scholarly journals Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators

2004 ◽  
Vol 47 (3) ◽  
pp. 445-455 ◽  
Author(s):  
A. Yu. Pirkovskii
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Convolution Product ◽  
Locally Compact ◽  
Convolution Algebra ◽  
Nuclear Operators ◽  
Convolution Algebras

AbstractFor a locally compact group G, the convolution product on the space 𝒩(Lp(G)) of nuclear operators was defined by Neufang [11]. We study homological properties of the convolution algebra 𝒩(Lp(G)) and relate them to some properties of the group G, such as compactness, finiteness, discreteness, and amenability.

scholarly journals ON HOMOLOGICAL PROPERTIES FOR SOME MODULES OF UNIFORMLY CONTINUOUS FUNCTIONS OVER CONVOLUTION ALGEBRAS

2011 ◽  
Vol 84 (2) ◽  
pp. 177-185
Author(s):  
RASOUL NASR-ISFAHANI ◽  
SIMA SOLTANI RENANI
Keyword(s):  
Compact Group ◽  
Group Algebra ◽  
Dual Space ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Convolution Algebras ◽  
Uniformly Continuous

AbstractFor a locally compact group G, let LUC(G) denote the space of all left uniformly continuous functions on G. Here, we investigate projectivity, injectivity and flatness of LUC(G) and its dual space LUC(G)* as Banach left modules over the group algebra as well as the measure algebra of G.

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 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.

Module Homomorphisms of the Dual Modules of Convolution Banach Algebras

1992 ◽  
Vol 35 (2) ◽  
pp. 180-185 ◽  
Author(s):  
F. Ghahramani ◽  
J. P. Mcclure
Keyword(s):  
Compact Group ◽  
Group Algebra ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Banach Module ◽  
Locally Compact ◽  
Arens Product ◽  
Convolution Algebra

AbstractSuppose that A is either the group algebra L1 (G) of a locally compact group G, or the Volterra algebra or a weighted convolution algebra with a regulated weight. We characterize: a) Module homomorphisms of A*, when A* is regarded an A** left Banach module with the Arens product, b) all the weak*-weak* continuous left multipliers of A**.

COMPACT ELEMENTS AND OPERATORS OF QUANTUM GROUPS

2016 ◽  
Vol 59 (2) ◽  
pp. 445-462 ◽  
Author(s):  
MASSOUD AMINI ◽  
MEHRDAD KALANTAR ◽  
ALIREZA MEDGHALCHI ◽  
AHMAD MOLLAKHALILI ◽  
MATTHIAS NEUFANG
Keyword(s):  
Compact Group ◽  
Quantum Groups ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Completely Continuous ◽  
Convolution Algebras ◽  
Locally Compact Quantum Groups ◽  
Function Algebras ◽  
Compact Quantum Groups

AbstractA locally compact group G is compact if and only if its convolution algebras contain non-zero (weakly) completely continuous elements. Dually, G is discrete if its function algebras contain non-zero completely continuous elements. We prove non-commutative versions of these results in the case of locally compact quantum groups.

Bipositive and Isometric Isomorphisms of Some Convolution Algebras

1965 ◽  
Vol 17 ◽  
pp. 839-846 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Compact Group ◽  
Lebesgue Space ◽  
Inductive Limit ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Natural Topology ◽  
Locally Compact ◽  
Convolution Algebras ◽  
Image Position ◽  
Complex Valued

Throughout this paper the term "space" will mean "Hausdorff locally compact space" and the term '"group" will mean "Hausdorff locally compact group." If G is a group and 1 ≤ p < ∞, Lp(G) denotes the usual Lebesgue space formed relative to left Haar measure on G. It is well known that L1(G) is an algebra under convolution, and that the same is true of Lp(G) whenever G is compact. We introduce also the space Cc(G) of complex-valued continuous functions f on G for each of which the support (supp f), is compact. The "natural" topology of CC(G) is obtained by regarding CC(G) as the inductive limit of its subspaces

scholarly journals Extreme non-Arens regularity of the group algebra

2018 ◽  
Vol 30 (5) ◽  
pp. 1193-1208
Author(s):  
Mahmoud Filali ◽  
Jorge Galindo
Keyword(s):  
Compact Group ◽  
Quotient Space ◽  
Closed Subspace ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Linear Isometry ◽  
Locally Compact ◽  
Convolution Algebra ◽  
Arens Regularity ◽  
Bounded Functions

AbstractThe Banach algebras of Harmonic Analysis are usually far from being Arens regular and often turn out to be as irregular as possible. This utmost irregularity has been studied by means of two notions: strong Arens irregularity, in the sense of Dales and Lau, and extreme non-Arens regularity, in the sense of Granirer. Lau and Losert proved in 1988 that the convolution algebra {L^{1}(G)} is strongly Arens irregular for any infinite locally compact group. In the present paper, we prove that {L^{1}(G)} is extremely non-Arens regular for any infinite locally compact group. We actually prove the stronger result that for any non-discrete locally compact group G, there is a linear isometry from {L^{\infty}(G)} into the quotient space {L^{\infty}(G)/\mathcal{F}(G)}, with {\mathcal{F}(G)} being any closed subspace of {L^{\infty}(G)} made of continuous bounded functions. This, together with the known fact that {\ell^{\infty}(G)/\mathscr{W\!A\!P}(G)} always contains a linearly isometric copy of {\ell^{\infty}(G)}, proves that {L^{1}(G)} is extremely non-Arens regular for every infinite locally compact group.

scholarly journals Interpolation in wavelet spaces and the HRT-conjecture

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Eirik Berge
Keyword(s):  
Compact Group ◽  
Wavelet Transforms ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Space Structure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Positive Type ◽  
Time Frequency ◽  
Hrt Conjecture

AbstractWe investigate the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })\subset L^{2}(G)$$ W g ( H π ) ⊂ L 2 ( G ) arising from square integrable representations $$\pi :G \rightarrow \mathcal {U}(\mathcal {H}_{\pi })$$ π : G → U ( H π ) of a locally compact group G. We show that the wavelet spaces are rigid in the sense that non-trivial intersection between them imposes strong restrictions. Moreover, we use this to derive consequences for wavelet transforms related to convexity and functions of positive type. Motivated by the reproducing kernel Hilbert space structure of wavelet spaces we examine an interpolation problem. In the setting of time–frequency analysis, this problem turns out to be equivalent to the HRT-conjecture. Finally, we consider the problem of whether all the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })$$ W g ( H π ) of a locally compact group G collectively exhaust the ambient space $$L^{2}(G)$$ L 2 ( G ) . We show that the answer is affirmative for compact groups, while negative for the reduced Heisenberg group.

The Superstability of the Generalized D'alembert Functional Equation

2003 ◽  
Vol 10 (3) ◽  
pp. 503-508 ◽  
Author(s):  
Elhoucien Elqorachi ◽  
Mohamed Akkouchi
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Compact Group ◽  
Locally Compact Group ◽  
Complex Numbers ◽  
Locally Compact

Abstract We generalize the well-known Baker's superstability result for the d'Alembert functional equation with values in the field of complex numbers to the case of the integral equation where 𝐺 is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of 𝐺.

scholarly journals Completely bounded bimodule maps and spectral synthesis

2017 ◽  
Vol 28 (10) ◽  
pp. 1750067 ◽  
Author(s):  
M. Alaghmandan ◽  
I. G. Todorov ◽  
L. Turowska
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Closed Subset ◽  
Spectral Synthesis ◽  
Locally Compact Group ◽  
Product Formula ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Multiplication Algebra

We initiate the study of the completely bounded multipliers of the Haagerup tensor product [Formula: see text] of two copies of the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text]. If [Formula: see text] is a closed subset of [Formula: see text] we let [Formula: see text] and show that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] then [Formula: see text] is a set of local spectral synthesis for [Formula: see text]. Conversely, we prove that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] and [Formula: see text] is a Moore group then [Formula: see text] is a set of spectral synthesis for [Formula: see text]. Using the natural identification of the space of all completely bounded weak* continuous [Formula: see text]-bimodule maps with the dual of [Formula: see text], we show that, in the case [Formula: see text] is weakly amenable, such a map leaves the multiplication algebra of [Formula: see text] invariant if and only if its support is contained in the antidiagonal of [Formula: see text].

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