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,\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}

On peak phenomena for non-commutative H∞H^\infty

A non-commutative extension of Amar and Lederer's peak set result is given. As its simple applications it is shown that any non-commutative $H^\infty$-algebra $H^\infty(M,\tau)$ has unique predual,and moreover some restriction in some of the results of Blecher and Labuschagne are removed, making them hold in full generality.Comment: final version (the presentation of some part is revised and one reference added

An Algebraic Spin and Statistics Theorem

A spin-statistics theorem and a PCT theorem are obtained in the context of the superselection sectors in Quantum Field Theory on a 4-dimensional space-time. Our main assumption is the requirement that the modular groups of the von Neumann algebras of local observables associated with wedge regions act geometrically as pure Lorentz transformations. Such a property, satisfied by the local algebras generated by Wightman fields because of the Bisognano-Wichmann theorem, is regarded as a natural primitive assumption.Comment: 15 pages, plain TeX, an error in the statement of a theorem has been corrected, to appear in Commun. Math. Phy