Derivations on generalized semidirect products of Banach algebras
| Autor: | Hasan Pourmahmood Aghababa |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Mathematics::Functional Analysis
Pure mathematics Semidirect product Algebra and Number Theory first Hochschild cohomology group 46H25 Approximation property Group (mathematics) Group cohomology 010102 general mathematics derivation 16E40 010103 numerical & computational mathematics Locally compact group 01 natural sciences Cohomology Algebra Banach algebra Homomorphism locally compact group 0101 mathematics 43A15 Analysis Mathematics |
| Zdroj: | Banach J. Math. Anal. 10, no. 3 (2016), 509-522 |
| ISSN: | 1735-8787 |
| DOI: | 10.1215/17358787-3607156 |
| Popis: | Let $A$ and $B$ be Banach algebras, let $\theta:A\to B$ be a continuous Banach algebra homomorphism, and let $I$ be a closed ideal in $B$ . Then the $l^{1}$ -direct sum of $A$ and $I$ with a special product becomes a Banach algebra, denoted by $A\bowtie^{\theta}I$ , which we call the generalized semidirect product of $A$ and $I$ . In this article, among other things, we first characterize derivations on $A\bowtie^{\theta}I$ and then we compute the first cohomology group of $A\bowtie^{\theta}I$ when the first cohomology groups of $A$ with coefficients in $A$ and $I$ are trivial. As an application we characterize the first cohomology group of second duals of dual Banach algebras. Then we provide a nice characterization of the first cohomology group of $A\bowtie^{\mathrm{id}}A$ . Furthermore, we calculate the first cohomology group of some concrete Banach algebras related to locally compact groups. |
| Databáze: | OpenAIRE |
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