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Traces on topological-graph algebras

Published online by Cambridge University Press:  02 May 2017

CHRISTOPHER SCHAFHAUSER
CHRISTOPHER SCHAFHAUSER*
Affiliation:
Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada, N2L 3G1 email [email protected]
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Abstract

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 $\text{K}_{0}(\text{C}^{\ast }(E))$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 5 , August 2018 , pp. 1923 - 1953
Copyright
© Cambridge University Press, 2017 

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