Uniform property $\Gamma$ and finite dimensional tracial boundaries

Abstract

We prove that a C${}^{\ast }$-algebra $A$ has uniform property $\Gamma$ if the set of extremal tracial states, ${\partial }_{e}T(A)$, is a non-empty compact space of finite covering dimension and for each $\tau \in {\partial }_{e}T(A)$, the von Neumann algebra ${\pi }_{\tau }(A{)}^{\prime \prime }$ arising from the GNS representation has property $\Gamma$.

24 pages; accepted version; Canad J. Math., to appear