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KMS states on the crossed product -algebra of a homeomorphism
Ergodic Theory and Dynamical Systems ( IF 0.9 ) Pub Date : 2021-02-15 , DOI: 10.1017/etds.2020.141
JOHANNES CHRISTENSEN 1 , KLAUS THOMSEN 2
Affiliation  

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 $\mathbb R$ .



中文翻译:

KMS 状态关于同胚的叉积代数

$\phi :X\to X$ 是紧度量空间X的同胚。对于任何连续函数 $F:X\to \mathbb {R}$ 在叉积 $C^*$ -algebra $上都有一个自同构(或流)的单参数组 $\alpha ^{F} $ C(X)\rtimes _{\phi }\mathbb {Z}$ 定义使得 $\alpha ^{F}_{t}(fU)=fUe^{-itF}$ $f \in C(X )$ U 是叉积构造中的规范酉。在本文中,我们通过发展与非奇异变换的遍历理论的密切关系来研究这些流动的 Kubo--Martin-Schwinger (KMS) 状态,并表明 KMS 状态的结构可以非常丰富和复杂。关于一组可能的逆温度,我们的结果是完整的;特别是,我们证明当 $C(X) \rtimes _{\phi } \mathbb Z$ 很简单时,这个集合要么是 $\{0\}$ 要么是整行 $\mathbb R$

更新日期:2021-02-15
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