KMS states on the crossed product C∗-algebra of a homeomorphism

Johannes Christensen; Klaus Thomsen

doi:10.1017/etds.2020.141

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

Johannes Christensen^*, Klaus Thomsen

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

Abstract

Let φ:X → X be a homeomorphism of a compact metric space X. For any continuous function F:X →R there is a one-parameter group αF C(X) ×φ Z of automorphisms (or a flow) on the crossed product C∗-algebra defined such that αFt (fU) = fUe-itf when f ∈ 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) ×φ Z is simple this set is either {0} or the whole line R.

Original language	English
Journal	Ergodic Theory and Dynamical Systems
Volume	42
Issue	4
Pages (from-to)	1373-1414
Number of pages	42
ISSN	0143-3857
DOIs	https://doi.org/10.1017/etds.2020.141
Publication status	Published - Apr 2022

Keywords

crossed products
ergodic theory
KMS states

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10.1017/etds.2020.141

https://arxiv.org/pdf/1912.06069.pdf

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title = "KMS states on the crossed product C∗-algebra of a homeomorphism",

abstract = "Let φ:X → X be a homeomorphism of a compact metric space X. For any continuous function F:X →R there is a one-parameter group αF C(X) ×φ Z of automorphisms (or a flow) on the crossed product C∗-algebra defined such that αFt (fU) = fUe-itf when f ∈ 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) ×φ Z is simple this set is either {0} or the whole line R.",

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author = "Johannes Christensen and Klaus Thomsen",

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AU - Christensen, Johannes

AU - Thomsen, Klaus

N1 - Publisher Copyright: © The Author(s), 2021.

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N2 - Let φ:X → X be a homeomorphism of a compact metric space X. For any continuous function F:X →R there is a one-parameter group αF C(X) ×φ Z of automorphisms (or a flow) on the crossed product C∗-algebra defined such that αFt (fU) = fUe-itf when f ∈ 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) ×φ Z is simple this set is either {0} or the whole line R.

AB - Let φ:X → X be a homeomorphism of a compact metric space X. For any continuous function F:X →R there is a one-parameter group αF C(X) ×φ Z of automorphisms (or a flow) on the crossed product C∗-algebra defined such that αFt (fU) = fUe-itf when f ∈ 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) ×φ Z is simple this set is either {0} or the whole line R.

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JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

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

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