Concrete realizations of quotients of operator spaces

TL;DR

This paper presents a new method to concretely realize quotients of operator spaces derived from C*-algebras using specific completely contractive maps, avoiding reliance on Ruan's theorem.

Contribution

It introduces a novel concrete realization technique for quotients of operator spaces from C*-algebras via maps involving Hermitian unitaries, bypassing Ruan's theorem.

Findings

Realization of A/B as a concrete operator space using a specific map

Extension of results to quotients A/V where V is a subspace

Application to operator systems and *-subspaces

Abstract

Let B be a unital C*-subalgebra of a unital C*-algebra 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 from A into the algebra of operators on a Hilbert space, of the form a maps to [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.