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 C*-Algebras of inverse semigroups

1985 ◽  
Vol 28 (1) ◽  
pp. 41-58 ◽  
Author(s):  
J. Duncan ◽  
A. L. T. Paterson
Keyword(s):  
Banach Algebra ◽  
Group Ring ◽  
Discrete Group ◽  
Regular Representation ◽  
Inverse Semigroups ◽  
C Algebras ◽  
Complex Group ◽  
Left Regular Representation ◽  
Left Regular ◽  
Image Position

There are various algebras which may be associated with a discrete group G. In particular we may consider the complex group ring ℂG, the convolution Banach algebra l1(G), the enveloping C*-algebra C*(G) of l1(G), and the reduced C*-algebra determined by the completion of l1(G) under the left regular representation on l2(G). There is a substantial literature on the circle of ideas associated with the embeddings

scholarly journals The Koopman Representation and Positive Rokhlin Entropy

2021 ◽  
Author(s):  
Brandon Seward
Keyword(s):  
Spectral Gap ◽  
Regular Representation ◽  
Positive Entropy ◽  
Completely Positive ◽  
Amenable Groups ◽  
Direct Sum ◽  
Left Regular Representation ◽  
Left Regular ◽  
Sofic Entropy ◽  
Final Consequence

Abstract In this paper, we study connections between positive entropy phenomena and the Koopman representation for actions of general countable groups. Following the line of work initiated by Hayes for sofic entropy, we show in a certain precise manner that all positive entropy must come from portions of the Koopman representation that embed into the left-regular representation. We conclude that for actions having completely positive outer entropy, the Koopman representation must be isomorphic to the countable direct sum of the left-regular representation. This generalizes a theorem of Dooley–Golodets for countable amenable groups. As a final consequence, we observe that actions with completely positive outer entropy must be mixing, and when the group is non-amenable they must be strongly ergodic and have spectral gap.

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 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.

scholarly journals Harmonic analysis for groups acting on triangle buildings

1994 ◽  
Vol 56 (3) ◽  
pp. 345-383 ◽  
Author(s):  
Donald I. Cartwright ◽  
Wojciech MŁotkowski
Keyword(s):  
Commutative Algebra ◽  
Von Neumann Algebra ◽  
Regular Representation ◽  
Positive Definiteness ◽  
Plancherel Formula ◽  
Von Neumann ◽  
Left Regular Representation ◽  
Left Regular ◽  
Multiplicative Functionals ◽  
Complex Valued

AbstractLet Δ be a thick building of type Ã2, and let be its set of vertices. We study a commutative algebra of ‘averaging’ operators acting on the space of complex valued functions on . This algebra may be identified with a space of ‘biradial functions’ on , or with a convolution algebra of bi-K-invariant functions on G, if G is a sufficiently large group of ‘type-rotating’ automorphisms of Δ, and K is the subgroup of G fixing a given vertex. We describe the multiplicative functionals on and the corresponding spherical functions. We consider the C*-algebra induced by on l2, find its spectrum Σ, prove positive definiteness of a kernel kz for each z ∈ Σ, find explicity the spherical Plancherel formula for any group G of type rotating automorphisms, and discuss the irreducibility of the unitary representations appearing therein. For the class of buildings ΔJ arising from the groups ΓJ introduced in [2], this involves proving that the weak closure of is maximal abelian in the von Neumann algebra generated by the left regular representation of ΓJ.

Sign in / Sign up

Export Citation Format

Share Document