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.

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 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?

scholarly journals Completely positive linear operators for Banach spaces

1994 ◽  
Vol 17 (1) ◽  
pp. 27-30
Author(s):  
Mingze Yang
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Extreme Point ◽  
General Setting ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Complete Positivity ◽  
Completely Positive

Using ideas of Pisier, the concept of complete positivity is generalized in a different direction in this paper, where the Hilbert spaceℋis replaced with a Banach space and its conjugate linear dual. The extreme point results of Arveson are reformulated in this more general setting.

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 Non-minimal tree actions and the existence of non-uniform tree lattices

2004 ◽  
Vol 70 (2) ◽  
pp. 257-266
Author(s):  
Lisa Carbone
Keyword(s):  
Invariant Measure ◽  
Compact Group ◽  
Connected Graph ◽  
Locally Compact Group ◽  
Minimal Tree ◽  
Locally Compact ◽  
Discrete Subgroups ◽  
Finite Tree ◽  
Locally Finite ◽  
Uniform Tree

A uniform tree is a tree that covers a finite connected graph. Let X be any locally finite tree. Then G = Aut(X) is a locally compact group. We show that if X is uniform, and if the restriction of G to the unique minimal G-invariant subtree X0 ⊆ X is not discrete then G contains non-uniform lattices; that is, discrete subgroups Γ for which Γ/G is not compact, yet carries a finite G-invariant measure. This proves a conjecture of Bass and Lubotzky for the existence of non-uniform lattices on uniform trees.

scholarly journals On a Certain Class of Semigroups of Operators

2011 ◽  
Vol 18 (02) ◽  
pp. 129-142 ◽  
Author(s):  
Paolo Aniello
Keyword(s):  
Banach Space ◽  
Compact Group ◽  
Banach Spaces ◽  
Locally Compact Group ◽  
Semigroups Of Operators ◽  
Probability Measures ◽  
Locally Compact ◽  
Quantum Dynamical ◽  
Interesting Class ◽  
Natural Way

We define an interesting class of semigroups of operators in Banach spaces, namely, the randomly generated semigroups. This class contains as a remarkable subclass a special type of quantum dynamical semigroups introduced in the early 1970s by Kossakowski. Each randomly generated semigroup is associated, in a natural way, with a pair formed by a representation or an antirepresentation of a locally compact group in a Banach space and by a convolution semigroup of probability measures on this group. Examples of randomly generated semigroups having important applications in physics are briefly illustrated.

The covariance algebra of an extended covariant system

1979 ◽  
Vol 85 (2) ◽  
pp. 271-280 ◽  
Author(s):  
Ronny Rousseau
Keyword(s):  
Hilbert Space ◽  
Compact Group ◽  
Unitary Group ◽  
Von Neumann Algebra ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Additional Axiom

Let M be a von Neumann algebra acting on a Hilbert space , and let G be a locally compact group. We consider an extension of G by , the unitary group of M. If the triple satisfies an additional axiom, we say that it is an extended covariant system. We define a Hilbert space and operators , acting on . The von Neumann algebra is then the covariance algebra of the extended covariant system , denoted by .

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