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,\cdot\Omega). 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 \omega=\omega\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 \overline{N_i\Omega} and \overline{N\Omega} respectively satisfy e=\bigwedge_{i\in I}e_i. This proves a conjecture of V.F.R. Jones and F. Xu.
Intersecting Jones projections
CARPI, Sebastiano
2005-01-01
Abstract
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,\cdot\Omega). 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 \omega=\omega\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 \overline{N_i\Omega} and \overline{N\Omega} respectively satisfy e=\bigwedge_{i\in I}e_i. This proves a conjecture of V.F.R. Jones and F. Xu.File in questo prodotto:
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