A Characterization of Nonunital Operator Algebras

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.