INTERSECTING JONES PROJECTIONS

Let M be a von Neumann algebra on a Hilbert space ℋ with a cyclic and separating unit vector Ω and let ω be the faithful normal state on M given by ω(·) = (Ω,·Ω). Moreover, let {N_i : i ∈ 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 ◦ E_i for all i ∈ I and let N = ∩_i∈I N_i. We show that the projections e_i, e of ℋ onto the closed subspaces and respectively satisfy e = ∧_i∈I e_i. This proves a conjecture of Jones and Xu in [1].