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A Characterization of Nonunital Operator Algebras

Zhong-Jin Ruan
Proceedings of the American Mathematical Society
Vol. 121, No. 1 (May, 1994), pp. 193-198
DOI: 10.2307/2160382
Stable URL: http://www.jstor.org/stable/2160382
Page Count: 6
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A Characterization of Nonunital Operator Algebras
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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∞-Banach pseudoalgebra with a contractive approximate identity, then the second dual A** of A is a unital L∞-Banach pseudoalgebra containing A as a subalgebra. It follows from the Blecher-Ruan-Sinclair characterization theorem for unital operator algebras that A** 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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