scholarly journals Integral operators in the theory of induced Banach representation II. The bundle approach

1981 ◽  
Vol 4 (4) ◽  
pp. 625-640 ◽  
Author(s):  
I. E. Schochetman
Keyword(s):  
Compact Group ◽  
Cross Sections ◽  
Closed Subgroup ◽  
Integral Operators ◽  
Locally Compact Group ◽  
Representation Space ◽  
Locally Compact

LetGbe a locally compact group,Ha closed subgroup andLa Banach representation ofH. SupposeUis a Banach representation ofGwhich is induced byL. Here, we continue our program of showing that certain operators of the integrated form ofUcan be written as integral operators with continuous kernels. Specifically, we show that: (1) the representation space of a Banach bundle; (2) the above operators become integral operators on this space with kernels which are continuous cross-sections of an associated kernel bundle.

scholarly journals Extension problems and non-Abelian duality for C*-algebras

2007 ◽  
Vol 75 (2) ◽  
pp. 229-238 ◽  
Author(s):  
Astrid an Huef ◽  
S. Kaliszewski ◽  
Iain Raeburn
Keyword(s):  
Compact Group ◽  
Unitary Representation ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Test Question ◽  
Locally Compact ◽  
Dual Representation ◽  
C Algebras

Suppose that H is a closed subgroup of a locally compact group G. We show that a unitary representation U of H is the restriction of a unitary representation of G if and only if a dual representation Û of a crossed product C*(G) ⋊ (G/H) is regular in an appropriate sense. We then discuss the problem of deciding whether a given representation is regular; we believe that this problem will prove to be an interesting test question in non-Abelian duality for crossed products of C*-algebras.

scholarly journals Amenability and semisimplicity for second duals of quotients of the Fourier algebraA(G)

1997 ◽  
Vol 63 (3) ◽  
pp. 289-296 ◽  
Author(s):  
Edmond E. Granirer
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Amenable Banach Algebra ◽  
Symmetric Set ◽  
Semisimple Banach Algebra

AbstractLetF ⊂ Gbe closed andA(F) = A(G)/IF. IfFis a Helson set thenA(F)**is an amenable (semisimple) Banach algebra. Our main result implies the following theorem: LetGbe a locally compact group,F ⊂ Gclosed,a ∈ G. Assume either (a) For some non-discrete closed subgroupH, the interior ofF ∩ aHinaHis non-empty, or (b)R ⊂ G, S ⊂ Ris a symmetric set andaS ⊂ F. ThenA(F)**is a non-amenable non-semisimple Banach algebra. This raises the question: How ‘thin’ canFbe forA(F)**to remain a non-amenable Banach algebra?

S-Subgroups of the Real Hyperbolic Groups

1980 ◽  
Vol 32 (1) ◽  
pp. 246-256 ◽  
Author(s):  
Thomas J. O'Malley
Keyword(s):  
Invariant Measure ◽  
Compact Group ◽  
Lie Group ◽  
Closed Subgroup ◽  
Density Theorem ◽  
Locally Compact Group ◽  
Hyperbolic Groups ◽  
Locally Compact ◽  
The Real ◽  
Solvable Lie Group

IfHis a closed subgroup of a locally compact groupG, withG/Hhaving finiteG-invariant measure, then, as observed by Atle Selberg [8], for any neighborhoodUof the identity inGand any elementginG, there is an integern >0 such thatgnis inU·H·U.A subgroup satisfying this latter condition is said to be anS-sub group,or satisfiesproperty (S).IfGis a solvable Lie group, then the converse of Selberg's result has been proved by S. P. Wang [10]: IfHis a closedS-subgroup ofG,thenG/His compact. Property(S)has been used by A. Borel in the important “density theorem” (see Section 2 or [1]).

A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM

2019 ◽  
Vol 100 (2) ◽  
pp. 317-322
Author(s):  
H. KUMUDINI DHARMADASA ◽  
WILLIAM MORAN
Keyword(s):  
Hilbert Space ◽  
Invariant Measure ◽  
Banach Spaces ◽  
Closed Subgroup ◽  
General Setting ◽  
Locally Compact Group ◽  
Reciprocity Theorem ◽  
Representation Space ◽  
Locally Compact ◽  
Frobenius Reciprocity

Let $G$ be a locally compact group and $K$ a closed subgroup of $G$. Let $\unicode[STIX]{x1D6FE},$$\unicode[STIX]{x1D70B}$ be representations of $K$ and $G$ respectively. Moore’s version of the Frobenius reciprocity theorem was established under the strong conditions that the underlying homogeneous space $G/K$ possesses a right-invariant measure and the representation space $H(\unicode[STIX]{x1D6FE})$ of the representation $\unicode[STIX]{x1D6FE}$ of $K$ is a Hilbert space. Here, the theorem is proved in a more general setting assuming only the existence of a quasi-invariant measure on $G/K$ and that the representation spaces $\mathfrak{B}(\unicode[STIX]{x1D6FE})$ and $\mathfrak{B}(\unicode[STIX]{x1D70B})$ are Banach spaces with $\mathfrak{B}(\unicode[STIX]{x1D70B})$ being reflexive. This result was originally established by Kleppner but the version of the proof given here is simpler and more transparent.

scholarly journals Weak Containment by Restrictions of Induced Representations

2018 ◽  
Vol 2020 (7) ◽  
pp. 2034-2053
Author(s):  
Matthew Wiersma
Keyword(s):  
Compact Group ◽  
Closed Subgroup ◽  
Discrete Group ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Locally Compact ◽  
Induced Representations ◽  
C Algebras ◽  
Weak Containment

Abstract A QSIN group is a locally compact group G whose group algebra $\mathrm L^{1}(G)$ admits a quasi-central bounded approximate identity. Examples of QSIN groups include every amenable group and every discrete group. It is shown that if G is a QSIN group, H is a closed subgroup of G, and $\pi \!: H\to \mathcal B(\mathcal{H})$ is a unitary representation of H, then $\pi$ is weakly contained in $\Big (\mathrm{Ind}_{H}^{G}\pi \Big )|_{H}$. This provides a powerful tool in studying the C*-algebras of QSIN groups. In particular, it is shown that if G is a QSIN group which contains a copy of $\mathbb{F}_{2}$ as a closed subgroup, then $\mathrm C^{\ast }(G)$ is not locally reflexive and $\mathrm C^{\ast }_{r}(G)$ does not admit the local lifting property. Further applications are drawn to the “(weak) extendability” of Fourier spaces $\mathrm A_{\pi }$ and Fourier–Stieltjes spaces $\mathrm B_{\pi }$.

scholarly journals Subgroup Correspondences

2018 ◽  
Vol 61 (4) ◽  
pp. 1127-1154 ◽  
Author(s):  
S. Kaliszewski ◽  
Nadia S. Larsen ◽  
John Quigg
Keyword(s):  
Compact Group ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Graph Algebras ◽  
Locally Compact ◽  
Induction Process

AbstractFor a closed subgroup of a locally compact group the Rieffel induction process gives rise to aC*-correspondence over theC*-algebra of the subgroup. We study the associated Cuntz–Pimsner algebra and show that, by varying the subgroup to be open, compact, or discrete, there are connections with the Exel–Pardo correspondence arising from a cocycle, and also with graph algebras.

scholarly journals Spaces of closed subgroups of a connected Lie group

1973 ◽  
Vol 14 (1) ◽  
pp. 77-79 ◽  
Author(s):  
N. Oler
Keyword(s):  
Compact Group ◽  
Compact Space ◽  
Lie Group ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Transformation Groups ◽  
Locally Compact ◽  
Discontinuous Groups ◽  
Groups Of Transformations ◽  
Connected Lie Group

In a sequence of two papers which appeared in 1968 and 1969 Herbert Abels [1, 2] has developed, from a method originated by Gerstenhaber [6], a means for extending the study of properly discontinuous groups of transformations to that of proper transformation groups in general. We recall that, if G is a Hausdorff locally compact group of transformations of a locally compact space X, then the action of Gis proper when, for any two compact subsets K and L, the subset G(K, L) = {g ɛ G: gL∩K # 0} of G is compact (see [3], p. 55). In what follows all groups and spaces will be Hausdorff and locally compact. If H is a closed subgroup of G, then it is clear that the property just defined is possessed by the action of H as a group of left translations of G.

scholarly journals Г-invariant operators associated with locally compact groups

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2721-2730
Author(s):  
Ali Ghaffari ◽  
Somayeh Amirjan
Keyword(s):  
Compact Group ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Usual Sense ◽  
Left Translation ◽  
Compact Groups ◽  
Locally Compact ◽  
Product Group

Let G be a locally compact group and let ? be a closed subgroup of G x G. In this paper, the concept of commutativity with respect to a closed subgroup of a product group, which is a generalization of multipliers under the usual sense, is introduced. As a consequence, we obtain characterization of operators on L2(G) which commute with left translation when G is amenable.

ON THE CONNECTEDNESS OF THE CHABAUTY SPACE OF A LOCALLY COMPACT PRONILPOTENT GROUP

2021 ◽  
pp. 1-10
Author(s):  
BILEL KADRI
Keyword(s):  
Compact Group ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Chabauty Topology

Abstract Let G be a locally compact group and let ${\mathcal {SUB}(G)}$ be the hyperspace of closed subgroups of G endowed with the Chabauty topology. The main purpose of this paper is to characterise the connectedness of the Chabauty space ${\mathcal {SUB}(G)}$ . More precisely, we show that if G is a connected pronilpotent group, then ${\mathcal {SUB}(G)}$ is connected if and only if G contains a closed subgroup topologically isomorphic to ${{\mathbb R}}$ .

Sign in / Sign up

Export Citation Format

Share Document