Semigroups of *-Endomorphisms, Dirichlet Series, and Phase Transitions☆
Semigroups of *-Endomorphisms, Dirichlet Series, and Phase Transitions☆
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*-自同态半群、狄利克雷级数和相变☆
DOI：
10.1006/jfan.1997.3166
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发表时间：
1998
影响因子：
1.7
通讯作者：
Marcelo Laca
中科院分区：
文献类型：
--
作者：
Marcelo Laca
本文研究C*-动力系统（A，σ）的相变，其中A是C*-代数与 *-自同态格半群的交叉积，σ是对偶作用的单参数子群，由半群上的实值标度决定。我们证明了KMS平衡条件等价于（预对偶）半群动力系统的马尔可夫型条件，并给出了平衡态和相变存在的半群作用的准则。当半群是一个整数格，分区功能的真实的规模是一个狄利克雷级数具有欧拉产品的扩展;在他们的收敛区域有一个相变的几何形状是完全由基态和独立的选择规模，推广了一个模型最近构建的博斯特和康纳斯的动机早期工作的朱莉娅。作为应用程序，我们简化了部分证明的Bost-Connes定理的相变与自发对称性破缺，我们讨论了推广到其他系统相关联的子集的素数和数字段的类数1。
Abstract We study phase transitions of C*-dynamical systems ( A , σ ) in which A is the crossed product of a C*-algebra by a lattice semigroup of *-endomorphisms, and σ is a one-parameter subgroup of the dual action, determined by a real-valued scale on the semigroup. We show that the KMS equilibrium condition is equivalent to a Markov-type condition on the (predual) semigroup dynamical system, and give criteria for the existence of equilibrium states and of phase transitions in terms of the semigroup action. When the semigroup is an integer lattice, the partition function of the real scale is a Dirichlet series having an Euler product expansion; on their convergence region there is a phase transition whose geometry is fully determined by the ground states and is independent of the chosen scale, generalizing a model recently constructed by Bost and Connes motivated by earlier work of Julia. As applications we simplify part of the proof the Bost–Connes theorem on phase transition with spontaneous symmetry breaking and we discuss generalizations to other systems associated to subsets of primes and to number fields of class number 1.