C*-Algebras Generated by Operator Systems

Abstract

An operator systemX, such thatX** is aC*-algebra and such that the canonical embedding ofXinX** is a unital complete isometry, is called aC*-system. IfAis a unitalC*-algebra andLis a closed left ideal ofA, thenA/(L+L*) has a natural operator-system structure relative to which it is aC*-system. It is shown that any separableC*-system is of this form for some separableA, and that an arbitrary inseparableC*-system is an inductive limit of separableC*-systems. Associated with any operator systemXis a universalC*-algebraC*u(X) which containsXas a generating subsystem and is maximal among suchC*-algebras in the sense that any other suchC*-algebra is canonically a*-homomorphic image ofC*u(X). IfXis additionally aC*-system, then theC*-subalgebraC*r(X) ofX** generated byXis minimal amongC*-algebras containingXas a generating subsystem in the sense thatC*r(X) is canonically a*-homomorphic image of any other suchC*-algebra. It follows that there is a canonical homomorphismσ:C*u(X)→C*r(X) which extends the identity map onX. WhenXis aC*-algebra of dimension greater than 1,σis never injective. One of the main results of the paper is that there exists a separable nuclearC*-systemXcontainingM2(C) for whichσis an isomorphism. This implies that for any embedding ofXas an operator subsystem of aC*-algebraA, theC*-subalgebra ofAgenerated byXis isomorphic toC*u(X). MoreoverC*u(M2(C)) is notC*-exact andC*u(M2(C)⊆C*u(X), from which it follows thatC*u(X), and henceA, are not exactC*-algebras. ThusX, though a nuclear operator system, cannot be embedded in a nuclearC*-algebra. © 1998 Academic Press.