Traces on the uniform tracial completion of $\mathcal{Z}$-stable C*-algebras (2407.00727v3)

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.