We study the KMS states of the C * -algebra of a strongly connected finite k-graph. We find that there is only one 1-parameter subgroup of the gauge action that can admit a KMS state. The extreme KMS states for this preferred dynamics are parameterised by the characters of an abelian group that captures the periodicity in the infinite-path space of the graph. We deduce that there is a unique KMS state if and only if the k-graph C * -algebra is simple, giving a complete answer to a question of Yang. When the k-graph C * -algebra is not simple, our results reveal a phase change of an unexpected nature in its Toeplitz extension.
Consider a higher-rank graph of rank k. Both the Cuntz-Krieger algebra and the Toeplitz-Cuntz-Krieger algebra of the graph carry natural gauge actions of the torus T k , and restricting these gauge actions to one-parameter subgroups of T k gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures β, the simplex of KMS β states on the Toeplitz-Cuntz-Krieger algebra has dimension d one less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a critical inverse temperature β c : for β larger than β c , there is a d-dimensional simplex of KMS states; when β = β c and the one-parameter subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz-Krieger algebra. As in previous studies for k = 1, our main tool is the Perron-Frobenius theory for irreducible nonnegative matrices, though here we need a version of the theory for commuting families of matrices.
We consider a finite directed graph E, and the gauge action on its Toeplitz-Cuntz-Krieger algebra, viewed as an action of R. For inverse temperatures larger than a critical value β c , we give an explicit construction of all the KMS β states. If the graph is strongly connected, then there is a unique KMS βc state, and this state factors through the quotient map onto C * (E). Our approach is direct and relatively elementary.
We construct a universal Cuntz–Krieger algebra ${\cal {AO}}_A$, which
is isomorphic to the usual Cuntz–Krieger algebra ${\cal O}_A$ when $A$
satisfies
condition $(I)$ of Cuntz and Krieger. The Cuntz classification of ideals in
${\cal O}_A$ when $A$ satisfies condition $(II)$ extends to a
classification of the gauge invariant ideals in ${\cal {AO}}_A$. We use this
to
describe the topology on the primitive ideal space of ${\cal {AO}}_A$.
We consider the dynamics on the C * -algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single 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.
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