EQUILIBRIUM STATES ON THE CUNTZ-PIMSNER ALGEBRAS OF SELF-SIMILAR ACTIONS
EQUILIBRIUM STATES ON THE CUNTZ-PIMSNER ALGEBRAS OF SELF-SIMILAR ACTIONS
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自相似作用的 CUNTZ-PIMSNER 代数的平衡态
DOI：
10.1016/j.jfa.2014.03.003
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发表时间：
2013
影响因子：
1.7
通讯作者：
Michael F. Whittaker
中科院分区：
文献类型：
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作者：
Marcelo Laca;I. Raeburn;Jacqui Ramagge;Michael F. Whittaker
我们考虑一族与自相似群作用相关的Cuntz-Pimsner代数及其Toeplitz类似物。这两个族都具有由真实的线的自守作用实现的自然动力学，我们研究了这些动力系统的平衡态（KMS态）。我们发现，对于所有的反温度超过一个临界值，KMS状态的Toeplitz代数给出，在一个非常具体的方式，由轨迹的全群代数的组。在临界逆温度下，KMS态通过Cuntz-Pimsner代数的态分解;如果自相似群收缩，则Cuntz-Pimsner代数只有一个KMS态。我们将这些结果应用到一些例子中，包括与整数扩张矩阵相关的自相似群作用，以及Basilica群和Grigorchuk群的典型自相似作用。
We consider a family of Cuntz–Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorphic actions of the real line, and we investigate the equilibrium states (the KMS states) for these dynamical systems. We find that for all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given, in a very concrete way, by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz–Pimsner algebra; if the self-similar group is contracting, then the Cuntz–Pimsner algebra has only one KMS state. We apply these results to a number of examples, including the self-similar group actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group.