Intersecting Jones projections

Abstract

Let M be a von Neumann algebra on a Hilbert space H with a cyclic and separating unit vector Ωand let ωbe the faithful normal state on M given by ω(\cdot)=(Ω,\cdotΩ). Moreover, let {N_i :i\in I} be a family of von Neumann subalgebras of M with faithful normal conditional expectations E_i of M onto N_i satisfying ω=ω\circ E_i for all i\in I and let N=\bigcap_{i\in I} N_i. We show that the projections e_i, e of H onto the closed subspaces \bar{N_iΩ} and \bar{NΩ} respectively satisfy e=\bigwedge_{i\in I}e_i.This proves a conjecture of V.F.R. Jones and F. Xu in \cite{JonesXu04}.