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. Several 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(ab)=F(a)b+ad(b), for all a,b∈A. (✓) F is a generalized derivation of the second type if there exists an element ξ∈M∗∗ satisfying F(ab)=F(a)b+aF(b)−aξb, for all a,b∈A. (✓) F is a generalized derivation of the third type (respectively, third+ type) if there exist two (not necessarily linear) mappings G,H:A→M (respectively, G:A→M and H:A→M∗∗) satisfying F(ab)=G(a)b+aH(b), for all a,b∈A.These different types of maps are not, in general, equivalent. Although the first two notions are well studied when A is a C∗-algebra, their connections with the third and third+ types have not yet been explored. In this note we prove that generalized derivations of the third and third+ types from a C∗-algebra A to a Banach A-bimodule M are automatically continuous. We also show that (continuous) generalized derivations of the third and third+ types from A to M are generalized derivations of the first and second type. Consequently, derivations of first, second and third+ types coincide in this case. We also explore some concepts of generalized Jordan derivations on a C∗-algebra and establish some continuity properties for them.