KMS states on finite-graph \(C^{\ast}\)-algebras (Q2846876)

The important contribution of this paper is that it shows how sources and sinks in a finite graph can give rise to KMS states on graph algebras endowed with the gauge-action. The authors show that for large inverse temperature \(\beta\), the set of extreme KMS\(_\beta\) states is parametrized by the set of sinks of the graph. If one compares \(C^*\)-algebras associated to finite graphs with \(C^*\)-algebras associated with complex dynamical systems of rational functions, then this description of the extremal KMS states indicates that sinks correspond to branch points of a rational function from the points of KMS states. The main technical tool is a generalization of a result of \textit{M. Laca} and \textit{S. Neshveyev} [J. Funct. Anal. 211, No. 2, 457--482 (2004; Zbl 1060.46049)]. More precisely, the authors extend the general Laca-Neshveyev theorem about the construction of KMS states on Pimsner algebras to the case of relative Cuntz-Pimsner algebras associated to \(C^*\)-correspondences with countable bases and a left action that is not necessarily injective.