On isotropic subspaces of operators in II1 factors

Abstract

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we firstly prove that given an X∈M, then there is a decomposition of the identity into N∈N⋃{+∞} mutually orthogonal nonzero projections {En}n=1N⊆M, I=∑n=1NEn, such that EnXEn=τ(X)En for all n=1,⋯,N. Secondly, we show that if R is the hyperfinite II1 factor and (Rω,τω) is the ultrapower of R, then for any X∈Rω, there is a finite family of mutually orthogonal nonzero projections {Ei}i=1N in Rω such that ∑i=1NEi=I and EnXEn=τ(X)En, for all n=1,2,⋯,N. Equivalently, there is a unitary operator U∈Rω such that 1N∑i=1N(U⁎)iXUi=τω(X)I.