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Journal of Operator Theory

Volume 82, Issue 2, Fall 2019  pp. 253-284.

Topological conjugacy of topological Markov shifts and Ruelle algebras

Authors:  Kengo Matsumoto
Author institution:Department of Mathematics, Joetsu University of Education, Joetsu, 943-8512, Japan

Summary: We will characterize topological conjugation for two-sided topological Markov shifts $(\overline{X}_A,\overline{\sigma}_A)$ in terms of the associated asymptotic Ruelle $C^*$-algebra ${\mathcal{R}}_A$ and its commutative $C^*$-subalgebra $C(\overline{X}_A)$ and the canonical circle action. We will also show that the extended Ruelle algebra $\widetilde{\mathcal{R}}_A$, which is a unital and purely infinite version of ${\mathcal{R}}_A$, together with its commutative $C^*$-subalgebra $C(\overline{X}_A)$ and the canonical torus action $\gamma^A$ is a complete invariant for topological conjugacy of $(\overline{X}_A,\overline{\sigma}_A)$. The diagonal action of $\gamma^A$ has a unique KMS-state on $\widetilde{\mathcal{R}}_A$, which is an extension of the Parry measure on $\overline{X}_A$.

Keywords: topological Markov shift, topological conjugacy, \'etale groupoid, Ruelle algebra, Cuntz--Krieger algebra, K-group, KMS-state

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