Cohomological properties and Arens regularity of Banach algebras

TL;DR

This paper investigates cohomological properties of Banach algebras, exploring conditions for Arens regularity and reflexivity, and analyzing the vanishing of Hochschild cohomology groups in various contexts.

Contribution

It provides new insights into the cohomological structure of Banach algebras, linking amenability, Arens regularity, and reflexivity with specific algebraic and topological conditions.

Findings

Vanishing of certain Hochschild cohomology groups for Banach algebras.

Conditions under which the second transpose of a derivation implies Arens regularity.

Characterization of reflexivity in dual left strongly irregular Banach algebras with amenable second dual.

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 $H_{w^*}^1(A^n,B^m)$, where $0\leq m,n\leq 3$. For amenable Banach algebra $A$, we show that there are Banach $A$-bimodules $C$, $D$ and elements $\mathfrak{a}, \mathfrak{b}\in A^{**}$ such that $$Z^1(A,C^*)=\{R_{D^{\prime\prime}(\mathfrak{a})}:~D\in Z^1(A,C^*)\}=\{L_{D^{\prime\prime}(\mathfrak{b})}:~D\in Z^1(A,D^*)\}.$$ where, for every $b\in B$, $L_{b}(a)=ba$ and $R_{b}(a)=a b,$ 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 $A^*$ i.e., a continuous linear map from $A^{**}$ into $A^{***}$, 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.