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Strong limit theorems for orthogonal sequences in von Neumann algebras


Author: R. Jajte
Journal: Proc. Amer. Math. Soc. 94 (1985), 229-235
MSC: Primary 46L50; Secondary 60B12, 82A15
DOI: https://doi.org/10.1090/S0002-9939-1985-0784169-7
MathSciNet review: 784169
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Abstract: Let $ A$ be a von Neumann algebra with a faithful normal state $ \phi $. It is shown that if a sequence $ ({x_n})$ in $ A$ is orthogonal relative to $ \phi $ and satisfies the condition

$\displaystyle \sum\limits_{k} {\phi (\vert{x_k}{\vert^2}){{\left( {\frac{{\log k}}{k}} \right)}^2} < \infty ,} $

then $ {}_n^1\sum\nolimits_{k = 1}^n {{x_k} \to 0} $ almost uniformly in $ A$. Some other results related to this theorem are also discussed.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0784169-7
Keywords: von Neumann algebras, faithful normal state, orthogonal sequences, independent sequences, almost uniform convergence, almost complete convergence, strong law of large numbers
Article copyright: © Copyright 1985 American Mathematical Society

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