The C*-algebra of a finite T_0 topological space

Abstract

We are concerned with the following motivating question: how can one extend the classical Gelfand-Naimark theorem to the simplest non-Hausdorff topological spaces? Our model space is a finite $T_0$ topological space, or equivalently, a finite poset. We construct a faithful functor from the category of finite posets with injective morphisms to the category $C^*$, whose objects are $C^*$-algebras and whose morphisms are isomorphism classes of Hilbert $C^*$-bimodules. Then we show in various ways how the construction of this functor fails to extend to the category of finite posets.