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KMS states on the crossed product $C^*$-algebra of a homeomorphism

Part of: Selfadjoint operator algebras Ergodic theory

Published online by Cambridge University Press:  15 February 2021

JOHANNES CHRISTENSEN Open the ORCID record for JOHANNES CHRISTENSEN [Opens in a new window]  and
KLAUS THOMSEN
JOHANNES CHRISTENSEN*
Affiliation:
Department of Mathematics, Aarhus University, Ny Munkegade, 8000Aarhus C, Denmark (e-mail: matkt@math.au.dk)
KLAUS THOMSEN
Affiliation:
Department of Mathematics, Aarhus University, Ny Munkegade, 8000Aarhus C, Denmark (e-mail: matkt@math.au.dk)
*
e-mail: johannes@math.au.dk
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Abstract

Let $\phi :X\to X$ be a homeomorphism of a compact metric space X. For any continuous function $F:X\to \mathbb {R}$ there is a one-parameter group $\alpha ^{F}$ of automorphisms (or a flow) on the crossed product $C^*$ -algebra $C(X)\rtimes _{\phi }\mathbb {Z}$ defined such that $\alpha ^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and U is the canonical unitary in the construction of the crossed product. In this paper we study the Kubo--Martin--Schwinger (KMS) states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when $C(X) \rtimes _{\phi } \mathbb Z$ is simple this set is either $\{0\}$ or the whole line $\mathbb R$ .

Keywords

KMS statescrossed productsergodic theory

MSC classification

Primary: 46L30: States
Secondary: 37A55: Relations with the theory of $C^*$-algebras
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 4 , April 2022 , pp. 1373 - 1414
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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