Traces on Topological Graph Algebras

TL;DR

This paper characterizes gauge-invariant tracial states on topological graph C*-algebras, linking them to invariant measures on the vertex space and analyzing their properties in acyclic and totally disconnected cases.

Contribution

It provides a complete description of gauge-invariant tracial states on topological graph C*-algebras and establishes their correspondence with invariant measures and K-theoretic states.

Findings

Gauge-invariant tracial states correspond to invariant Radon measures on vertices.

If the graph has no cycles, all tracial states are gauge invariant.

In totally disconnected cases, these states correspond to K_0 algebra states.

Abstract

Given a topological graph $E$, we give a complete description of tracial states on the C*-algebra $\mathrm{C}^*(E)$ which are invariant under the gauge action; there is an affine homeomorphism between the space of gauge invariant tracial states on $\mathrm{C}^*(E)$ and Radon probability measures on the vertex space $E^0$ which are, in a suitable sense, invariant under the action of the edge space $E^1$. It is shown that if $E$ has no cycles, then every tracial state on $\mathrm{C}^*(E)$ is gauge invariant. When $E^0$ is totally disconnected, the gauge invariant tracial states on $\mathrm{C}^*(E)$ are in bijection with the states on $\mathrm{K}_0(\mathrm{C}^*(E))$.