Traces on ultrapowers of C*-algebras

Abstract

Using Cuntz semigroup techniques, we characterize when limit traces are dense in the space of all traces on a free ultrapower of a C*-algebra. More generally, we consider density of limit quasitraces on ultraproducts of C*-algebras. Quite unexpectedly, we obtain as an application that every simple C*-algebra that is (m,n)-pure in the sense of Winter is already pure. As another application, we provide a partial verification of the first Blackadar–Handelman conjecture on dimension functions. Crucial ingredients in our proof are new Hahn–Banach type separation theorems for noncancellative cones, which in particular apply to the cone of extended-valued traces on a C*-algebra.