$\sigma$-Derivations in Banach Algebras

TL;DR

This paper introduces and studies $\sigma$-derivations, $\sigma$-endomorphisms, and their dynamics in Banach algebras, establishing their properties, generators, and extending classical theorems to this new setting.

Contribution

It defines $\sigma$-derivations and $\sigma$-dynamics in Banach algebras, and extends key results like the Leibniz formula and Kleinenckr--Sirokov theorem to this framework.

Findings

$\sigma$-infinitesimal generator of inner $\sigma$-endomorphisms is an inner $\sigma$-derivation

Established a generalized Leibniz formula for $\sigma$-derivations

Extended Kleinenckr--Sirokov theorem to $\sigma$-derivations

Abstract

Introducing the notions of (inner) $\sigma$-derivation, (inner) $\sigma$-endomorphism and one-parameter group of $\sigma$-endomorphisms ($\sigma$-dynamics) on a Banach algebra, we correspond to each $\sigma$-dynamics a $\sigma$-derivation named as its $\sigma$-infinitesimal generator. We show that the $\sigma$-infinitesimal generator of a $\sigma$-dynamics of inner $\sigma$-endomorphisms is an inner $\sigma$-derivation and deal with the converse. We also establish a nice generalized Leibniz formula and extend the Kleinenckr--Sirokov theorem for $\sigma$-derivations under certain conditions.