Traces on the uniform tracial completion of $\mathcal{Z}$-stable C*-algebras

TL;DR

This paper investigates the trace extension problem for the uniform tracial completion of *-algebras, providing a positive answer for *-algebras absorbing the Jiang-Su algebra, which are central to classification theory.

Contribution

It proves that every trace on the uniform tracial completion extends continuously from the original *-algebra for *-algebras absorbing the Jiang-Su algebra.

Findings

Every trace on the uniform tracial completion is a continuous extension.

The result applies to *-algebras tensorially absorbing the Jiang-Su algebra.

Supports the classification program for *-algebras.

Abstract

The uniform tracial completion of a C*-algebra A with compact non-empty trace space T(A) is obtained by completing the unit ball with respect to the uniform 2-seminorm $\|a\|_{2,T(A)}=\sup_{\tau \in T(A)} \tau(a^*a)^{1/2}$. The trace problem asks whether every trace on the uniform tracial completion is the $\|\cdot\|_{2,T(A)}$-continuous extension of a trace on A. We answer this question positively in the case of C*-algebras that tensorially absorb the Jiang-Su algebra, such as those studied in the Elliott classification programme.