Hecke C*-algebras and semidirect products

C*-量子群的半直积：乘法酉算子方法

Semidirect products of C*-quantum groups: multiplicative unitaries approach

该研究针对带投影C*-量子群的分解问题，借助Radford定理的思路，在可管理乘法酉算子层面证明了带投影C*-量子群可唯一分解为普通与辫子C*-量子群的结论。

On representations of semidirect products of a compact quantum group with a finite group

We study unitary representations of semidirect products of a compact quantum group with a finite group. We give a classification of all irreducible unitary representations, a description of the conjugate representation of irreducible unitary representations in terms of this classification, and the fusion rules for the semidirect product.

Representations of Hecke algebras and dilations of semigroup crossed products

We consider a family of Hecke C*-algebras which can be realised as crossed products by semigroups of endomorphisms. We show by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group.

Crossed products by Hecke pairs I: *-algebraic theory

We develop a theory of crossed products by "actions" of Hecke pairs $(G, Γ)$, motivated by applications in non-abelian $C^*$-duality. Our approach gives back the usual crossed product construction whenever $G / Γ$ is a group and retains many of the aspects of crossed products by groups. In this first of two articles we lay the $^*$-algebraic foundations of these crossed products by Hecke pairs and we explore their representation theory.

A characterization of relative Kazhdan Property T for semidirect products with abelian groups

Let A be a locally compact abelian group, and H a locally compact group acting on A. Let G=HA be the semidirect product. We prove that the pair (G,A) has Kazhdan's Property T if and only if the only countably approximable H-invariant mean on the Borel subsets of the Pontryagin dual of A, supported at the neighbourhood of the trivial character, is the Dirac measure.

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…

Crossed products by Hecke pairs II: C*-completions

In this second article on crossed products by "actions" of Hecke pairs we study their different C*-completions, namely we show how reduced and full C*-crossed products can be defined. We also establish that our construction coincides with that of Laca, Larsen and Neshveyev whenever they are both definable. As an application of our theory, we prove a Stone-von Neumann theorem for Hecke pairs which encompasses the work of an Huef, Kaliszewski and Raeburn and we lay down the foundations for obtaining a form of Katayama duality with respect to the Echterhoff-Quigg crossed product.

Graph product Khintchine inequalities and Hecke C*-algebras: Haagerup inequalities, (non)simplicity, nuclearity and exactness

Graph products of groups were introduced by Green in her thesis. They have an operator algebraic counterpart introduced and explored by Fima and the first-named author. In this paper we prove Khintchine type inequalities for general C$^{\ast}$-algebraic graph products which generalize results by Ricard and Xu on free products of C$^{\ast}$-algebras. We apply these inequalities in the context of (right-angled) Hecke C$^{\ast}$-algebras, which are deformations of the group algebra of Coxeter groups. For these we deduce a Haagerup inequality. We further use this to study the simplicity and trace …

Semidirect product rigidity of group von Neumann algebras arising from class $\mathscr{S}$, inductive limits and fundamental group

In this article we study property (T) groups arising from Rips construction in geometric group theory in the spirit of \cite{CDK19} and certain inductive limit groups from this class. Using interplay between Popa's deformation/rigidity and methods in geometric group theory we are able to extend the class of groups considered in \cite{CDK19} that remembers semidirect product features while passing to the group von Neumann algebras. Combining these results with the method developed in \cite{CDHK20} we are able to produce more examples of property (T) group factors with trivial fundamental group.…

由某些半直积产生的群的 von Neumann 代数的不变子代数刚性

Invariant subalgebras rigidity for von Neumann algebras of groups arising as certain semidirect products

本文研究由 $\mathbb{Z}_2$ 直积等 2-挠群上代数作用形成的半直积群的 ISR 性质，构造了首个带非平凡阿贝尔正规子群的 amenable ISR 群，并分析特殊反例及标准 lamplighter 群的不变子代数。

半直积的上循环扭化与Transmutation（嬗变）

Cocycle twisting of semidirect products and transmutation

该研究将Majid的嬗变程序应用于到紧阿贝尔群函数代数的Hopf代数映射，得到商群上的辫子Hopf代数，统一推广了辫子紧量子群的相关构造并给出其玻色化描述。

Classification of Thompson related groups arising from Jones technology II

In this second article, we continue to study classes of groups constructed from a functorial method due to Vaughan Jones. A key observation of the author shows that these groups have remarkable diagrammatic properties that can be used to deduce their properties. Given any group and two of its endomorphisms, we construct a semidirect product. In our first article dedicated to this construction, we classify up to isomorphism all these semidirect products when one of the endomorphisms is trivial and described their automorphism group. In this article we focus on the case where both endomorphisms …