Journal of Operator Theory

Volume 92, Issue 2, Autumn 2024 pp. 363-411.

$C^*$-dynamical invariants and Toeplitz algebras of graphs

Authors: Chris Bruce (1) and Takuya Takeishi (2)
Author institution: (1) School of Mathematics, Statistics and Physics, Herschel Building, Newcastle University, Newcastle upon Tyne, NE1 7RU, U.K.
(2) Faculty of Arts and Sciences, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, Japan

Summary: In recent joint work of the authors with Laca, we precisely formulated the notion of partition function in the context of $C^*$-dynamical systems. Here, we compute the partition functions of $C^*$-dynamical systems arising from Toeplitz algebras of graphs, and we explicitly recover graph-theoretic information in terms of $C^*$-dynamical invariants. In addition, we compute the type for KMS states on $C^*$-algebras of finite (reducible) graphs and prove that the extremal KMS states at critical inverse temperatures give rise to type III$_\lambda$ factors. Our starting point is an independent result parameterising the partition functions of a certain class of $C^*$-dynamical systems arising from groupoid $C^*$-algebras in terms of $\beta$-summable orbits.

DOI: http://dx.doi.org/10.7900/jot.2022oct26.2394
Keywords: Toeplitz algebra, graph $C^*$-algebra, KMS state, partition function, $C^*$-dynamical system

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