On (m,n,l)-Jordan Centralizers of Some Algebras

TL;DR

This paper investigates a generalized form of Jordan centralizers in certain algebras, establishing conditions under which these mappings are actually centralizers, thus extending understanding of algebraic symmetries.

Contribution

It introduces and studies weak (m,n,l)-Jordan centralizers on generalized matrix and reflexive algebras, proving they are centralizers under specific parameter conditions.

Findings

Weak (m,n,l)-Jordan centralizers are centralizers when m+l≥1 and n+l≥1.

The study extends to algebras like CSL and certain reflexive algebras.

Provides new insights into the structure of algebraic mappings in operator algebras.

Abstract

Let $\mathcal{A}$ be a unital algebra over the complex field $\mathbb{C}$. A linear mapping $\delta$ from $\mathcal{A}$ into itself is called a weak (\textit{m,n,l})-Jordan centralizer if $(m+n+l)\delta(A^2)-m\delta(A)A-nA\delta(A)-lA\delta(I)A\in \mathbb{C}I$ for every $A\in \mathcal{A}$, where $m\geq0, n\geq0, l\geq0$ are fixed integers with $m+n+l\neq 0$. In this paper, we study weak (\textit{m,n,l})-Jordan centralizer on generalized matrix algebras and some reflexive algebras alg$\mathcal{L}$, where $\mathcal{L}$ is CSL or satisfies $\vee\{L: L\in \mathcal{J}(\mathcal{L})\}=X$ or $\wedge\{L_-: L\in \mathcal{J}(\mathcal{L})\}=(0)$, and prove that each weak (\textit{m,n,l})-Jordan centralizer of these algebras is a centralizer when $m+l\geq1$ and $n+l\geq1$.