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

Abstract

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