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Invariance of KMS states on graph C*-algebras under classical and quantum symmetry

Part of: Graph theory Selfadjoint operator algebras

Published online by Cambridge University Press:  20 September 2021

Soumalya Joardar  and
Arnab Mandal
Soumalya Joardar
Affiliation:
Department of Mathematics and Statistics, IISER Kolkata, Mohanpur, West Bengal741246, India (soumalya.j@gmail.com)
Arnab Mandal
Affiliation:
Presidency University, College Street, Kolkata, West Bengal700073, India (arnabmaths@gmail.com)
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Abstract

We study the invariance of KMS states on graph $C^{\ast }$-algebras coming from strongly connected and circulant graphs under the classical and quantum symmetry of the graphs. We show that the unique KMS state for strongly connected graphs is invariant under the quantum automorphism group of the graph. For circulant graphs, it is shown that the action of classical and quantum automorphism groups preserves only one of the KMS states occurring at the critical inverse temperature. We also give an example of a graph $C^{\ast }$-algebra having more than one KMS state such that all of them are invariant under the action of classical automorphism group of the graph, but there is a unique KMS state which is invariant under the action of quantum automorphism group of the graph.

Keywords

KMS statesGraph C*-algebraquantum automorphism group

MSC classification

Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Secondary: 46L30: States 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 4 , November 2021 , pp. 762 - 778
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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