Tracial weights on topological graph algebras

TL;DR

This paper characterizes all extremal tracial weights on topological graph C*-algebras, showing they are gauge-invariant under certain conditions, which simplifies their analysis.

Contribution

It introduces a description of invariant measures on boundary path spaces to classify tracial weights on topological graph C*-algebras, especially for free graphs.

Findings

All tracial weights are gauge-invariant for free second countable topological graphs.

In simple, separable cases, all tracial weights are gauge-invariant.

The paper provides a complete description of extremal tracial weights not gauge-invariant.

Abstract

We describe two kinds of regular invariant measures on the boundary path space of a second countable topological graph, which allows us to describe all extremal tracial weights on the graph C$^{*}$-algebra which are not gauge-invariant. Using this description we prove that all tracial weights on the C$^{*}$-algebra of a second countable topological graph are gauge-invariant when the graph is free. This in particular implies that all tracial weights are gauge-invariant when the graph C$^{*}$-algebra is simple and separable.