Uniform property $\Gamma$ and finite dimensional tracial boundaries

TL;DR

This paper establishes that a C*-algebra has uniform property Gamma when its extremal tracial states form a finite-dimensional compact space and the associated von Neumann algebras have property Gamma.

Contribution

It links the uniform property Gamma of C*-algebras to the topological and von Neumann algebraic properties of their tracial states.

Findings

Uniform property Gamma characterized by finite-dimensional extremal tracial boundary.

Connection between property Gamma in von Neumann algebras and C*-algebraic structure.

Extension of property Gamma criteria to broader classes of C*-algebras.

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 $\Gamma$.