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(cdot)=(Omega,cdotOmega). Moreover, let {N_i :iin I} be a family of von Neumann subalgebras of M with faithful normal conditional expectations E_i of $M$ onto N_i satisfying omega=omegacirc E_i for all iin I and let N=igcap_{iin I} N_i. We show that the projections e_i, e of H onto the closed subspaces overline{N_iOmega} and overline{NOmega} respectively satisfy e=igwedge_{iin I}e_i. This proves a conjecture of V.F.R. Jones and F. Xu.