Linear Maps that are (g,h)-Derivations or (g,t,h)-Ternary Derivations at a Point on *-Module Extension Banach Algebras

Abstract

We introduce a *-module extension Banach algebras to generalized the results of Essaleh and Peralta. Precisely, let g,t and h are bounded homomorphism maps on an unital *-module extension Banach algebra, if a bounded linear map D on an unital *-module extension Banach algebra is (g,t,h)-ternary derivation at the unit element, then the next statements are hold:

1) D is (g,h)-generalized derivation;

2) D is *-(g,h)-derivation and (g,t,h)-triple (ternary) derivation, whenever D(1,0)=(0,0);

3) D is (g,t,h)-ternary derivation.

In addition, we prove that a bounded linear map on *-module extension Banach algebra which is (g,h)-derivation or (g,t,h)-ternary derivation at the zero element is (g,h)-generalized derivation.