Partial Dynamical Systems and the KMS Condition
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Given a countably infinite 0–1 matrix A without identically zero rows, let 𝒪 A be the Cuntz–Krieger algebra recently introduced by the authors and 𝒯 A be the Toeplitz extension of 𝒪 A , once the latter is seen as a Cuntz–Pimsner algebra, as recently shown by Szymański. We study the KMS equilibrium states of C * -dynamical systems based on 𝒪 A and 𝒯 A , with dynamics satisfying \(\) for the canonical generating partial isometries s x and arbitrary real numbers N x > 1. The KMSβ states on both 𝒪 A and 𝒯 A are completely characterized for certain values of the inverse temperature β, according to the position of β relative to three critical values, defined to be the abscissa of convergence of certain Dirichlet series associated to A and the N(x). Our results for 𝒪 A are derived from those for 𝒯 A by virtue of the former being a covariant quotient of the latter. When the matrix A is finite, these results give theorems of Olesen and Pedersen for 𝒪 n and of Enomoto, Fujii and Watatani for 𝒪 A as particular cases.
KeywordsDynamical System Real Number Equilibrium State Inverse Temperature Dirichlet Series
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