Traces on Topological Graph Algebras

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 $\mathrm{K}_0(\mathrm{C}^*(E))$.