An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite 𝑤*-algebras

Abstract

Suppose that (s%)if¡, is a family of semifinite w*-algebras and that /¿(is a normal state of sft with /¿¡(1) = 1 for each i el. Let s/=0ls, (s/u /¿¡) and let AK-+Ax denote the natural injection of ^ into si. (The notation is explained in §3 below; si is the (/¿^-incomplete direct product of (s¿¡): see [9], [12] or [1].) Given a normal state vt of s/t for each i e I, a normal state v of s/ is written (x)je/ vt when " in ï) = n "«(^o for all Aiesii and all finite subsets F of /. Our main result (Theorem 4.1) is that Ç §ie,vi exists on sé if and only if 2»6í \d(p,, v¡)]2 converges, or, equivalently, if and only if Y}ie, p(p¡, v¡) converges. Here d is a metric on the set of normal states of a w*-algebra 38. d is defined essentially by d(p, v) = inf{\\x-y\\}, the infimum being taken over all vectors x and v inducing p. and v relative to a representation of ¿% as a von Neumann algebra. p is a kind of inner product defined by 2p(p,v) = p(l) + v(l)-[d(p,V)]2. We show that d and p correspond to Kakutani's d and p [6] when J1 is abelian (and normal states are made, in the usual fashion, to correspond to measures absolutely continuous with respect to a fixed measure). Thus our result reduces to Kakutani's [6] when each sfx is abelian. We give two applications of our main result. First, suppose that fa is an iso-morphism of the w*-algebra s/¡ onto the w*-algebra ^¡. Then we show that an isomorphism from (g) (s/t, p¡) to (g) (äS{, v¡) such that (Â~i) = i(Ai) for all At e s/t and all i e I exists if and only if 2 [d(p¡, vt ° fa)]2 < oo. This result generalizes results in [1] and [7].