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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Irreducible algebras of operators which contain a minimal idempotent.

Author: Bruce A. Barnes
Journal: Proc. Amer. Math. Soc. 30 (1971), 337-342
MSC: Primary 46.65
MathSciNet review: 0290118
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Abstract: We prove that when A is a closed subalgebra of the bounded operators on a reflexive Banach space X, which acts irreducibly on X and contains a minimal idempotent, then every bounded operator with finite dimensional range on X is in A. We use this result to prove that every continuous irreducible representation of a GCR-algebra on a Hilbert space $ \mathcal{H}$ is similar to a $ ^ \ast $-representation on $ \mathcal{H}$.

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Keywords: Algebras of operators, irreducible algebra, irreducible representation, $ {B^ \ast }$-algebra
Article copyright: © Copyright 1971 American Mathematical Society

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