Uniform property $Γ$ and finite dimensional tracial boundaries

Abstract

We prove that a C$^*$-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 $\pi_\tau(A)''$ arising from the GNS representation has property $\Gamma$.