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 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.

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]).

Unit Elements in the Double Dual of a Subalgebra of the Fourier Algebra A(G)

2009 ◽  
Vol 61 (2) ◽  
pp. 382-394
Author(s):  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Arens Product ◽  
The Right ◽  
Topological Centers ◽  
The Relationship

Abstract. Let 𝒜 be a Banach algebra with a bounded right approximate identity and let ℬ be a closed ideal of 𝒜. We study the relationship between the right identities of the double duals ℬ** and 𝒜** under the Arens product. We show that every right identity of ℬ** can be extended to a right identity of 𝒜** in some sense. As a consequence, we answer a question of Lau and Ülger, showing that for the Fourier algebra A(G) of a locally compact group G, an element ϕ ∈ A(G)** is in A(G) if and only if A(G)ϕ ⊆ A(G) and Eϕ = ϕ for all right identities E of A(G)**. We also prove some results about the topological centers of ℬ** and 𝒜**.

scholarly journals AN ARBITRARY INTERSECTION OF Lp-SPACES

2012 ◽  
Vol 85 (3) ◽  
pp. 433-445 ◽  
Author(s):  
F. ABTAHI ◽  
H. G. AMINI ◽  
H. A. LOTFI ◽  
A. REJALI
Keyword(s):  
Banach Space ◽  
Compact Group ◽  
Banach Algebra ◽  
Locally Compact Group ◽  
Convolution Product ◽  
Special Choice ◽  
Arbitrary Subset ◽  
Locally Compact ◽  
Lp Spaces ◽  
Algebraic Properties

AbstractFor a locally compact group G and an arbitrary subset J of [1,∞], we introduce ILJ(G) as a subspace of ⋂ p∈JLp(G) with some norm to make it a Banach space. Then, for some special choice of J, we investigate some topological and algebraic properties of ILJ(G) as a Banach algebra under a convolution product.

scholarly journals On Cauchy-type functional equations

2004 ◽  
Vol 2004 (16) ◽  
pp. 847-859
Author(s):  
Elqorachi Elhoucien ◽  
Mohamed Akkouchi
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Functional Equations ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Cauchy Type ◽  
Matrix Functions ◽  
Continuous Solutions ◽  
Locally Compact

LetGbe a Hausdorff topological locally compact group. LetM(G)denote the Banach algebra of all complex and bounded measures onG. For all integersn≥1and allμ∈M(G), we consider the functional equations∫Gf(xty)dμ(t)=∑i=1ngi(x)hi(y),x,y∈G, where the functionsf,{gi},{hi}:G→ℂto be determined are bounded and continuous functions onG. We show how the solutions of these equations are closely related to the solutions of theμ-spherical matrix functions. WhenGis a compact group andμis a Gelfand measure, we give the set of continuous solutions of these equations.

scholarly journals Multinorms and Approximate Amenability of Weighted Group Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Saman Ghaderkhani
Keyword(s):  
Weight Function ◽  
Compact Group ◽  
Banach Algebra ◽  
Locally Compact Group ◽  
Group Algebras ◽  
Locally Compact ◽  
Approximate Amenability

Let G be a locally compact group, and take p,q with 1≤p,q<∞. We prove that, for any left (p,q)-multiinvariant functional on L∞(G) and for any weight function ω≥1 on G, the approximate amenability of the Banach algebra L1(G,ω) implies the left (p,q)-amenability of G, but in general the opposite is not true. Our proof uses the notion of multinorms. We also investigate the approximate amenability of M(G,ω).

scholarly journals Approximate Connes-biprojectivity of dual Banach algebras

2020 ◽  
pp. 2150025
Author(s):  
A. Sahami ◽  
S. F. Shariati ◽  
A. Pourabbas
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Convolution Product ◽  
Locally Compact ◽  
Triangular Banach Algebras ◽  
Connes Amenability

In this paper, we introduce a notion of approximate Connes-biprojectivity for dual Banach algebras. We study the relation between approximate Connes-biprojectivity, approximate Connes amenability and [Formula: see text]-Connes amenability. We propose a criterion to show that certain dual triangular Banach algebras are not approximately Connes-biprojective. Next, we show that for a locally compact group [Formula: see text], the Banach algebra [Formula: see text] is approximately Connes-biprojective if and only if [Formula: see text] is amenable. Finally, for an infinite commutative compact group [Formula: see text], we show that the Banach algebra [Formula: see text] with convolution product is approximately Connes-biprojective, but it is not Connes-biprojective.

scholarly journals Completely continuous elements of Banach algebras related to locally compact groups

2007 ◽  
Vol 76 (1) ◽  
pp. 49-54 ◽  
Author(s):  
M. J. Mehdipour ◽  
R. Nasr-Isfahani
Keyword(s):  
Banach Space ◽  
Compact Group ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Completely Continuous ◽  
Measurable Functions

Let G be a locally compact group and be the Banach space of all essentially bounded measurable functions on G vansihing an infinity. Here, we study some families of right completely continuous elements in the Banach algebra equipped with an Arens type product. As the main result, we show that has a certain right completely continuous element if and only if G is compact.

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 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.

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