Arens Regularity of Tensor Products and Weak Amenability of Banach Algebras

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${}^{\ast }$-weak convergence property [$L{w}^{\ast }wc-$property] and right-weak${}^{\ast }$-weak convergence property [$R{w}^{\ast }wc-$property] and for Banach algebra $A$, suppose that $A^{\ast }$ and $A^{\ast \ast }$, respectively, have $R{w}^{\ast }wc-$property and $L{w}^{\ast }wc-$property. Then if $A^{\ast \ast }$ 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 $D:A\to A^{\ast }$. We obtain some conclusions in the Arens regularity of Banach algebras.