Characterizations of * and *-left derivable mappings on some algebras

A linear mapping d from a *-algebra A into a *-A-bimodule M is a *-derivable mapping at G. A if Ad(B)* + d( A) B = d(G) for each A, B in A with AB* = G. We prove that every (continuous) *-derivablemapping at G from a (unital C *-algebra) factor von Neumann algebra into its Banach *-bimodule is a *-derivation if and only if G is a left separating point. A linear mapping d from a *-algebra A into a *-left A-module M is a *-left derivable mapping at G. A if Ad( B)* + Bd( A) = d( G) for each A, B in A with AB* = G. We prove that every continuous *-left derivable mapping at a left separating point from a unital C*-algebra or von Neumann algebra into its Banach *-left A-module is identical with zero under certain conditions.