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

Irreducible Representations of Bost-Connes systems

The classification problem of Bost-Connes systems was studied by Cornellissen and Marcolli partially, but still remains unsolved. In this paper, we will give a representation-theoretic approach to this problem. We generalize the result of Laca and Raeburn, which concerns with the primitive ideal space on the Bost-Connes system for $\mathbb{Q}$. As a consequence, the Bost-Connes $C^*$-algebra for a number field $K$ has $h_K^1$-dimensional irreducible representations and does not have finite-dimensional irreducible representations for the other dimensions, where $h_K^1$ is the narrow class numbe…

源自Bost-Connes型系统的冯·诺依曼代数

Von Neumann algebras arising from Bost-Connes type systems

该研究证明数域Bost-Connes型系统与Connes-Marcolli GL₂系统特定β区间的KMS_β态为III₁型，等价于阿代尔空间上相关作用的遍历性，并给出GL₂系统β=2时的对应实例。

On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev

This paper has two parts. In the first part we construct arithmetic models of Bost-Connes systems for arbitrary number fields, which has been an open problem since the seminal work of Bost and Connes [3]. In particular our construction shows how the class field theory of an arbitrary number field can be realized through the dynamics of a certain operator algebra. This is achieved by working in the framework of Endomotives, introduced by Connes, Marcolli and Consani [5], and using a classification result of Borger and de Smit [1] for certain Λ-rings in terms of the Deligne-Ribet monoid. Moreove…

Weak amenability and simply connected Lie groups

Following an approach of Ozawa, we show that several semidirect products are not weakly amenable. As a consequence, we are able to characterize the simply connected Lie groups that are weakly amenable.

与函数域相关的Bost-Connes系统

Bost-Connes systems associated with function fields

该研究构造了与整体函数域、素理想有限集及阿贝尔扩张相关的C*-动力系统，统一了已有构造，证明了相变定理并刻画了KMS态对应的因子类型与权重流性质。

Complete Characterization of K-Theory for C*-algebras Associated to Locally Finite Unoriented Graphs

In this paper we give a complete description of K-theory groups for Cuntz-Krieger C*-algebras associated to general locally-finite (topologically connected) graphs via Bass-Hashimoto operator. Our result generalizes the one obtained by the second author for the case of graphs with not necessarily finite first Betti numbers. On the basis of purely graph-theoretical method introduced by G. Cornelissen, O. Lorscheid, M. Marcolli and developed further by N.Iyudu, we prove that for the algebra O_E associated to an infinite graph E of the above form holds K_0(O_E)=Z^{β(E)} \oplus Z^{γ(E)} and K_1(O_…

K-homology of certain group C*-algebras

Motivated by the search for new examples of ``noncommutative manifolds'', we study the noncommutative geometry (in the sense of Connes) of the group C*-algebras of various discrete groups. The examples we consider are the infinite dihedral group Z \times_σ $Z_2$ and the semidirect product group $Z \times_σ Z$. We present a unified treatment of the K-homology and cyclic cohomology of these algebras.

The Higson-Mackey analogy for finite extensions of complex semisimple groups

In the 1970's, George Mackey pointed out an analogy that exists between tempered representations of semisimple Lie groups and unitary representations of associated semidirect product Lie groups. More recently, Nigel Higson refined Mackey's analogy into a one-to-one correspondence for connected complex semisimple groups, and in doing so obtained a novel verification of the Baum-Connes conjecture with trivial coefficients for such groups. Here we extend Higson's results to any Lie group with finitely many connected components whose connected component of the identity is complex semisimple. Our m…

Classification of Thompson related groups arising from Jones technology I

In the quest in constructing conformal field theories (CFT) Jones has discovered a beautiful and deep connection between CFT, Richard Thompson's groups and knot theory. This led to a powerful functorial framework for constructing actions of particular groups arising from categories such as Thompson's groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product where the largest Thompson's group $V$ is acting. These semidirect products have remarkable diagrammatic descriptions which were previously used to provide new examples of grou…

Phase transitions on Hecke C*-algebras and class-field theory over Q

We associate a canonical Hecke pair of semidirect product groups to the ring inclusion of the algebraic integers $\oo$ in a number field $\kk$, and we construct a C*-dynamical system on the corresponding Hecke C*-algebra, analogous to the one constructed by Bost and Connes for the inclusion of the integers in the rational numbers. We describe the structure of the resulting Hecke C*-algebra as a semigroup crossed product and then, in the case of class number one, analyze the equilibrium (KMS) states of the dynamical system. The extreme KMS$_β$ states at low-temperature exhibit a phase transitio…

Complex quantum groups and a deformation of the Baum-Connes assembly map

We define and study an analogue of the Baum-Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. Our starting point is the deformation picture of the Baum-Connes assembly map for a complex semisimple Lie group $ G $, which allows one to express the $ K $-theory of the reduced group $ C^* $-algebra of $ G $ in terms of the $ K $-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup $ K $ acting on $ \mathfrak{k}^* $ via the coad…

关于精确离散群和$\ell^p$一致Roe代数的一些$p$-逼近性质

On some $p$-approximation properties of discrete groups and $\ell^p$ uniform Roe algebras

研究$\ell^p$一致Roe代数的$p$-逼近性质及其与粗几何和群论的联系，引入$p$-不变平移逼近性质，并证明精确群与$p$-算子逼近性质的等价性。