Traces on the uniform tracial completion of $\mathcal{Z}$-stable C*-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 $\Vert a{\Vert }_{2,T(A)}={\sup }_{\tau \in T(A)}\tau (a^{\ast }a{)}^{1/2}$. The trace problem asks whether every trace on the uniform tracial completion is the $\Vert \cdot {\Vert }_{2,T(A)}$-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.

22 pages; accepted version; J. Lond. Math. Soc., to appear