On the equivalence of all notions of generalized derivations whose domain is a C${}^{\ast }$-algebra

Abstract

Let $\mathcal{M}$ be a Banach bimodule over an associative Banach algebra $\mathcal{A}$, and let $F:\mathcal{A}\to \mathcal{M}$ be a linear mapping. Three main uses of the term \emph{generalized derivation} are identified in the available literature, namely, ($✓$) $F$ is a generalized derivation of the first type if there exists a derivation $d:\mathcal{A}\to {\mathcal{M}}^{\ast \ast }$ satisfying $F(ab)=F(a)b+ad(b),$ for all $a,b\in \mathcal{A}$. ($✓$) $F$ is a generalized derivation of the second type if there exists an element $\xi \in {\mathcal{M}}^{\ast \ast }$ satisfying $F(ab)=F(a)b+aF(b)-a\xi b,$ for all $a,b\in \mathcal{A}$. ($✓$) $F$ is a generalized derivation of the third type if there exist two (non-necessarily linear) mappings $G,H:\mathcal{A}\to \mathcal{M}$ satisfying $F(ab)=G(a)b+aH(b),$ for all $a,b\in \mathcal{A}$. These three types of maps are not, in general, equivalent. Although the first two notions are well studied when $\mathcal{A}$ is a C${}^{\ast }$-algebra, their connections with the third one have not yet been explored. In this note we prove that every generalized derivation of the third type from a C${}^{\ast }$-algebra $\mathcal{A}$ to a Banach $\mathcal{A}$-bimodule $\mathcal{M}$ is automatically continuous. We also show that every (continuous) generalized derivation of the third type from $\mathcal{A}$ to $\mathcal{M}$ is a generalized derivation of the first and second type. Consequently, the three notions coincide in this case. We also explore some concepts of generalized Jordan derivations on a C${}^{\ast }$-algebra and establish some continuity properties for them.