Some functional identities characterizing two-sided centralizers and two-sided generalized derivations on triangular algebras
Amin Hosseini
TL;DR
The paper addresses when additive maps on a unital triangular algebra $\mathcal{T}=Tri(\mathcal{A},\mathcal{M},\mathcal{B})$ act as two-sided centralizers or two-sided generalized derivations by enforcing power-type functional identities with a central parameter $\gamma$. The authors develop a method based on functional identities and a Vandermonde-determinant argument to linearize the identities, deducing that $\Psi$ and $\Omega$ are two-sided centralizers and satisfy $\Psi=\gamma\Omega$. They also derive a functional-identity characterization for two-sided generalized derivations on triangular rings, showing that symmetric identities force a decomposition into centralizers and derivations, and they obtain automatic-continuity results in the triangular normed setting. These results extend the understanding of centralizers and generalized derivations in triangular algebras and provide tools for automatic-continuity analysis in noncommutative ring settings.
Abstract
Let T be a unital triangular algebra, let n > 1 be an integer, let gamma be an invertible element of Z(T), the center of T, and let Psi, Omega:\mathcal{T}\rightarrow \mathcal{T}$ be additive mappings satisfying \begin{align*} Ψ(X^n) = γX^{n - 1}Ω(X) = γΩ(X) X^{n - 1}\end{align*} for all $X \in \mathcal{T}$. If $Ω(\textbf{1}) \in Z(\mathcal{T})$, then $Ψ$ and $Ω$ are two-sided centralizers on $\mathcal{T}$ and also $Ψ= γΩ$. Moreover, using a functional identity, a characterization of two-sided generalized derivations is presented. Some other related results are also discussed.