Intersecting Jones projections

Let M be a von Neumann algebra on a Hilbert space H with a cyclic and separating unit vector Omega and let omega be the faithful normal state on M given by omega(center dot) = (Omega, center dot Omega). Moreover, let {N-i : i is an element of I} be a family of von Neumann subalgebras of M with faithful normal conditional expectations E-i of M onto N-i satisfying omega = omega circle E-i for all i is an element of I and let N = boolean AND(i is an element of I)N(i). We show that the projections e(i), e of H onto the closed subspaces (N(i)Omega) over bar and (N Omega) over bar respectively satisfy e = Lambda(i is an element of I) e(i). This proves a conjecture of Jones and Xu in [1].