An operator system X, such that X** is a C*-algebra and such that the canonical embedding of X in X** is a unital complete isometry, is called a C*-system. If A is a unital C*-algebra and L is a closed left ideal of A, then Aร(L+L*) has a natural operator-system structure relative to which it is a C*-system. It is shown that any separable C*-system is of this form for some separable A, and that an arbitrary inseparable C*-system is an inductive limit of separable C*-systems.
Associated with any operator system X is a universal C*-algebra C u *(X) which contains X as a generating subsystem and is maximal among such C*-algebras in the sense that any other such C*-algebra is canonically a *-homomorphic image of C u *(X). If X is additionally a C*-system, then the C*-subalgebra C r *(X ) of X** generated by X is minimal among C*-algebras containing X as a generating subsystem in the sense that C r *(X ) is canonically a *-homomorphic image of any other such C*-algebra. It follows that there is a canonical homomorphism _: C u *(X ) ร C r *(X) which extends the identity map on X. When X is a C*-algebra of dimension greater than 1, _ is never injective. One of the main results of the paper is that there exists a separable nuclear C*-system X containing M 2 (C) for which _ is an isomorphism. This implies that for any embedding of X as an operator subsystem of a C*-algebra A, the C*-subalgebra of A generated by X is isomorphic to C u *(X). Moreover C u *(M 2 (C)) is not C*-exact and C u *(M 2 (C) C u *(X ), from which it follows that C u *(X ), and hence A, are not exact C*-algebras. Thus X, though a nuclear operator system, cannot be embedded in a nuclear C*-algebra.
1998 Academic Press
1. Introduction
Informally an operator system is a closed unital self-adjoint linear subspace of a C*-algebra with unit. More precisely, an operator system X is a complex vector space with involution x ร x*, a distinguished self-adjoint article no.