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A characterization of thin operators in a von Neumann algebra


Author: Catherine L. Olsen
Journal: Proc. Amer. Math. Soc. 39 (1973), 571-578
MSC: Primary 46L10
DOI: https://doi.org/10.1090/S0002-9939-1973-0341121-5
MathSciNet review: 0341121
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Abstract: Let $ \mathcal{A}$ be a von Neumann algebra, $ \mathcal{J}$ a uniformly closed, weakly dense, two-sided ideal in $ \mathcal{A},\mathcal{L}$ the center of $ \mathcal{A}$, and $ \mathcal{P}$ the lattice of projections in $ \mathcal{J}$. An operator $ A \in \mathcal{A}$ is thin relative to $ \mathcal{J}$ if $ A = Z + K$, for some $ Z \in \mathcal{L},K \in \mathcal{J}$. The thin operators relative to $ \mathcal{J}$ are characterized as those $ A \in \mathcal{A}$ satisfying $ {\lim _{P \in \mathcal{P}}}\vert\vert AP - PA\vert\vert = 0$. It is also shown that

$\displaystyle \mathop {\lim \sup }\limits_{P \in \mathcal{P}} \vert\vert PAP - ... ... \mathop {\lim \sup }\limits_{P \in \mathcal{P}} \vert\vert PAP - PA\vert\vert.$


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DOI: https://doi.org/10.1090/S0002-9939-1973-0341121-5
Article copyright: © Copyright 1973 American Mathematical Society

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