Invariance of KMS states on graph C*-algebras under classical and quantum symmetry
Soumalya Joardar, Arnab Mandal
TL;DR
This work analyzes how KMS states on graph C*-algebras respond to classical and quantum graph symmetries. It proves that for strongly connected graphs the unique KMS state at the critical temperature is invariant under the quantum automorphism group, with implications for ergodicity. For circulant graphs, although multiple KMS states can occur at the critical temperature, a single Aut(Γ)-invariant (and hence Q^{aut}_{Ban}-invariant) KMS state is enforced by transitivity. The paper also presents a Mermin-Peres–type LBCS-based example where many Aut-invariant KMS states exist but there is a unique KMS state invariant under the quantum symmetry, illustrating the necessity of quantum symmetry for fixing uniqueness.
Abstract
We study invariance of KMS states on graph C*-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 quantum automorphism group of the graph. For circulant graphs, it is shown that the action of classical and quantum automorphism group preserves only one of the KMS states occurring at the critical inverse temperature. We also give an example of a graph C*-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.