On the equivalence of all notions of generalized derivations whose domain is a C^∗-algebra

Let ℳ be a Banach bimodule over an associative Banach algebra 𝒜, and let F: 𝒜→ℳ be a linear mapping. Three main uses of the term generalized derivation are identified in the available literature, namely, () F is a generalized derivation of the first type if there exists a derivation d : 𝒜→ℳ satisfying F(a b ) = F(a) b + a d(b) for all a,b∈𝒜. () F is a generalized derivation of the second type if there exists an element ξ∈ℳ^** satisfying F(a b ) = F(a) b + a F(b) - a ξ b for all a,b∈𝒜. () F is a generalized derivation of the third type if there exist two (non-necessarily linear) mappings G,H : 𝒜→ℳ satisfying F(a b ) = G(a) b + a H(b) for all a,b∈𝒜. There are examples showing that these three definitions are not, in general, equivalent. Despite that the first two notions are well studied when 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 𝒜 into a general Banach 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.