Left derivable or Jordan left derivable mappings on Banach algebras

Let d be a linear mapping from a unital Banach algebra A into a unital left A-module M, and w in Z(A) be a left separating point of M. We show that the following three conditions are equivalent: (i) d is a Jordan left derivation; (ii) d is left derivable at w; (iii) d is Jordan left derivable at w. Let A be a Banach algebra with the property (B), and M be a Banach left A-module. We consider the relations between generalized (Jordan) left derivations and (Jordan) left derivable mappings at zero from A into M.

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