Characterization of linear mappings on (Banach) $*$-algebras by similar properties to derivations

Year, pages: 2020, 1003 - 1011

Let $\mathcal{A}$ be a $*$-algebra, $ δ: \mathcal{A} \to \mathcal{A}$ be a linear map, and $z\in \mathcal{A}$ be fixed. We consider the condition that $δ$ satisfies $xδ(y)^*+δ(x)y^*=δ(z)$ ( $x^*δ(y)+δ(x)^*y=δ(z)$) whenever $xy^*=z$ ($x^*y=z$), and under several conditions on $\mathcal{A}$, $δ$ and $z$ we characterize the structure of $δ$. In particular, we prove that if $\mathcal{A}$ is a Banach $*$-algebra, $δ$ is a continuous linear map, and $z$ is a left (right) separating point of $\mathcal{A}$, then $δ$ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map $δ$ satisfying the above conditions with $z=0$ on two classes of $*$-algebras: zero product determined algebras and standard operator algebras.

Fadaee, B., Fallahi, K., Ghahramani, H. 2020. Characterization of linear mappings on (Banach) $*$-algebras by similar properties to derivations. In Mathematica Slovaca, vol. 70, no.4, pp. 1003-1011. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0409

Fadaee, B., Fallahi, K., Ghahramani, H. (2020). Characterization of linear mappings on (Banach) $*$-algebras by similar properties to derivations. Mathematica Slovaca, 70(4), 1003-1011. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0409

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava