Traces on topological-graph algebras

Given a topological graph $E$, we give a complete description of tracial states on the $\text{C}^{\ast }$-algebra $\text{C}^{\ast }(E)$ which are invariant under the gauge action; there is an affine homeomorphism between the space of gauge invariant tracial states on $\text{C}^{\ast }(E)$ 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 $\text{C}^{\ast }(E)$ is gauge invariant. When $E^{0}$ is totally disconnected, the gauge invariant tracial states on $\text{C}^{\ast }(E)$ are in bijection with the states on $\text{K}_{0}(\text{C}^{\ast }(E))$.