Intersecting Jones projections

Sebastiano Carpi
December 22, 2004
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 $\bar{N_i\Omega}$ and $\bar{N\Omega}$ respectively satisfy $e=\bigwedge_{i\in I}e_i$.This proves a conjecture of V.F.R. Jones and F. Xu in \cite{JonesXu04}.
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