KMS states on the \(C^*\)-algebras of reducible graphs. (English) Zbl 1348.46067
Summary: We consider the dynamics on the \(C^{*}\)-algebras of finite graphs obtained by lifting the gauge action to an action of the real line. M. Enomoto et al. [Math. Jap. 29, 607–619 (1984; Zbl 0557.46039)] proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single Kubo-Martin-Schwinger (KMS) state. We have previously studied the dynamics on the Toeplitz algebra, and explicitly described a finite-dimensional simplex of KMS states for inverse temperatures above a critical value. Here we study the KMS states for graphs with reducible vertex matrix, and for inverse temperatures at and below the critical value. We prove a general result which describes all the KMS states at a fixed inverse temperature, and then apply this theorem to a variety of examples. We find that there can be many patterns of phase transition, depending on the behaviour of paths in the underlying graph.
Citations:
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