
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 , we give a complete description of tracial states on the C*-algebra which are invariant under the gauge action; there is an affine homeomorphism between the space of gauge invariant tracial states on and Radon probability measures on the vertex space which are, in a suitable sense, invariant under the action of the edge space . It is shown that if has no cycles, then every tracial state on is gauge invariant. When is totally disconnected, the gauge invariant tracial states on are in bijection with the states on .
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