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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The slice map problem for $\sigma$-weakly closed subspaces of von Neumann algebras
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by Jon Kraus PDF
Trans. Amer. Math. Soc. 279 (1983), 357-376 Request permission

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 ${\text {alg}}({\mathcal {L}_{1}} \otimes {\mathcal {L}_2}) = {\text {alg}}\;{\mathcal {L}_{1}} \overline \otimes {\text {alg}}\;{\mathcal {L}_2}$.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 279 (1983), 357-376
  • MSC: Primary 46L10; Secondary 46L55, 46M05, 47D25
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0704620-0
  • MathSciNet review: 704620