Cones of traces arising from AF C*-algebras

TL;DR

This paper characterizes certain topological cones as projective limits and relates them to traces on AF C*-algebras, extending duality concepts to a non-cancellative setting.

Contribution

It introduces a duality between non-cancellative topological cones and Cuntz semigroups, generalizing existing dualities to a broader context.

Findings

Characterization of cones as projective limits of finite powers of [0,∞]

Duality established between non-cancellative cones and Cuntz semigroups

Extension of duality between convex sets and order unit spaces

Abstract

We characterize the topological non-cancellative cones that are expressible as projective limits of finite powers of $[0,\infty]$. These are also the cones of lower semicontinuous extended-valued traces on AF C*-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.