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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


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

Author: Jon Kraus
Journal: Trans. Amer. Math. Soc. 279 (1983), 357-376
MSC: Primary 46L10; Secondary 46L55, 46M05, 47D25
MathSciNet review: 704620
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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... ...hi \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}$.

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Additional Information

PII: S 0002-9947(1983)0704620-0
Keywords: Tensor product, slice map, Property $ {S_\sigma }$, semidiscrete, $ {W^{\ast}}$-dynamical system, spectral subspace, subspace lattice, reflexive algebra
Article copyright: © Copyright 1983 American Mathematical Society

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