Concrete realizations of quotients of operator spaces

Abstract

Let B be a unital C-subalgebra of a unital Calgebra A, so that A/B is an abstract operator space. We show how to realize A/B as a concrete operator space by means of a completely contractive map fromA into the algebra of operators on a Hilbert space, of the form A & [Z,A] where Z is a Hermitian unitary operator. We do not use Ruan's theorem concerning concrete realization of abstract operator spaces. Along the way we obtain corresponding results for abstract operator spaces of the form A/V where V is a closed subspace of A, and then for the more special cases in which V is a subspace or an operator system.