Uniform property $ \Gamma $ 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 $ .