A characterization of operator algebras

An operator algebra is a uniformly closed algebra of bounded operators on a Hilbert space. In this paper we give a characterization of unital operator algebras in terms of their matricial norm structure. More precisely if A is an L^∞-matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1, then there is a completely isometric isomorphism of A onto a unital operator algebra. Indeed the multiplication on A need not be assumed to be associative for this conclusion to follow. Examples are given to show that the condition on the identity is necessary. It follows from the above that the quotient of an operator algebra by a closed two-sided ideal (with the natural matricial structure) is again an operator algebra up to complete isometric isomorphism.