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A characterization of nonunital operator algebras

Author: Zhong-Jin Ruan
Journal: Proc. Amer. Math. Soc. 121 (1994), 193-198
MSC: Primary 47D25; Secondary 46L05, 47D15
MathSciNet review: 1186994
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Abstract: We give an abstract matrix norm characterization for operator algebras with contractive approximate identities by using the second dual approach. We show that if A is an $ {L^\infty }$-Banach pseudoalgebra with a contractive approximate identity, then the second dual $ {A^{ \ast \ast }}$ of A is a unital $ {L^\infty }$-Banach pseudoalgebra containing A as a subalgebra. It follows from the Blecher-Ruan-Sinclair characterization theorem for unital operator algebras that $ {A^{ \ast \ast }}$ is completely isometrically unital isomorphic to a concrete unital operator algebra on a Hilbert space. Thus A is completely isometrically isomorphic to a concrete nondegenerate operator algebra with a contractive approximate identity.

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Keywords: Operator algebras, $ {L^\infty }$-Banach pseudoalgberas, $ {L^\infty }$-matricially normed spaces, operator duals, completely bounded maps
Article copyright: © Copyright 1994 American Mathematical Society

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