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**.

scholarly journals Fourier Spaces and Completely Isometric Representations of Arens Product Algebras

2019 ◽  
Vol 71 (03) ◽  
pp. 717-747
Author(s):  
Ross Stokke
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Bounded Operators ◽  
Locally Compact ◽  
Type Representation ◽  
Arens Product ◽  
Isometric Representation ◽  
Product Algebra

AbstractMotivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra $A$ . We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$ . Examples include all left Arens product algebras over $A$ , but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$ -module action $Q$ on a space $X$ , we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak $^{\ast }$ -continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.

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 Johnson pseudo-Connes amenability of dual Banach algebras

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 551-559
Author(s):  
Amir Sahami ◽  
Seyedeh Shariati ◽  
Abdolrasoul Pourabbas
Keyword(s):  
Compact Group ◽  
Finite Index ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Hereditary Properties ◽  
Connes Amenability

We introduce the notion of Johnson pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion with the various notions of Connes amenability like Connes amenability, approximate Connes amenability and pseudo Connes amenability. We also investigate some hereditary properties of this new notion. We prove that for a locally compact group G,M(G) is Johnson pseudo-Connes amenable if and only if G is amenable. Also we show that for every non-empty set I,MI(C) under this new notion is forced to have a finite index. Finally, we provide some examples of certain dual Banach algebras and we study their Johnson pseudo-Connes amenability.

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 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 Non commutative convolution measure algebras with no proper L-ideals

1989 ◽  
Vol 40 (1) ◽  
pp. 13-23 ◽  
Author(s):  
Sahl Fadul Albar
Keyword(s):  
Compact Group ◽  
Group Algebra ◽  
Locally Compact Group ◽  
Dimensional Representation ◽  
Locally Compact ◽  
Finite Dimensional Representation ◽  
Finite Dimensional

We study non-commutative convolution measure algebras satisfying the condition in the title and having an involution with a non-degenerate finite dimensional *-representation. We show first that the group algebra L1(G) of a locally compact group G satisfies these conditions. Then we show that to a given algebra A with the above conditions there corresponds a locally compact group G such that A is a * and L-subalgebra of M(G) and such that the enveloping C*-algebra of A is *isomorphic to C*(G). Finally we show for certain groups that L1(G) is the only example of such algebras, thus giving a characterisation of L1(G).

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 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.

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