The smallness problem for C*-algebras

Abstract

Akemann showed that any von Neumann algebra with a weak * separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C * -algebra has a weak * separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C * -algebra has the weak * separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C * -algebra almost separably representable. We say that a unital C * -algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C * -algebra is small if and only if the state spaces of all n by n matrix algebras over the C * -algebra are weak * -separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C * -algebras but the general question remains open.