A characterization of nonunital operator algebras

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 Hubert space. Thus A is completely isometrically isomorphic to a concrete nondegenerate operator algebra with a contractive approximate identity.