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Proceedings of the American Mathematical Society

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



Commutativity in operator algebras

Author: David P. Blecher
Journal: Proc. Amer. Math. Soc. 109 (1990), 709-715
MSC: Primary 46L05; Secondary 47D25
MathSciNet review: 1009985
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Abstract: By an 'operator algebra' we shall mean a subalgebra $ \mathcal{A}$ of the algebra $ \mathcal{B}(\mathcal{H})$ of bounded operators on a Hilbert space $ \mathcal{H}$, together with the matrix normed structure $ \mathcal{A}$ inherits from $ \mathcal{B}(\mathcal{H})$. A unital operator algebra is an operator algebra with an identity of norm 1. Note that we do not require the algebra to be self-adjoint or uniformly closed. Such algebras were characterized abstractly by Ruan, Sinclair, and the author up to complete isometric isomorphism. In this paper we study commutativity for operator algebras, and we give a characterization of commutative unital operator algebras and a characterization of unital uniform algebras.

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