Arens Regularity of Tensor Products and Weak Amenability of Banach Algebras

Details Arens Regularity of Tensor Products and Weak Amenability of Banach Algebras Arens Regularity of Tensor Products and Weak Amenability of Banach Algebras

2010-11-02

arxiv.org/abs/1011.0762v1

Mathematics

Functional Analysis

Scientific paper

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.

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