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arXiv:2403.18773v1 Announce Type: new
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,
($\checkmark$) $F$ is a generalized derivation of the first type if there exists a derivation $ d : \mathcal{A}\to \mathcal{M}$ satisfying $F(a b ) = F(a) b + a d(b)$ for all $a,b\in \mathcal{A}$.
($\checkmark$) $F$ is a generalized derivation of the second type if there exists an element $\xi\in \mathcal{M}^{**}$ satisfying $F(a b ) = F(a) b + a F(b) - a \xi b$ for all $a,b\in \mathcal{A}$.
($\checkmark$) $F$ is a generalized derivation of the third type if there exist two (non-necessarily linear) mappings $G,H : \mathcal{A}\to \mathcal{M}$ satisfying $F(a b ) = G(a) b + a H(b)$ for all $a,b\in \mathcal{A}$.
There are examples showing that these three definitions are not, in general, equivalent. Despite that the first two notions are well studied when $\mathcal{A}$ is a C$^*$-algebra, it is not known if the three notions are equivalent under these special assumptions. In this note we prove that every generalized derivation of the third type whose domain is a C$^*$-algebra is automatically continuous. We also prove that every (continuous) generalized derivation of the third type from a C$^*$-algebra $\mathcal{A}$ into a general Banach $\mathcal{A}$-bimodule is a generalized derivation of the first and second type. In particular, the three notions coincide in this case. We also explore the possible notions of generalized Jordan derivations on a C$^*$-algebra and establish some continuity properties for them.