,) is an analytic subset of X ) then, definingf to bef, on X , , as a subset of the sum space X , , f is a continuous mapping of X,, a csm space, onto Uf(X,). Thus the countable union of analytic subsets of a space is analytic.

a.

t.

1038

14. DIRECT INTEGRALS AND DECOMPOSITIONS

Again, the product Y of the spaces X , is a csm space. If xi is the jth coordinate mapping of Y onto X j , the set Y,, of points of Y at which the two (continuous) mappings f , o x , and f m 0 7 t m agree is closed. Thus K, ( = Yo)is a closed subset of Y: Hence Yo is a csm space; and all the mappings f.0 x, coincide on Yo to determine a single continuous mapping of Yo onto n f n ( X n ) .Therefore a countable intersection of analytic sets is analytic. It follows that the family of analytic subsets of a csm space X whose complements are also analytic subsets of X is a Borel structure on X containing the closed sets. (Note, for this, that, with cautious and strict interpretation, 0is a csm space and is the continuous image of itself under the identity mapping on it.) Hence each Borel subset of a csm space is analytic. Our next goal is to prove a result approximating a converse to the preceding conclusion: If ( X , p) is a complete a-compact csrn Borel measure space and A is an analytic subset of X , then A is measurable. We shall prove this by showing that there is a compact csm space I: a descending sequence { Y,} of a-compact subsets of Y with intersection B, and a continuous mapping f of Y into X , such that f ( B ) = A . We first show that, under these circumstances, A is measurable after a few easy measure-theoretic observations that will be needed for the argument. If A’ is a subset of the finite measure space ( X , p ) , we define p*(A’) as inf(p(S) : A‘ c S , S measurable). Of course p*(A‘) 5 p*(B’) if A’ G B’. If S , is a measurable set containing A‘ such that p(S,) I p*(A’) l/n, then S, is a measurable set containing A‘ with measure not exceeding p*(A’). Thus p ( n S,) = p*(A’). If { A , } is an ascending sequence of sets with union A’, S , is a measurable set containing A , such that p(S,) = p*(A,), and T. = S, n n...,then p(T,) = p(S,) = p*(A,) since T, is a measurable set and A , c T, c S,. Moreover, { T,} is an ascending sequence of measurable sets with (measurable) union Tcontaining A’; and p ( T ) = lirn p(T,) = lim p*(A,). Thus p*(A’) I lim p*(A,). Since A , c A’ for each n, lim p*(A,) I p*(A’); and p*(A’) = lim p*(A,). If the measure space ( X , p) is complete and, for each positive m, there is a measurable subset C , of A’ such that p*(A‘) - l/m _< p(C,), then A’ is measurable. To see this, choose S measurable such that p ( S ) = p*(A‘) and A’ c S . Then 0 C, (= C ) is measurable and p*(A’) I p ( C ) I p ( S ) = p*(A‘) while C s A’ G S . Thus p(S\C) = 0 and A’ is the union of C and the p-null set A‘\C. We are assuming that X = K , with K , compact and p ( K , ) finite. We want to show that the analytic subset A of X is measurable. Since A n K, is analytic and A = U ( A n K,), we may assume that p ( X ) is finite and that X is compact. With these assumptions in force and with the notation adopted earlier ( A = f ( B ) and B = T), we use the discussion of the preceding paragraph to show that A is measurable. There is an ascending sequence ( K , J of compact sets with union Y,. Since B s Y,, f(B) = Ujf(B n Klj).

n,,,

+

u

n

14.3. APPENDIX-BOREL

Thus p * ( f ( B ) ) = lim ,u*(f(B n K

MAPPINGS AND ANALYTIC SETS j));

1039

and there is a j , such that

P * ( ~ ( B) ) l / m < p*(f(B n KIj1)) 5 P(f(Kljl)). Since B n Klj1 c Y,, B n Klj, = U , ( B n K , , , n K,j) and, again, there is a j , such that

p * ( f ( B ) >- l/m < A f ( K 1jl n . .. n Knj,,)).

Knjn) is a descending sequence of compact sets with Now (KIjl n (compact) intersection K so that {f(KIjl n ... n Knj,)} is a descending sequence of compact sets with (compact) intersection S containing f ( K ) . If p E S, there is a q, in K j , n . . . n K n j , such that f(q,) = p . The closures of the sets { q j :j = n, n + I , . ..), n = I , 2,. .. form a descending sequence of non-null compact subsets of Y and their intersection is non-null. With q in this intersection: q E K and f ( q ) = p . Thus S = f(K). Hence p * ( f ( B ) ) - l/m I p ( f ( K ) ) . But K E K n j , E Y, for all n; so that K G B and f(K) E f ( B ) . From the discussion of the preceding paragraph, f(B)( = A ) is measurable. It and f. remains to lind X { We have: seen that there is a continuous mapping g of S into X such that g(S) = A . Let R, be a copy of the one-point compactification of R for n = 1, 2,. ..; and let W be the product of the w,. If Y = W x X,then Y is a compact (csm) space and S can be viewed as a subset of W. Let B be the graph of g (in Y ) and f the projection of Y onto X . Then f ( B ) = A and f is continuous. Let F,, be the union of all closed intervals of lengths l/m with centers 0 or a positive integer in R,. If .

.

e

n

x},

F,

= (Fin X , , F 2 , x

... x F,,

x

R,,,

x R,,,

x ...) x

x,

then ( F , ) is, a descending sequence of 0-compact sets in Y with intersection S x X . Since y is a continuous mapping of S into X its graph B is closed in S x X ; so that B - n (S x X ) = B, where B - is the closure of B in Y. Thus B = B- n I:, n F , n .-.. Since B - is compact (closed in Y ) , it follows that ( B - n F,) is a descending sequence of 0-compact sets in Y with intersection B. (In the notation of the preceding paragraph, Y, = B - n F , and K,, = B - n C,,, where { C n j }is an ascending sequence of compact subsets of Y with union F,.) We summarize the information of the preceding discussion, needed for reference, in the theorem that follows.

1040

14. DIRECT INTEGRALS AND DECOMPOSITIONS

14.3.5. THEOREM. If X is a separable topological space metrizable as a complete metric space ( X is a csm space), then each Borel set is analytic. If in addition ( X , p ) is a complete, Borel measure space, X = K, with K , compact and p ( K , ) finite, then each analytic set is measurable. We are now in a position to prove the measurable-selection principle used in establishing the last results of Section 14.1. For this purpose, we introduce the total (Iexicographical) order on S in which ( n l , n,, .. .) < (ml,m,, . . .) when, for some k , n1 = m,, . .. ,nk = mk and nk+ < mk+ '. Each closed set C in S has a smallest element in this ordering. To see this, let c , be an element of C with smallest first coordinate. Among the elements of C with the same first coordinate as c , choose one, c 2 , with smallest second coordinate. Continuing in this way, we construct a sequence { c , } of elements of C tending to the element c formed from these smallest coordinates. Since C is closed, c E C and c is the smallest element in C. If ( X . p ) is a complete Borel measure space. X = I) K, with K,, compact and p ( K , ) finite, and X is a csm space, then each analytic subset A of X is p-measurable (from Theorem 14.3.5). There is a continuous mapping f of S into X with image A . A prototype (and test) of our measurable-selection principle might involve finding a measurable "cross-section" for f;that is, a measurable mapping y of A into S such that .fog is the identity on A . (In effect, we want to select a point g ( p ) inf-'(p) for each p in A in such a way that g is measurable.) Since f is continuous, f - '(p) is closed in S. Let g ( p ) be its smallest element (in the lexicographical ordering). We show that g is measurable by proving that the inverse image under g of each set in a base for the open sets of S is measurable. As our base, we use sets of the form {nl} x { n 2 } x -.. x {nk} x Z, x Z+x . . I . If s and s' are, respectively, the points ( n l , n2,... ,nkr 0, 0, ...) and ( n l , n, ,... ,nk + 1, 0, 0,...), this set is { s o : s,ES, s I so < s'} (=(s)). (It is closed as well as open.) Now p E g - ' ( ( s ) ) if and only if f - ' ( p ) has its smallest element in (s). Thus g-'((s)) = f(Ss,)\f(Ss) where S, = { s o : S,E S , so < s}. Since S , and Ss, are open in S , they are analytic subsets of S . Hencef(Ss.) andf(Ss) are analytic subsets of X . It follows that g-'((s)) is measurable, and g is a measurable mapping. If Yis a csm space and 9 is an analytic subset of X x Y, the image A of Y under the projection mapping no of X x Yonto X is an analytic subset of X . The set A consists of precisely those points p of X such that the set Y,, of points q in Y with ( p , q ) in Y is non-empty. The measurable-selection principle asserts that, in these circumstances, we can select a point from each Y, in a measurable manner-that is, there is a measurable mapping v of A into Y such that v ( p ) ~Y,, for each p . We know that there is a continuous Thus n o u f o is a continuous mapping fo of S into X x Y with image 9.

14.4. EXERCISES

1041

mapping f of S onto A. From the discussion of the preceding paragraph, there is a measurable mapping g from A into S such that f o g is the identity on A . Let n, be the projection of X x Y onto YI Then n , o f o o g is a measurable mapping q of A into Y ; and, for each p in A , fo(g(p)) (= 4)) E y’. Now P’ = nofo(g(P))= f ( g ( p ) ) = P , so that V ( P ) = nl(fo(g(p)))== 7c,(p, 4 ) = q~ 5;and q is the desired measurable mapping of A into Y. We summarize this discussion in our measurable-selection principle.

w,

If ( X , u 14.3.6. THEOREM. K, X Y X , 9’

p) is a complete Bore1 measure space, X = K,, with compact and p ( K , ) finite, X and Y are csm spaces, n is the projection of x onto is an analytic subset of X x and A = n(.Y), then there is LI measurable mapping q of A into Y such that (p, q ( p ) )E Y , f o reach p in A .

14.4.

Exercises

14.4.1. If % is the direct integral of the Hilbert spaces {X,,} over (X, p) and q, q‘ are two decomposable representations of a C*-algebra ‘11 on %, show that cp = q’ if qp= ql, almost everywhere. 14.4.2. I[f % is the direct integral of the Hilbert spaces { X P over } ( X , p) and p + q p , p + q b are decompositions of the representation q of the norm-separable C*-algebra ‘11, prove that q p= ql, almost everywhere. In the five exercises that follow, we outline some basic results of the theory of locally compact, abelian groups. Our goal, attained in Exercise 14.4.10,is a strengthening of the result of Exercise 13.4.23(iii).Our starting point is Haar measure on such groups [H : pp. 250-2631. We indicate the group operations additively (the group “product” of s and t is s + t and the group “inverse” to t is - t ) . the group identity by 0, and the integral off (in L,(G)) relative to Haar measure by J f ( t )dt.

14.4.3. Let G be a a-compact, locally compact, abelian group. (i) Examine the discussion beginning with Definition 3.2.21 and through to the statement of Proposition 3.2.23, with G in place of R, and conclude that it remains valid. (ii) Show that the non-zero linear functional p on L , ( G ) is multiplicative if and only if there is a character x of G (that is, a continuous

1042

14. DIRECT INTEGRALS AND DECOMPOSITIONS

homomorphism of G into 8,) such that, for each f in L,(G), P(f)

=

s

f ( t ) x ( O dt

=Rx>.

[Hint. See the proof of Theorem 3.2.26.1 (iii) Let {%, : a E A} be a neighborhood base of 0 in G such that each @, has compact closure, and let u, be a positive function in L,(G) such that IIu,1(, = 1 andu,(t) = Owhent$&,.Showthat Ilf*u, -ffll,+Ooverthenet A (directed by inclusion of the neighborhoods %J for each f in L,(G). [Hint. See Lemma 3.2.24.1 (iv) Show that, for each Y in G, different from 0, there is a character x of G such that ~ ( r#) 1. [Hint. Use the pattern of the construction in the comments following the proof of Lemma 3.2.24, in conjunction with (iii), to obtain an abelian C*-algebra 21U,(G)(acting on L,(G)). The pure states of 'U,(G) give rise to characters of G through the construction indicated in the proof of Theorem 3.2.26.1

14.4.4. Let G be a a-compact, locally compact, abelian group. Show that (i) the mapping t + f , ( t E G,f~ L,(G)) is a continuous mapping from G into the normed space L,(G) [ H i n t . Establish this first, when f is continuous and vanishes outside a compact set.]; (ii) f * g is continuous whenfE L,(G) and g E L , ( G ) ; (iii) S - S (= {s - s' : s, S'E S } ) contains an (open) neighborhood of 0 when S is a measurable subset of G having positive measure. [Hint. Use (ii) with f and g replaced by the characteristic functions of S and - S.] 14.4.5. A mapping cp from a measure space into a topological space X is said to be measurable when cp-'(U) is a measurable set for each opeq subset 8 of X . Let G be a o-compact, locally compact, abelian group, and let cp be a measurable homomorphism of G into a topological group H that contains a countable dense subset. Show that cp is continuous. [Hint. Use Exercise 14.4.4(iii).] 14.4.6. Let E be a topological group, C be a closed normal subgroup of E, x be a continuous, idempotent (x(x(g)) = x(g)) homomorphism of E onto C, H be the kernel of x, and cp be the quotient mapping of E onto EIC. Show that (i) ~ ( c =) c for each c in C; (ii) s = hc for each s in E, where

h = sx(s)-'( =( ( S ) ) E H ,

c = X(S)E

c;

14.4. EXERCISES

r

(iii) cp(4P)

-

YW;

= cp(0) for

1043

each (relative open) subset 42 of H , where 0 =

(iv) cp restricts to a topological group isomorphism (homeomorphism and group isomorphism) of H onto E/C provided with the quotient topology (the open subsets of E/C are the images under cp of the open subsets of E). 14.4.7. Let E be a topological group, C be a compact normal subgroup of E, and cp be the quotient mapping of E onto E/C. Suppose E/C is locally compact. (i) With Va compact subset of E/C containing the unit C of E/C and (0, :a E A} an open covering of cp- '(V ) ( = U ) , let (0,(g), ... O,(g)) be a finite subcovering of gC for g in U . Let 0 be u3=10j(g). Find an open neighborhood 0, of the unit e of E such that gC0, G Lo. [Hint. For each h in gC, find an open neighborhood V, of e such that hV, V, E 0.1 (ii) With the notation of (i), show that (cp(g0,) : gE U > is an open covering of '% (iii) With the notation of (i) and (ii), let {cp(g(l)Og(lJ,.. . ,cp(g(m)0,(,,)} be a finite subcovering of I/; Show that

is a covering of U and a finite subcovering of (0, : a E A}. (iv) Conclude that U is compact and that E is locally compact. 14.4.8. Let A be a factor, a be a continuous automorphic representation of R on A by inner automorphisms, G be the group of unitary operators in A! that implement the automorphisms a(t) ( t R), ~ C be { c l : CET,},and q(t) be UC, where U in G implements a(t). (i) Show that q is well defined and is a homomorphism of R onto G/C. Suppose q in (i) is continuous, where G is provided with its weak-operator topology and G/C with its quotient topology (see Exercise 14.4.6(iv)). Let E be the topological group { ( U , t ) : U E G, t E R, U implements ~ ( t ) }

as a subgroup of G 0 R with the product topology on G x R, C, be the (closed) subgroup ( ( c l , 0) : c E Ul}of E, and n( U , t ) be t for each ( U , t ) in E. Show that (ii) 7t is a continuous, open homomorphism of E onto R with kernel C, and conclude that E / C , is isomorphic and homeomorphic to R [ H i n t To

1044

14. DIRECT INTEGRALS AND DECOMPOSITIONS

show that 7~ is open, note that a basis for the open sets of E consists of sets of the form ( ( U , t ) : a < t < b, q ( t ) = U C , U E ~0 ,open in G} and use the assumption that q is continuous.]; (iii) E is a o-compact, locally compact, abelian group [ H i n t . Use Exercises 13.4.22 and 14.4.7.1; (iv) each character of C, is the restriction of a character of E [Hint. Use Exercise 3.5.38to show that the group of characters of C, is Z.With the aid of Exercise 14.4.3, note that the subgroup ol 7 consisting of restrictions of characters of E “separates” points of C , . Conclude that this subgroup is Z.]; (v) there is a continuous, idempotent homomorphism x of E onto C , and a closed subgroup H of E that is homeomorphic and isomorphic to R [Hint. Use (iv) to extend the identity mapping on C, to a homomorphism x with the desired properties. Use Exercise 14.4.6.); (vi) there is a (continuous) one-parameter unitary group t -+ U , that implements c1 such that U , is in A? for each t in R.

14.4.9. Let 9 be a von Neumann algebra acting on a separable Hilbert space &? Show that

is a csm space when is endowed with its strong-operator * (i) (a), topology (determined by the semi-norms T+ )ITxII (IT*xI))[Hint. Introduce a metric of the type described in Exercise 2.8.35. Use the argument of Proposition 2.5.11.3; (ii) the restriction of the strong-operator * topology to the unitary group @(a) of k%? coincides with the strong- (and weak-)operator topology on %(a) [Hint. Use Remark 2.5.10 and Exercise 5.7.5.1; (iii) %(a) is a closed subset of (a), and conclude that &(a) is a csm space when %(B)is provided with its strong- (or weak-)operator topology.

+

14.4.10. Let A be a factor acting on a separable Hilbert space 2 and CI be a continuous automorphic representation of iW by inner automorphisms of M . (i) With E as in Exercise 14.4.8, show that E is a closed subset of

%(&) x R, where %(A), the unitary group of A?, is endowed with its

strong-operator topology. (ii) Conclude from (i) and Exercise 14.4.9that @(A?) x R is a csm space and E is an analytic subset of it. (iii) Show that there is a measurable mapping t -+V, of R into %(A?) such that (K, ~ ) E for E each t in R. [Hint. Use Theorem 14.3.6.1 (iv) With q as in Exercise 14.4.8, show that q is continuous. [Hint. Use Exercise 14.4.5.1

14.4. EXERCISES

1045

(v) Shsow that there is a (continuous) one-parameter unitary group

t + LI, that implements CL and such that each U , is in A.

14.4.11. Let A be a factor acting on a separable Hilbert space S and u be a separating and generating unit vector for A’. Let t + (r, be the modular automorphism group corresponding to (A’,u). Suppose each ot is inner. Show that (i) A’ is semi-finite [Hint. Use Theorem 9.2.21 and Exercise 14.4.10.1; (ii) W ( 4 , a) is * isomorphic to A?0d,where d is the multiplication algebra corresponding to Lebesgue measure on [w [Hint. Use Exercise 1 3.4.17.1; (iii) W(&, a) is semi-finite. 14.4.12. Let A’ be a factor of type I11 acting on a separable Hilbert space. ShoR that (i) & has a separating and generating vector [Hint. Use Proposition 9.1.6.1; admits an outer automorphism. [Hint. Use Exercise 14.4.1 1.1 (ii) 14.4.13. Let B be a von Neumann algebra of type 111 acting on a separable Hilbert space and let a be the * automorphism of 9 0W described in Exercise 11.5.25(iii). Show that a is outer. [Hint. Use Exercises 12.4.19, 12.4.20, and 14.4.12.1 14.4.14. Let X and Y be csm spaces,f be a continuous mapping of X into Y, and A be an analytic subset of Y. Show thatf-’(A) is an analytic subset of X . [Hint. Choose V a csm space and g a continuous mapping of V onto A . Study n(B), where B is the inverse image of the diagonal in Y x Y under the mapping (x, u ) + ( f ( x ) , g(u)) of X x Vinto Y x Y and 7-c is the projection of X x Vonto X.] 14.4.15. Let W be a von Neumann algebra acting on a separable Hilbert space S.Show that (= ai(9)) is an analytic subset of the (i) { I rU’ : U E @(W),U’ E @(W’)} where @(B), @(8), and @ ( X are ) the groups of unitary (csm) space %(S), and 9#(#), respectively, each provided with its strongoperators in R, W’, operator tspology [Hint. Use Exercise 14.4.9, and consider the mapping ( U , U’)+ CrU’ of %(a) x a(%?’) into %(X).];

(ii) art automorphism of 9 implemented by a unitary operator Von S is inner if and only if V€0iYi(W);

1046

14. DIRECT INTEGRALS AND DECOMPOSITIONS

(iii) T ( W ) is an analytic subset of R. [Hint. Use the result of Exercise 14.4.14 in conjunction with (i) and (ii).] 14.4.16. Let A be a factor of type 111 acting on a separable Hilbert space 2.Show that (i) T ( A ) is a subset of R having Lebesgue measure 0 [Hint. Use Exercise 14.4.15(iii), Theorem 14.3.5, Exercise 14.4.4(iii), and Exercise 14.4.1l(i).]; (ii) J1p admits an automorphism a such that a" is outer for all positive integers n. [Hint. Use (i) and note that U,"=I[n-'. T(&)] has Lebsegue measure 0.1 14.4.17. Let .A! be a factor of type I11 acting on a separable Hilbert space 2,and let tl be a * automorphism of A implemented by a unitary operator U on 2 Suppose that CI" ( =a,) is outer for each non-zero integer n. Show that (i) W ( A ,a) is a factor of type I11 [Hint. Use Proposition 13.1.5 and Exercise 13.4.2.1; nB(&, a) = { c l : CE@) [Hint. Recall the matrix descrip(ii) @(A)' and of 9(A,a). Use Exercise 12.4.17(iv).]; tions of the elements of @(A) is * isomorphic to &' [Hint. Use Proposition 9.1.6 and (iii) @(A)' Theorem 7.2.9.1; (iv) ,%(A, a)' is a proper subset of @(A)' [Hint. Show that U @ 1, is in &?(A, a) and not in @(A).]; is not normal in the sense of Exercise 12.4.31. (v) @(A)' 14.4.18. Show that a von Neumann algebra acting on a separable Hilbert space is normal (in the sense of Exercise 12.4.31) if and only if it is a factor of type I. [Hint. Use Exercises 12.4.31, 14.4.16, and 14.4.17.1 14.4.19. Let 9 acting on a Hilbert space 2 be a von Neumann algebra,

Y be a von Neumann subalgebra of 9,and t -+ V, be a (continuous)

one-parameter unitary group on 2 that implements one-parameter groups t + oi and t -+ orof automorphisms of W and Y ,respectively, where t + 6,is the modular automorphism group of Y corresponding to a faithful normal state o of .Y. Let @' be a faithful, ultraweakly continuous conditional expectation of W onto Y and Y be a group of unitary operators in 9 such that 9 and Y generate ,% as a von Neumann algebra. Suppose that V Y V * = Y , o ( V / A V * )= o(A)(A €94o;(V) , = Yand @ ( V ) = 0,foreach V ( # I ) in 9.Show that

14.4. EXERCISES

1047

is a self-adjoint sub(i) {VIA4,+ ... + V b A , : V J E ~ A, j e Y } (=a) algebra of W and '9- = 9; (ii) t + $7;is the modular automorphism group of &? corresponding to o @'. [Hint. Use (i) and Lemma 9.2.17.1 0

14.4.20. Let A, acting on the Hilbert space 2,be the factor of type I11 constructed in Exercise 13.4.12, w o be a faithful normal state of A, ( J , A ) be the modular structure and t -+ at be the modular automorphism group of A Corresponding to wo. Thus t + A" (= U ( t ) )implements t + gt. Let &?(A?,5 ) be the (implemented) crossed product of A by 5 considered as an automorphic representation of R (as a discrete group) on A?. With the notation of Definition 13.1.3, let 9 be the von Neumann algebra (which acts on 3? 0 12(R)) generated by ( Y = ) @ ( Aand ) { V ( t ): t~ G}, where G is a given subgroup of R (as a discrete group). Show that a), where a = 0 1 G [Hint. Show that the (i) 9 is * isomorphic to &?(A, projection of Ct.& onto CBEG@ A?' commutes with 9, and the restricY? is a * isomorphism of B onto 9(A,a). Consider tion of 9 to generators and matrix representations.] ; (ii) 9 is a factor of type 111. [ H i n t . Use Exercises 13.4.2, 13.4.12, and Proposition 13.1S(ii).] Let a;(T) be V(t)TV(t)*( t E R, T E B).With o the faithful normal state of Y such that w @ = coo, and @' the conditional expectation of .@(A', a) onto Y described in Exercise 13.4.1, let w' be (w W ) I R Show that (iii) o' is a faithful normal state of 9 and t +oi is the modular automorphis.m group of 92 corresponding to o' [ H i n t . Use the Exercises 13.4.1 and 14.4.19.1; (iv) Y' n B ( A ,a) = { c l : C E C }[ H i n t . Use Exercise 12.4.17(iv).]; (v) G == T ( W ) [ H i n t . Use (iii) and (iv). Study the matrix representation of V(t).]; (vi) there is a countably decomposable factor of type I11 for which the modular automorphism group consists of inner automorphisms. (Compare the result of Exercise 14.4.16(i).)

xgEG@ 0

0