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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Regularity for operator algebras on a Hilbert space


Author: John Froelich
Journal: Proc. Amer. Math. Soc. 119 (1993), 1269-1277
MSC: Primary 47D25; Secondary 46L99
MathSciNet review: 1181164
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Abstract: Four notions of regularity for operator algebras are introduced. An algebra $ A$ is called $ 1$-regular if for any two linearly independent vectors $ x,y \in H$ there is an $ a \in A$ such that $ ax = 0$ and $ ay \ne 0$. We show that the only weakly closed transitive $ 1$-regular algebra is $ B(H)$, thus providing a natural generalization of the Rickart-Yood density theorem. We construct an example of a $ 1$-regular algebra which contains no nonzero compact operators. This example is related to the "thin set" phenomena of classical harmonic analysis.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1181164-8
PII: S 0002-9939(1993)1181164-8
Article copyright: © Copyright 1993 American Mathematical Society