Commutativity in operator algebras

Blecher, David P.

doi:10.1090/S0002-9939-1990-1009985-X

Commutativity in operator algebras
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Proc. Amer. Math. Soc. 109 (1990), 709-715 Request permission

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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