The Gelfand–Naimark–Segal theorem gives an abstract characterization of the Banach ∗-algebras that can be represented ∗-isomorphically as C*-subalgebras of B(H) for some Hilbert space H. Thus, the GNS theorem frees us from always having to regard C*-algebras as concrete subalgebras of some Hilbert space. At the same time we may continue to regard them as concrete C*-subalgebras when that might aid us in a proof. For example, proving that the quotient of a concrete C*-subalgebra of some B(H) by a two-sided ideal can again be regarded as a C*-subalgebra of some B(K) would be quite difficult without the GNS theorem. On the other hand defining the norm on Mn(A) and many other constructions are made considerably easier by regarding A as a concrete C*-subalgebra of some B(H).

In this chapter we shall develop the Choi–Effros [49] abstract characterization of operator systems and Ruan's [203] abstract characterization of operator spaces. In analogy with the GNS theory, these characterizations will free us from being forced to regard operator spaces and systems as concrete subspaces of operators.

We begin with the theory of abstract operator systems. We wish to characterize operator systems up to complete order isomorphism. To this end, let S be a complex vector space, and assume that there exists a conjugate linear map s → s* on S with (s*)* = s for all s in S.