$\sigma$-Mappings of triangular algebras

Abstract

Let $A$ be an algebra and $\sigma$ an automorphism of $A$. A linear map $d$ of $A$ is called a $\sigma$-derivation of $A$ if $d(xy) = d(x)y + \sigma(x)d(y)$, for all $x, y \in A$. A linear map $D$ is said to be a generalized $\sigma$-derivation of $A$ if there exists a $\sigma$-derivation $d$ of $A$ such that $D(xy) = D(x)y + \sigma(x)d(y)$, for all $x, y \in A$. An additive map $\Theta$ of $A$ is $\sigma$-centralizing if $\Theta(x)x - \sigma(x)\Theta(x) \in Z(A)$, for all $x \in A$. In this paper, precise descriptions of generalized $\sigma$-derivations and $\sigma$-centralizing maps of triangular algebras are given. Analogues of the so-called commutative theorems, due to Posner and Mayne, are also proved for the triangular algebra setting.