|
This article is cited in 1 scientific paper (total in 1 paper)
What can be expected from the image of $\sigma$-derivations on Banach algebras?
A. Hosseini Kashmar Higher Education Institute, Kashmar, Iran
Abstract:
The main purpose of this paper is to obtain some results on the image of $\sigma$-derivations on Banach algebras. One of the main results of this paper is to prove that if $\mathcal{A}$ is a commutative Banach algebra and $d:\mathcal{A} \rightarrow \mathcal{A}$ is a continuous $\sigma$-derivation such that $\sigma$ is a continuous homomorphism, $d \sigma = \sigma d = d$ and $\sigma^{2} = \sigma$, then $d(\mathcal{A}) \subseteq {\rm rad}(\mathcal{A})$, where ${\rm rad}(\mathcal{A})$ denotes the Jacobson radical of $\mathcal{A}$. Moreover, we obtain Sinclair's theorem for $\sigma$-derivations without assuming continuity. Indeed, under certain conditions, we prove that if $d$ is a $\sigma$-derivation on a Banach algebra $\mathcal{A}$, then $d(\mathcal{P}) \subseteq \mathcal{P}$ for every primitive ideal $\mathcal{P}$ of $\mathcal{A}$. Some other related results are also discussed.
Keywords:
derivation, $\sigma$-derivation, $(\sigma, \tau)$-derivation, Sinclair's theorem, Singer-Wermer theorem.
Received: 31.08.2021 Revised: 01.11.2021 Accepted: 23.12.2021
Citation:
A. Hosseini, “What can be expected from the image of $\sigma$-derivations on Banach algebras?”, Izv. Vyssh. Uchebn. Zaved. Mat., 2022, no. 7, 44–57; Russian Math. (Iz. VUZ), 66:7 (2022), 38–50
Linking options:
https://www.mathnet.ru/eng/ivm9792 https://www.mathnet.ru/eng/ivm/y2022/i7/p44
|
Statistics & downloads: |
Abstract page: | 81 | Full-text PDF : | 17 | References: | 23 | First page: | 7 |
|