A Unified Approach to the Arens Regularity and Related Problems for a Class of Banach Algebras Associated With Locally Compact Groups

Based on Katznelson–Tzafriri Theorem on power bounded operators, we prove in this paper a theorem, which applies to the most of the classical Banach algebras of harmonic analysis associated with locally compact groups, to deal with the problems when a given Banach algebra A is Arens regular and when A is an ideal in its bidual. In the second part of the paper, we study the topological center of the bidual of a class of Banach algebras with a multiplier bounded approximate identity.

ARENS REGULARITY OF PROJECTIVE TENSOR PRODUCT

In this paper, we study approximate identity properties, some propositions from Baker, Dales, Lau in general situations and we establish some relationships between the topological centers of module actions and factorization properties with some results in group algebras. We consider under which sufficient and necessary conditions the Banach algebra $A\widehat{\otimes}B$ is Arens regular.

Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers

Let $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$. In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$. Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$-module homomorphisms on $I^{\ast \ast }$ and on $I^{\ast }$.

ARENS REGULARITY AND STRONG IRREGULARITY OF CERTAIN BILINEAR MAPPINGS

In this paper, the relations between the topological centers of bounded bilinear mappings and some of their higher rank adjoints are investigated. Particularly, for a Banach algebra A, some results about the Banach A−modules and Arens regularity and strong Arens irregularity of module actions will be obtained.

ARENS PRODUCTS, ARENS REGULARITY AND RELATED PROBLEMS

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.

Extreme non-Arens regularity of the group algebra

The 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.

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.