scholarly journals Regularity of induced representations and a theorem of Quigg and Spielberg

2002 ◽  
Vol 133 (2) ◽  
pp. 249-259
Author(s):  
ASTRID AN HUEF ◽  
IAIN RAEBURN
Keyword(s):  
Compact Group ◽  
Closed Subgroup ◽  
Morita Equivalence ◽  
Locally Compact Group ◽  
Unitary Representations ◽  
Crossed Products ◽  
Locally Compact ◽  
Induced Representations ◽  
Morita Equivalent ◽  
Special Cases

Mackey's imprimitivity theorem characterizes the unitary representations of a locally compact group G which have been induced from representations of a closed subgroup K; Rieffel's influential reformulation says that the group C*-algebra C*(K) is Morita equivalent to the crossed product C0(G/K)×G [14]. There have since been many important generalizations of this theorem, especially by Rieffel [15, 16] and by Green [3, 4]. These are all special cases of the symmetric imprimitivity theorem of [11], which gives a Morita equivalence between two crossed products of induced C*-algebras.

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 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 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?

scholarly journals Crossed products of Hilbert pro-C-bimodules and associated pro-C-algebras

2016 ◽  
Vol 32 (2) ◽  
pp. 195-201
Author(s):  
MARIA JOITA ◽  
◽  
RADU-B. MUNTEANU ◽  
Keyword(s):  
Compact Group ◽  
Inverse Limit ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Locally Compact ◽  
C Algebras

An action (γ, α) of a locally compact group G on a Hilbert pro-C∗-bimodule (X, A) induces an action γ × α of G on A ×X Z the crossed product of A by X. We show that if (γ, α) is an inverse limit action, then the crossed product of A ×α G by X ×γ G respectively of A ×α,r G by X ×γ,r G is isomorphic to the full crossed product of A ×X Z by γ × α respectively the reduced crossed product of A ×X Z by γ × α.

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]).

scholarly journals Group actions on topological graphs

2011 ◽  
Vol 32 (5) ◽  
pp. 1527-1566 ◽  
Author(s):  
VALENTIN DEACONU ◽  
ALEX KUMJIAN ◽  
JOHN QUIGG
Keyword(s):  
Compact Group ◽  
Universal Covering ◽  
Locally Compact Group ◽  
Skew Product ◽  
Topological Graph ◽  
Topological Graphs ◽  
Locally Compact ◽  
Natural Action ◽  
Morita Equivalent ◽  
Dual Action

AbstractWe define the action of a locally compact groupGon a topological graphE. This action induces a natural action ofGon theC*-correspondence ℋ(E) and on the graphC*-algebraC*(E). If the action is free and proper, we prove thatC*(E)⋊rGis strongly Morita equivalent toC*(E/G) . We define the skew product of a locally compact groupGby a topological graphEvia a cocyclec:E1→G. The group acts freely and properly on this new topological graphE×cG. IfGis abelian, there is a dual action onC*(E) such that$C^*(E)\rtimes \hat {G}\cong C^*(E\times _cG)$. We also define the fundamental group and the universal covering of a topological graph.

scholarly journals Rigidity theory for C∗-dynamical systems and the “Pedersen Rigidity Problem”, II

2019 ◽  
Vol 30 (08) ◽  
pp. 1950038
Author(s):  
S. Kaliszewski ◽  
Tron Omland ◽  
John Quigg
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Compact Group ◽  
Related Problem ◽  
Locally Compact Group ◽  
Crossed Products ◽  
Locally Compact ◽  
Fixed Point Algebra ◽  
Multiplier Algebras

This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen’s theorem, which does hold for an arbitrary locally compact group [Formula: see text], saying that two actions [Formula: see text] and [Formula: see text] of [Formula: see text] are outer conjugate if and only if the dual coactions [Formula: see text] and [Formula: see text] of [Formula: see text] are conjugate via an isomorphism that maps the image of [Formula: see text] onto the image of [Formula: see text] (inside the multiplier algebras of the respective crossed products). We do not know of any examples of a pair of non-outer-conjugate actions such that their dual coactions are conjugate, and our interest is therefore exploring the necessity of latter condition involving the images; and we have decided to use the term “Pedersen rigid” for cases where this condition is indeed redundant. There is also a related problem, concerning the possibility of a so-called equivariant coaction having a unique generalized fixed-point algebra, that we call “fixed-point rigidity”. In particular, if the dual coaction of an action is fixed-point rigid, then the action itself is Pedersen rigid, and no example of non-fixed-point-rigid coaction is known.

Sign in / Sign up

Export Citation Format

Share Document