Generalized Jordan derivations of unital algebras
Dominik Benkovič, Mateja Grašič
TL;DR
This work develops a framework for generalized Jordan derivations on unital algebras with $ ext{char}(F) eq 2$, introducing quasi Jordan centralizers and quasi Jordan derivations as fundamental components. It proves that every generalized Jordan derivation decomposes as a sum of a quasi Jordan centralizer and a quasi Jordan derivation, and identifies when these maps reduce to familiar centralizers or derivations in various algebra classes. A precise characterization is given for quasi Jordan centralizers via the centers $Z_J( abla A)$ and $Z_Q( abla A)$, with extensive examples across matrix, triangular, nest, and semiprime algebras to illustrate when equalities like $ ext{QJCent}= ext{Cent}$ or $ ext{JCent}= ext{Cent}$ hold or fail. The paper also analyzes the structure of maps in $ ext{QJDer}( abla A)$, establishing conditions under which $ ext{QJDer}( abla A)= ext{Cent}( abla A)+ ext{Der}( abla A)$, notably for matrix algebras and algebras generated by idempotents, thereby unifying several strands of Jordan derivation theory and clarifying the relationships among centralizers, Jordan centralizers, and their generalized counterparts.
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.