  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.

Functional Analysis;Operator Algebras