Around traces and quasitraces

Abstract

This paper presents a survey of results on traces and quasitraces on C${}^{\ast }$-algebras, and it provides some new results on traces on ultrapowers and on the existence of faithful traces. As for the former, we exhibit a sequence of traceless simple, separable, unital, nuclear C${}^{\ast }$-algebras whose ultrapower does admit a quasitrace (and likely also a trace). We characterize in different ways C${}^{\ast }$-algebras that admit a faithful trace, respectively, where each quotient of the C${}^{\ast }$-algebra admits a faithful trace.

21 pages