Faithful tracial states on quotients of C*-algebras

TL;DR

This paper investigates conditions under which C*-algebras and their quotients admit faithful tracial states, linking these conditions to the Cuntz semigroup and strong quasidiagonality, with implications for group C*-algebras.

Contribution

It provides a necessary and sufficient condition for faithful tracial states on C*-algebras using the Cuntz semigroup and relates this to strong quasidiagonality and group properties.

Findings

Characterization of faithful tracial states via the Cuntz semigroup

Equivalence condition for all quotients to admit faithful tracial states

Link between faithful tracial states on group C*-algebras and strong quasidiagonality

Abstract

We study the existence of faithful tracial states on C*-algebras as well as the stronger proerty that all quotients admit faithful tracial states. We provide a sufficient and necessary condition for when C*-algebras admit faithful tracial states in terms of the Cuntz semigroup and use this to give an equivalent formulation for all quotients to admit faithful tracial states. We relate this to the notion of strong quasidiagonality, and show that any amenable discrete group with faithful tracial states on all quotients of the corresponding group-C*-algebra is strongly quasidiagonal under the condition that all quotients satisfy the Universal Coefficient Theorem.