Annals of Functional Analysis

Some results on $\sigma$-derivations

‎M‎. ‎Hassani‎, ‎S‎. ‎Hejazian, A‎. ‎Hosseini‎, and ‎A‎. ‎Niknam‎

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‎Let $\mathcal{A}$ and $\mathcal{B}$ be two Banach algebras and let‎ ‎$\mathcal{M}$ be a Banach $\mathcal{B}$-bimodule‎. ‎Suppose that‎ ‎$\sigma:\mathcal{A} \rightarrow \mathcal{B}$ is a linear mapping and‎ ‎$d:\mathcal{A} \rightarrow \mathcal{M}$ is a $\sigma$-derivation‎. ‎We‎ ‎prove several results about automatic continuity of‎ ‎$\sigma$-derivations on Banach algebras‎. ‎In addition‎, ‎we define a‎ ‎notion for m-weakly continuous linear mapping and show that‎, ‎under‎ ‎certain conditions‎, ‎$d$ and $\sigma$ are m-weakly continuous‎. ‎Moreover‎, ‎we prove that if $\mathcal{A}$ is commutative and $\sigma‎: ‎\mathcal{A} \rightarrow \mathcal{A}$ is a continuous homomorphism‎ ‎such that $\sigma^{2} = \sigma$ then $\sigma d \sigma (\mathcal{A})‎ ‎\subseteq \sigma(Q(\mathcal{A})) \subseteq rad(\mathcal{A})$‎.

Article information

Ann. Funct. Anal., Volume 2, Number 2 (2011), 75- 84.

First available in Project Euclid: 12 May 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B47: Commutators, derivations, elementary operators, etc.
Secondary: 17B40: Automorphisms, derivations, other operators

$\sigma$-derivation ‎derivation‎ ‎$m$-weakly continuous‎ ‎linear mapping ‎quasi-nilpotent


‎Hosseini‎, A‎.; ‎Hassani‎, ‎M‎.; ‎Niknam‎, ‎A‎.; ‎Hejazian, ‎S‎. Some results on $\sigma$-derivations. Ann. Funct. Anal. 2 (2011), no. 2, 75-- 84. doi:10.15352/afa/1399900196.

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