Completeness of *-symmetric Gelfand-Naimark-Segal inner product spaces

Abstract

Every state ϱ on a C*-algebra A induces a *-symmetric semi-inner product (x, y)↦ ϱ(y* x) + ϱ(xy*) (x, y ∈ A). The main scope of the paper is to characterize those states for which the induced *-symmetric Gelfand–Naimark–Segal inner product space is complete. It is shown that this happens precisely when ϱ is a finite convex combination of pure states. (It is well known that the same conclusion follows if one considers the non-symmetric semi-inner product (x, y) ↦ ϱ(y* x).) In so doing, we exhibit an interesting connection between convexity properties of states, the transitivity of irreducible representations and Banach space properties of the quotients of C*-algebras by kernels of states.