Marcelo Laca, Nadia S. Larsen
We consider group-subgroup pairs in which the group is a semidirect product and the subgroup is contained in the normal part. We give conditions for the pair to be a Hecke pair and we show that the enveloping Hecke algebra and Hecke C*-algebra are canonically isomorphic to semigroup crossed products, generalizing earlier results of Arledge, Laca and Raeburn and of Brenken.
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We analyze Hecke pairs (G,H) and the associated Hecke algebra when G is a semidirect product N x Q and H = M x R for subgroups M of N and R of Q with M normal in N. Conditions are given in terms of N, Q, M, and R which are equivalent to the Hecke condition on (G,H), and the Schlichting completion of (G,H) is identified in terms of completions of N, Q, M, and R. Our main result shows that (assuming (G,H) coincides with its Schlichting completion) when R is normal in Q, the closure of the Hecke algebra in C*(G) is Morita-Rieffel equivalent to a crossed product I x Q/R, where I is a certain ideal…
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。
Given an action of a groupoid by isomorphisms on a Fell bundle (over another groupoid), we form a semidirect-product Fell bundle, and prove that its $C^{*}$-algebra is isomorphic to a crossed product.
The Hecke algebra H(G,H) of a Hecke pair (G,H) is studied using the Schlichting completion (G',H'), which is a Hecke pair whose Hecke algebra is isomorphic to H(G,H) and which is topologized so that H' is a compact open subgroup of G'. In particular, the representation theory and C*-completions of H(G,H) are addressed in terms of the projection p in C*(G') corresponding to the characteristic function of H', using both Fell's and Rieffel's imprimitivity theorems and the identity H(G,H) = p C_c(G') p. An extended analysis of the case where H is contained in a normal subgroup of G (and in particu…
For discrete Hecke pairs $(G,H)$, we introduce a notion of covariant representation which reduces in the case where $H$ is normal to the usual definition of covariance for the action of $G/H$ on $c_0(G/H)$ by right translation; in many cases where $G$ is a semidirect product, it can also be expressed in terms of covariance for a semigroup action. We use this covariance to characterise the representations of $c_0(G/H)$ which are multiples of the multiplication representation on $\ell^2(G/H)$, and more generally, we prove an imprimitivity theorem for regular representations of certain crossed pr…
Given a free and proper action of a groupoid on a Fell bundle (over another groupoid), we give an equivalence between the semidirect-product and the generalized-fixed-point Fell bundles, generalizing an earlier result where the action was by a group. As an application, we show that the Stabilization Theorem for Fell bundles over groupoids is essentially another form of crossed-product duality.
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。
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.
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.
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.
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.
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…
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 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 …
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.…
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。
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。
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 …
We introduce the notion of continuous twisted partial actions of a locally compact group on a C*-algebra. With such, we construct an associated C*-algebraic bundle called the semidirect product bundle. Our main theorem shows that, given any C*-algebraic bundle which is second countable and whose unit fiber algebra is stable, there is a continuous twisted partial action of the base group on the unit fiber algebra, whose associated semidirect product bundle is isomorphic to the given one.
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