scholarly journals On the K-theory of the C*-algebra generated by the left regular representation of an Ore semigroup

2015 ◽  
Vol 17 (3) ◽  
pp. 645-687 ◽  
Author(s):  
Joachim Cuntz ◽  
Siegfried Echterhoff ◽  
Xin Li
Keyword(s):  
Regular Representation ◽  
C Algebra ◽  
Left Regular Representation ◽  
Left Regular ◽  
K Theory

Topological Grading of Semigroup C*-Algebras

2020 ◽  
pp. 44-55
Author(s):  
R.N. Gumerov ◽  
E.V. Lipacheva
Keyword(s):  
Banach Space ◽  
Fourier Coefficients ◽  
Regular Representation ◽  
Main Tool ◽  
Linear Operators ◽  
Natural Object ◽  
C Algebra ◽  
C Algebras ◽  
Left Regular Representation ◽  
Left Regular

The paper deals with the abelian cancellative semigroups and the reduced semigroup C*-algebras. It is supposed that there exist epimorphisms from the semigroups onto the group of integers modulo n. For these semigroups we study the structure of the reduced semigroup C*-algebras which are also called the Toeplitz algebras. Such a C*-algebra can be defined for any non-abelian left cancellative semigroup. It is a very natural object in the category of C*-algebras because this algebra is generated by the left regular representation of a semigroup. In the paper, by a given epimorphism σ we construct the grading of a semigroup C*-algebra. To this aim the notion of the σ-index of a monomial is introduced. This notion is the main tool in the construction of the grading. We make use of the σ-index to define the linear independent closed subspaces in the semigroup C*-algebra. These subspaces constitute the C*-algebraic bundle, or the Fell bundle, over the group of integers modulo n. Moreover, it is shown that this grading of the reduced semigroup C*-algebra is topological. As a corollary, we obtain the existence of the contractive linear operators that are non-commutative analogs of the Fourier coefficients. Using these operators, we prove the result on the geometry of the underlying Banach space of the semigroup C*-algebra

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

scholarly journals GAUSSIAN FLUCTUATIONS OF REPRESENTATIONS OF WREATH PRODUCTS

2006 ◽  
Vol 09 (04) ◽  
pp. 529-546
Author(s):  
PIOTR ŚNIADY
Keyword(s):  
Finite Group ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Regular Representation ◽  
Wreath Products ◽  
Young Diagrams ◽  
Reducible Representation ◽  
Irreducible Components ◽  
Left Regular Representation ◽  
Left Regular

We study the asymptotics of the reducible representations of the wreath products G≀Sq = Gq ⋊ Sq for large q, where G is a fixed finite group and Sq is the symmetric group in q elements; in particular for G = ℤ/2ℤ we recover the hyperoctahedral groups. We decompose such a reducible representation of G≀Sq as a sum of irreducible components (or, equivalently, as a collection of tuples of Young diagrams) and we ask what is the character of a randomly chosen component (or, what are the shapes of Young diagrams in a randomly chosen tuple). Our main result is that for a large class of representations, the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian. The considered class consists of the representations for which the characters asymptotically almost factorize and it includes, among others, the left regular representation therefore we prove the analogue of Kerov's central limit theorem for wreath products.

scholarly journals Random products of automorphisms of Heisenberg nilmanifolds and Weil’s representation

2010 ◽  
Vol 31 (5) ◽  
pp. 1277-1286 ◽  
Author(s):  
BACHIR BEKKA ◽  
JEAN-ROMAIN HEU
Keyword(s):  
Probability Measure ◽  
Spectral Gap ◽  
Regular Representation ◽  
Metaplectic Representation ◽  
Matrix Coefficients ◽  
Left Regular Representation ◽  
Left Regular ◽  
Random Products ◽  
Image Position ◽  
Dense Set

AbstractForn≥1, letHbe the (2n+1)-dimensional real Heisenberg group, and let Λ be a lattice inH. Let Γ be the group of automorphisms of the corresponding nilmanifold Λ∖HandUthe associated unitary representation of Γ onL2(Λ∖H) . Denote byTthe maximal torus factor associated to Λ∖H. Using Weil’s representation (also known as the metaplectic representation), we show that a dense set of matrix coefficients of the restriction ofUto the orthogonal complement ofL2(T) inL2(Λ∖H) belong toℓ4n+2+ε(Γ) for every ε>0 . We give the following application to random walks on Λ∖Hdefined by a probability measureμon Aut (Λ∖H) . Denoting by Γ(μ) the subgroup of Aut (Λ∖H) generated by the support ofμand byU0andV0the restrictions ofUto, respectively, the subspaces ofL2(Λ∖H) andL2(T) with zero mean, we prove the following inequality:whereλis the left regular representation of Γ(μ) onℓ2(Γ(μ)) . In particular, the action of Γ(μ) on Λ∖Hhas a spectral gap if and only if the corresponding action of Γ(μ) onThas a spectral gap.

scholarly journals Automorphisms of Full II1 Factors, II

1978 ◽  
Vol 21 (3) ◽  
pp. 325-328 ◽  
Author(s):  
John Phillips
Keyword(s):  
Regular Representation ◽  
Conjugacy Classes ◽  
Type Ii ◽  
Nonabelian Group ◽  
Left Regular Representation ◽  
Left Regular ◽  
Free Generators

The purpose of this note is to continue the author's study of the automorphisms of certain factors of type II1 Namely, those factors arising from the left regular representation of a free nonabelian group. Our main result shows that the outer conjugacy classes of automorphisms of such a factor are not countably separated. This had previously been shown only when the number of free generators was assumed to be infinite.

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