Around traces and quasitraces

TL;DR

This paper surveys key results on traces and quasitraces in C*-algebras, introduces new findings on ultrapowers and faithful traces, and characterizes conditions for their existence.

Contribution

It provides new results on traces in ultrapowers and characterizes C*-algebras that admit faithful traces, advancing understanding in the field.

Findings

Existence of quasitrace in ultrapowers of traceless C*-algebras

Characterizations of C*-algebras with faithful traces

Construction of a sequence of traceless algebras with quasitraces in ultrapowers

Abstract

This paper presents a survey of results on traces and quasitraces on C$^*$-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$^*$-algebras whose ultrapower does admit a quasitrace (and likely also a trace). We characterize in different ways C$^*$-algebras that admit a faithful trace, respectively, where each quotient of the C$^*$-algebra admits a faithful trace.