Cones of traces arising from AF $C^{\ast }$-algebras

Abstract

We characterize the topological non-cancellative cones that can be expressed as projective limits of finite powers of $[0,\infty ]$. For metrizable cones, these are also the cones of lower semicontinuous extended-valued traces on approximately finite-dimensional (AF) $C^{\ast }$-algebras. Our main result may be regarded as a generalization of the fact that any Choquet simplex is a projective limit of finite-dimensional simplices. To obtain our main result, we first establish a duality between certain non-cancellative topological cones and Cuntz semigroups with real multiplication. This duality extends the duality between compact convex sets and complete order unit vector spaces to a non-cancellative setting.