Operator valued weights without structure theory

A result of Haagerup, generalizing a theorem of Takesaki, states the following: If K C M are von Neumann algebras, then there exists a faithful, normal and semi-finite (fns) operator valued weight T: M?+ N + if and only if there exist fns weights b on M and cp on K satisfying r't (x) = oa (x) Vx E K, t E R. In fact, T can be chosen such that f = p o T; T is then uniquely determined by this condition. We present a proof of the above which does not use any structure theory. 0. INTRODUCTION Consider the following theorem of Haagerup: Theorem. Let KV c M be von Neumann algebras. There exists a faithful, semifinite normal operator valued weight T: M+ -S K+ if and only if there exist faithful semi-finite normal weights (p on M and (p on KV such that a,' (x) = UCf (x), x E J\r. If this condition is satisfied, then T can be chosen in such a way that 0 = 0 o T; moreover, T is uniquely determined by this identity. Haagerup's proof of the above depends upon the use of structure theory, i.e., the existence of a crossed product decomposition of a general von Neumann algebra [Haal], [Haa2]. However, it is possible to demonstrate this without recourse to the crossed product. The method for showing this depends critically on a result of Connes-Masuda which provides a converse to the theorem purporting the existence of the Radon-Nikodym cocycle derivative. (Masuda's result does not use structure theory, unlike that of Connes.) This result implies that, given an invariance condition like that in the hypothesis, a weight on KV induces one on M. With the ability to associate weights on JKto weights on M, it is possible to construct an operator valued weight T: M+ -+ K+. The desire to recast the proof of the above in a form which does not depend on any structure theory is motivated by the belief that it is a result which is fundamental in the theory of non-commutative integration. Specifically, application of an operator valued weight to a positive element of M (which results in an element in Received by the editors January 30, 1997. 1991 Mathematics Subject Classification. Primary 46L50; Secondary 22D25. This work is supported, in part, by NSF Grant DMS95-00882. ?1999 American Mathematical Society 323 This content downloaded from 157.55.39.27 on Wed, 07 Sep 2016 05:33:38 UTC All use subject to http://about.jstor.org/terms 324 TONY FALCONE AND MASAMICHI TAKESAKI the extended positive cone of A) can and should be thought of as "partial integration." The "space" over which we are integrating is not always explicit; however, just as we often think of von Neumann algebras as "non-commutative L?-spaces," we sense the presence of an underlying measure space indirectly, by observing the interactions of function-like objects defined on this "space." Moreover, this point of view allows us to interpret the expression = o T as a form of Fubini's theorem (or more precisely, Tonelli's theorem). Because we believe that structure theory should, in some sense, be a consequence of the results of integration theory, and not vice-versa, a proof which does not involve the crossed product is indicated. It should also be noted that work in this area was done by Hirakawa [Hir]. The work presented herein addresses the problem from a different direction than he did, however, and therefore represents a new approach. 1. SOME PRELIMINARIES In order to discuss the subject of operator valued weights, we begin with a review of some of the relevant terms and concepts. When studying weights, we consider the extended positive real numbers R+ U {oo}. To study "unbounded conditional expectations" (i.e., operator valued weights), we need to consider the "extended positive part" MAf+ of the von Neumann subalgebra H of M. We begin with the following: Definitlion 1.1. For a von Neumann algebra M, the extended positive cone M+ of M is the set of maps m: +[0, oo] with the following properties: (i) m(A(p) = Am((p), p E MA+ A > 0, (iii) m is lower semi-continuous. Clearly, the positive part M+ of M is a subset of M+. It is easy to see that M+ is closed under addition, multiplication by non-negative scalars and increasing limits. Example 1.2. Let {M, S} be a von Neumann algebra and A a positive selfadjoint operator on Si affiliated with M. Suppose that A j / A de(A) is the spectral decomposition of A. For each S? e M+, put mA (.p) A d(p(e(A)) Then mA satisfies the conditions (i), (ii) and (iii) of Definition 1.1. The last condition, the lower semi-continuity, follows from n mA((o) = sup ,o(A,) with AnjAde(A) e M+. n It now follows that mA(wa) A d(e-(A), = A /241121 E?(A,1/2) If B is another positive self-adjoint operator on Sj affiliated with M, then the equality mA mB means precisely A = B. Hence the map A F mA C M+ is This content downloaded from 157.55.39.27 on Wed, 07 Sep 2016 05:33:38 UTC All use subject to http://about.jstor.org/terms OPERATOR VALUED WEIGHTS WITHOUT STRUCTURE THEORY 325 injective. Thus, the set of positive self-adjoint operators affiliated with M can be identified with a subset of the extended positive cone M+. Definition 1.3. For m, n E M+, A > 0 and a E M, we define the following operations: (Am)p) :-Am((p), EC M*+, (Tm + n) ((p) Tn m(() + n ((p), fo E M*+, (a*ma)((p) :-m(a(pa*), ,o E M*. We also note here that supi mi of an increasing net in M+ can be naturally defined. Lemma 1.4. Let {M,5i} be a von Neumann algebra. To each m E M+, there corresponds uniquely a pair {A, A} of a closed subspace A of S) and a positive selfadjoint operator A on A such that (i) A is affiliated with M, in the sense that the projection to A belongs to M, and A is affiliated with M; (ii) (1) ~ ~ ~~~ m(Wt,) IIIA1/(l E: E-(A/2 1 +oo, otherwise. Here wz means, of course, the functional x C M A4 (x (x ). The proof of the above lemma, as well as the next theorem and its corollary, is standard, and is therefore omitted. (See, for instance, [Haal].) We say that an element m E M+ is semi-finite if { E M+ : m((p) 0 for every non-zero o E M*. Theorem 1.5. Let M be a von Neumann algebra. Each m E M+ has a unique spectral decomposition of the form (2) m(A) j A\d(p(e(A)) + oo(p(p), EM+, where {e(A): A E R+} is an increasing family of projections in A/I which is ostrongly continuous from the right, and p = -limAOe(A). Furthermore, e(O) = 0 if and only if m is faithful, and p 0 O if and only if m is semi-finite. To simplify the notation, we write (2') m-H+oop, H= A de (A), when m has the form of (2). We keep the convention 0 (+oo) = 0. Although we consider H as an operator affiliated with M, we use the following notation: D(H1/2) = {d E : m(WS) O, m cM+, o(m?+rn) =z((m)-+Vp(n), m,n E M+, W(sup mi) = sup W(mi) i i for any increasing net {mi} in M+, where we say the net {mi} is increasing if {mi(w)} is increasing for every w c M+. So, we may think of the extended positive cone of M as M+, along with a myriad of "points at infinity" adjoined to it. There will be a different infinite point for each projection in M. When we think about the extended positive cone in this way, it is clear why we adopt the notation in (2'). Now, for each m = H + oop, we put mno =(I + H)-,(l-P), m? = H(1 + H)I(1-p) + -p > Or We note that both mo and m, are bounded. Lemma 1.7. (i) For each m, n A M+, we have the following equivalence: m no X~ m? 0. (ii) Let {mi} H= Hi + oopi} be an increasing net in M+ and m = H + oop. Then we have mi / m X (mi)o \ mO X (mi)/ m/ , ? > 0. Proof. (i) Suppose m -14) so that ((Ei + K)-1) |) ((I + K)-141 ) S5 Setting (1 + K)-'1 = 0 for ( c ?' n i, we view (1 + K)-1 as an operator on A and have (1 + H)-1 > (1 + K)-1. Then we have (1 + H)-1/2 > (1 + K)-1/2, and D(H1/2) (1 + H)--1/2.q D (1 + K)'/2-q =D(K1/2) The argument in the first paragraph shows that (1 + H)(1 + E(1 + H)- lim 11 (1 + H)/2(1 + E(1 + H)-1)-1/2(112 =11 (1 + H) 1/2( 112 so we conclude that 1 + n > 1 + rn; equivalently n > m. Now we have seen the equivalence: m no. For a fixed E > 0, we have then m (En)o X 1 (Em)o 1(En)o X m = -(1 (em)0) < --(1E-(n)o) = n(ii) By (i), the net {(mj)o} is decreasing. If = infi(mi)o, then there exists n c AM+ such that no = X, because (mi)o < 1 implies e < 1. If m = supi mi, then we have mo < (mj)o, so mo ? no, which implies n < m by (i). Hence mo = infi(mi)o = lim(mi)o. Thus we prove the equivalence: mi / m X (mi)o \i mO. Finally, the equality