The slice map problem for $\sigma$-weakly closed subspaces of von Neumann algebras

Abstract:

A $\sigma$-weakly closed subspace $\mathcal {S}$ of $B(\mathcal {H})$ is said to have Property ${S_\sigma }$ if for any $\sigma$-weakly closed subspace $\mathcal {T}$ of a von Neumann algebra $\mathcal {N},\{ x \in \mathcal {S}\;\overline \otimes \mathcal {N}:{R_\varphi }(x) \in \mathcal {T}\; {\text {for all}}\;\varphi \in B{(\mathcal {H})_{\ast }}\} = \mathcal {S} \overline \otimes \mathcal {T}$, where ${R_\varphi }$ is the right slice map associated with $\varphi$. It is shown that semidiscrete von Neumann algebras have Property ${S_\sigma }$, and various stability properties of the class of $\sigma$-weakly closed subspaces with Property ${S_\sigma }$ are established. It is also shown that if $(\mathcal {M},G,\alpha )$ is a ${W^{\ast }}$-dynamical system such that $\mathcal {M}$ has Property ${S_\sigma }$ and $G$ is compact abelian, then all of the spectral subspaces associated with $\alpha$ have Property ${S_\sigma }$. Some applications of these results to the study of tensor products of spectral subspaces and tensor products of reflexive algebras are given. In particular, it is shown that if ${\mathcal {L}_1}$ is a commutative subspace lattice with totally atomic core, and ${\mathcal {L}_2}$ is an arbitrary subspace lattice, then ${\text {alg}}({\mathcal {L}_{1}} \otimes {\mathcal {L}_2}) = {\text {alg}}\;{\mathcal {L}_{1}} \overline \otimes {\text {alg}}\;{\mathcal {L}_2}$.