scholarly journals Topological Boundaries of Unitary Representations

2020 ◽  
Author(s):  
Alex Bearden ◽  
Mehrdad Kalantar
Keyword(s):  
Invariant Subspace ◽  
Unitary Representation ◽  
Discrete Group ◽  
Regular Representation ◽  
Unitary Representations ◽  
Algebra Structure

Abstract We introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group $\Gamma $ to the setting of a general unitary representation $\pi : \Gamma \to B(\mathcal H_\pi )$. This space, which we call the “Furstenberg–Hamana boundary” (or "FH-boundary") of the pair $(\Gamma , \pi )$, is a $\Gamma $-invariant subspace of $B(\mathcal H_\pi )$ that carries a canonical $C^{\ast }$-algebra structure. In many natural cases, including when $\pi $ is a quasi-regular representation, the Furstenberg–Hamana boundary of $\pi $ is commutative but can be noncommutative in general. We study various properties of this boundary and discuss possible applications, for example in uniqueness of certain types of traces.

MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP

2008 ◽  
Vol 19 (10) ◽  
pp. 1187-1201 ◽  
Author(s):  
MASAYASU MORIWAKI
Keyword(s):  
Unitary Representation ◽  
Orthogonal Group ◽  
Irreducible Unitary Representation ◽  
Unitary Representations ◽  
Minimal Representation ◽  
Infinite Dimensional ◽  
Irreducible Unitary Representations ◽  
Direct Product Group ◽  
Multiplicity Free ◽  
Kirillov Dimension

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G′ of G, the restriction π|G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p1) × U(p2) × U(q) for p1, p2 ≥ 1 with p1 + p2 = p is multiplicity-free, whereas the restriction to U(p1) × U(p2) × U(q1) × U(q2) for q1, q2 ≥ 1 with q1 + q2 = q has infinite multiplicities.

scholarly journals On Jones’ subgroup of R. Thompson’s group T

2018 ◽  
Vol 28 (05) ◽  
pp. 877-903
Author(s):  
Jordan Nikkel ◽  
Yunxiang Ren
Keyword(s):  
Unitary Representation ◽  
Unitary Representations ◽  
Thompson Groups ◽  
Diagram Groups ◽  
Thompson’S Group ◽  
Planar Algebra ◽  
Thompson Group

Jones introduced unitary representations for the Thompson groups [Formula: see text] and [Formula: see text] from a given subfactor planar algebra. Some interesting subgroups arise as the stabilizer of certain vector, in particular the Jones subgroups [Formula: see text] and [Formula: see text]. Golan and Sapir studied [Formula: see text] and identified it as a copy of the Thompson group [Formula: see text]. In this paper, we completely describe [Formula: see text] and show that [Formula: see text] coincides with its commensurator in [Formula: see text], implying that the corresponding unitary representation is irreducible. We also generalize the notion of the Stallings 2-core for diagram groups to [Formula: see text], showing that [Formula: see text] and [Formula: see text] are not isomorphic, but as annular diagram groups they have very similar presentations.

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 A Stone-Weierstrass theorem for group representations

1978 ◽  
Vol 1 (2) ◽  
pp. 235-244 ◽  
Author(s):  
Joe Repka
Keyword(s):  
Compact Group ◽  
Unitary Representation ◽  
Regular Representation ◽  
Discrete Series ◽  
Series Representation ◽  
Tensor Products ◽  
Weil Representation ◽  
Weierstrass Theorem ◽  
Group Representations ◽  
Discrete Series Representation

It is well known that ifGis a compact group andπa faithful (unitary) representation, then each irreducible representation ofGoccurs in the tensor product of some number of copies ofπand its contragredient. We generalize this result to a separable typeIlocally compact groupGas follows: letπbe a faithful unitary representation whose matrix coefficient functions vanish at infinity and satisfy an appropriate integrabillty condition. Then, up to isomorphism, the regular representation ofGis contained in the direct sum of all tensor products of finitely many copies ofπand its contragredient.We apply this result to a symplectic group and the Weil representation associated to a quadratic form. As the tensor products of such a representation are also Weil representations (associated to different forms), we see that any discrete series representation can be realized as a subrepresentation of a Weil representation.

The Type I Part of the Regular Representation

1974 ◽  
Vol 26 (5) ◽  
pp. 1086-1089 ◽  
Author(s):  
Edward Formanek
Keyword(s):  
Discrete Group ◽  
Regular Representation ◽  
Complex Numbers ◽  
Type I ◽  
Bounded Operators ◽  
Weak Operator Topology ◽  
Weak Operator ◽  
Image Position ◽  
The Right

Let G be a discrete group and let H = L2(G), with norm | |. Let B(H) be the ring of bounded operators on H with the normThe right regular representation of G on H induces an injection ρ : C[G] → B(H), and W(G) is the closure of the image of ρ in the weak operator topology on B(H) (C = complex numbers). Using ρ, we identify C[G] with its image in W(G).

Countable Amenable Identity Excluding Groups

2004 ◽  
Vol 47 (2) ◽  
pp. 215-228 ◽  
Author(s):  
Wojciech Jaworski
Keyword(s):  
Probability Measure ◽  
Unitary Representation ◽  
Discrete Group ◽  
Polynomial Growth ◽  
Irreducible Unitary Representation ◽  
Compact Groups ◽  
Groups Of Polynomial Growth ◽  
Identity Representation ◽  
Dimensional Identity

AbstractA discrete group G is called identity excluding if the only irreducible unitary representation of G which weakly contains the 1-dimensional identity representation is the 1-dimensional identity representation itself. Given a unitary representation π of G and a probability measure μ on G, let Pμ denote the μ-average ∫π(g)μ(dg). The goal of this article is twofold: (1) to study the asymptotic behaviour of the powers , and (2) to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure μ on an identity excluding group and every unitary representation π there exists and orthogonal projection Eμ onto a π-invariant subspace such that for every a ∈ supp μ. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of FC-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic.

Unitary representations of the unitary group

1969 ◽  
Vol 65 (2) ◽  
pp. 377-386 ◽  
Author(s):  
D. B. Hunter
Keyword(s):  
Unitary Representation ◽  
Unitary Group ◽  
Unitary Representations ◽  
Unitary Matrices

It is well known that every representation of the group Un of unitary matrices of order n × n is equivalent to a unitary representation (see e.g. Little-wood (6), ch. XI). Our object in the present paper is to discuss some properties of those representations, and to construct a specific unitary representation.

scholarly journals Certain Unitary Representations of the Infinite Symmetric Group, I

1987 ◽  
Vol 105 ◽  
pp. 121-128 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Regular Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Type I ◽  
Discrete Topology ◽  
Type Ii ◽  
Infinite Symmetric Group ◽  
Von Neumann ◽  
Group I

Let X be the set of all natural numbers and let be the group of all finite permutations of X. The group equipped with the discrete topology, is called the infinite symmetric group. It was discussed in F. J. Murray and J. von Neumann as a concrete example of an ICC-group, which is a discrete group with infinite conjugacy classes. It is proved that the regular representation of an ICC-group is a factor representation of type II1. The infinite symmetric group is, therefore, a group not of type I. This may be the reason why its unitary representations have not been investigated satisfactorily. In fact, only few results are known. For instance, all indecomposable central positive definite functions on , which are related to factor representations of type IIl, were given by E. Thoma. Later on, A. M. Vershik and S. V. Kerov obtained the same result by a different method in and gave a realization of the representations of type II1 in. Concerning irreducible representations, A. Lieberman and G. I. Ol’shanskii obtained a characterization of a certain family of countably many irreducible representations by introducing a particular topology in However, irreducible representations have been studied not so actively as factor representations.

scholarly journals Property (T), finite-dimensional representations, and generic representations

2019 ◽  
Vol 22 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Michal Doucha ◽  
Maciej Malicki ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Unitary Representation ◽  
Unitary Group ◽  
Discrete Group ◽  
Coefficient Function ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Invariant Vectors ◽  
Property T

Abstract Let G be a discrete group with Property (T). It is a standard fact that, in a unitary representation of G on a Hilbert space {\mathcal{H}} , almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation σ, then the vector is close to a sub-representation isomorphic to σ: this makes quantitative a result of P. S. Wang. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot, that a group G with Property (T) and such that {C^{*}(G)} is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in {Rep(G,\mathcal{H})} under the unitary group {U(\mathcal{H})} is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in {\mathrm{Rep}(G,\mathcal{H})} .

scholarly journals On the uniqueness of unitary representations of the non-commutative Heisenberg–Weyl algebra

2009 ◽  
Vol 87 (9) ◽  
pp. 995-997 ◽  
Author(s):  
L. Gouba ◽  
F. G. Scholtz
Keyword(s):  
Unitary Representation ◽  
Weyl Algebra ◽  
Critical Line ◽  
Unitary Representations ◽  
Von Neumann ◽  
Neumann Theorem ◽  
Von Neumann Theorem

In this paper, we discuss the uniqueness of the unitary representations of the noncommutative Heisenberg–Weyl algebra. We show that, apart from a critical line for the noncommutative position and momentum parameters, the Stone–von Neumann theorem still holds, which implies uniqueness of the unitary representation of the Heisenberg–Weyl algebra.

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