Arens Regularity of Tensor Products and Weak Amenability of Banach Algebras

TL;DR

This paper investigates the conditions under which the Arens regularity of Banach algebras is preserved in their tensor products and explores the implications for weak amenability, introducing new convergence properties.

Contribution

It establishes new criteria linking the Arens regularity of tensor products to that of individual Banach algebras and introduces novel convergence properties affecting weak amenability.

Findings

Arens regularity of tensor products characterized by algebra properties

New convergence properties linked to weak amenability

Conditions under which $A^{**}$ weakly amenable implies $A$ is weakly amenable

Abstract

In this note, we study the Arens regularity of projective tensor product $A\hat{\otimes}B$ whenever $A$ and $B$ are Arens regular. We establish some new conditions for showing that the Banach algebras $A$ and $B$ are Arens regular if and only if $A\hat{\otimes}B$ is Arens regular. We also introduce some new concepts as left-weak$^*$-weak convergence property [$Lw^*wc-$property] and right-weak$^*$-weak convergence property [$Rw^*wc-$property] and for Banach algebra $A$, suppose that $A^*$ and $A^{**}$, respectively, have $Rw^*wc-$property and $Lw^*wc-$property. Then if $A^{**}$ is weakly amenable, it follows that $A$ is weakly amenable. We also offer some results concerning the relation between these properties with some special derivation $D:A\rightarrow A^*$. We obtain some conclusions in the Arens regularity of Banach algebras.