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A note on compact ideal perturbations in semifinite von Neumann algebras


Author: Florin Pop
Journal: Proc. Amer. Math. Soc. 119 (1993), 843-847
MSC: Primary 46L10
DOI: https://doi.org/10.1090/S0002-9939-1993-1184084-8
MathSciNet review: 1184084
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Abstract: Let $ M$ be a semifinite von Neumann algebra and denote by $ J(M)$ the closed two-sided ideal generated by the finite projections in $ M$. A subspace $ S \subset M$ is called local if it is equal to the ultraweak closure of $ S \cap J(M)$. If $ M = B(H)$ and $ J(M) = K(H)$, Fall, Arveson, and Muhly proved that $ S + J(M)$ is closed for every local subspace $ S$.

In this note we prove that if $ M$ is a type $ {\text{I}}{{\text{I}}_\infty }$, factor, then there exist local subspaces in $ M$ which fail to have closed compact ideal perturbations; thus answering in the negative a question of Kaftal, Larson, and Weiss.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1184084-8
Article copyright: © Copyright 1993 American Mathematical Society

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