Generalized Jordan derivations of unital algebras

TL;DR

This paper introduces a new concept called generalized Jordan derivations for unital algebras, extending existing notions like Jordan derivations and centralizers, and characterizes their structure in terms of quasi Jordan derivations.

Contribution

It defines generalized Jordan derivations in unital algebras and provides their explicit form and characterization, expanding the understanding of derivation-like maps.

Findings

Characterization of generalized Jordan derivations in unital algebras.

Representation of these maps via quasi Jordan centralizers and derivations.

Extension of classical derivation concepts to a broader class of maps.

Abstract

Let $A$ be a unital algebra over a field $F$ with $\operatorname*{char} (F)\neq2$. In this paper we introduce a new concept of a generalized Jordan derivation, covering Jordan centralizers and Jordan derivations, as follows: a linear map $f:A\rightarrow A$ is a generalized Jordan derivation if there exist linear maps $g;h:A\rightarrow A$ such that $f\left( x\right) \circ y+x\circ g\left( y\right) =h\left( x\circ y\right) $ for all $x,y\in A$ (here $x\circ y=xy+yx$). Our aim is to give the form of map $f$ in terms of the so called quasi Jordan centralizers and quasi Jordan derivations. In addition, a characterization of such maps is presented.