Intersecting Jones projections
Abstract
Let M be a von Neumann algebra on a Hilbert space H with a cyclic and separating unit vector and let ω be the faithful normal state on M given by ω(·)=(,·). Moreover, let Ni :i∈ I be a family of von Neumann subalgebras of M with faithful normal conditional expectations Ei of M onto Ni satisfying ω=ω Ei for all i∈ I and let N=i∈ I Ni. We show that the projections ei, e of H onto the closed subspaces Ni and N respectively satisfy e=i∈ Iei.This proves a conjecture of V.F.R. Jones and F. Xu in JonesXu04.
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