Cohomological properties and Arens regularity of Banach algebras

Abstract

In this paper, we study some cohomlogical properties of Banach algebras. For a Banach algebra $A$ and a Banach $A$-bimodule $B$, we investigate the vanishing of the first Hochschild cohomology groups $H^1(A^n,B^m)$ and $H_{w^{\ast }}^1(A^n,B^m)$, where $0\le m,n\le 3$. For amenable Banach algebra $A$, we show that there are Banach $A$-bimodules $C$, $D$ and elements $\mathfrak{a},\mathfrak{b}\in A^{\ast \ast }$ such that $$Z^1(A,C^{\ast })=\{R_{D^{\prime \prime }(\mathfrak{a})}:D\in Z^1(A,C^{\ast })\}=\{L_{D^{\prime \prime }(\mathfrak{b})}:D\in Z^1(A,D^{\ast })\}.$$ where, for every $b\in B$, $L_b(a)=ba$ and $R_b(a)=ab,$ for every $a\in A$. Moreover, under a condition, we show that if the second transpose of a continuous derivation from the Banach algebra $A$ into $A^{\ast }$ i.e., a continuous linear map from $A^{\ast \ast }$ into $A^{\ast \ast \ast }$, is a derivation, then $A$ is Arens regular. Finally, we show that if $A$ is a dual left strongly irregular Banach algebra such that its second dual is amenable, then $A$ is reflexive.

arXiv commentarXiv admin note: text overlap with arXiv:1008.2655, arXiv:1007.3110, arXiv:1007.3106