Cohomological properties and Arens regularity of Banach algebras

• Published in: arXiv.org 2020-06
• Main Authors: Hossein Eghbali Sarai, Azar, Kazem Haghnejad, Jabbari, Ali
• Format: Article
• Language: English
• Subjects: Algebra; Banach spaces; Derivation; Homology
• ISSN: 2331-8422

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.