Connes-Marcolli GL2-系统中的相变

Bost-Connes-Marcolli systems for Shimura varieties. I. Definitions and formal analytic properties

We construct a quantum statistical mechanical system $(A,s)$ analogous to the systems constructed by Bost-Connes and Connes-Marcolli in the case of Shimura varieties. Along the way, we define a new Bost-Connes system for number fields which has the ``correct'' partition function and the ``correct'' symmetries. We give a formalism that applies to general Shimura data $(G,X)$. The object of this series of papers is to show that these systems exhibit phase transitions with spontaneous symmetry breaking, and to classify their KMS states, at least for low temperature.

The $\mathrm{GL}_n$-Connes-Marcolli Systems

In this paper, we generalize the results of Laca, Larsen, and Neshveyev on the $\mathrm{GL}_2$-Connes-Marcolli system to the $\mathrm{GL}_n$ systems. We introduce the $\mathrm{GL}_n$-Connes-Marcolli systems and discuss the question of the existence and the classification of KMS equilibrium states at different inverse temperatures $β$. In particular, using an ergodicity argument, we prove that in the range $n-1 <β\leq n$, there is only one KMS state. We show that there are no KMS states for $β<n-1$ and not an integer, while we construct KMS states for integer values of $β$ in the range $1\leqβ\…

与Connes-Marcolli $GSp_4$-系统相关的von Neumann代数中的类型转换

Transition of type in the von Neumann algebras associated to the Connes-Marcolli $GSp_4$-system

本文研究了Connes-Marcolli $GSp_4$-系统产生的von Neumann代数，发现其类型在逆温度 $\beta= 4$ 处发生从 $I_\infty$ 型到 $III_1$ 型的相变。

半直积的Hecke代数与Connes-Marcolli C*-代数的有限部分

Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra

该研究针对2×2整数矩阵环嵌入有理矩阵环产生的C*-动力系统，分析其Hecke代数结构、对称群与不动点代数分解，完整刻画了一类自然平衡态，推进了相关非交换几何动力系统研究。

Reconstructing the Bost--Connes semigroup actions from K-theory

We complete the classification of Bost--Connes systems. We show that two Bost--Connes C*-algebras for number fields are isomorphic if and only if the original semigroups actions are conjugate. Together with recent reconstruction results in number theory by Cornelissen--de Smit--Li--Marcolli--Smit, we conclude that two Bost--Connes C*-algebras are isomorphic if and only if the original number fields are isomorphic.

数域的Bost-Connes型系统研究

On Bost-Connes type systems for number fields

本文刻画了两类学者新提出的数域Bost-Connes型系统的相变，引入K-格概念并从一维K-格角度给出该系统的诠释，推进了相关数论与算子代数交叉研究。