Characterization of cubic derivations on various class of Banach algebras

Let A be a Banach algebra and X be a Banach A-bimodule. A mapping D:A--->X is called a cubic derivation if, for all a,bin A we have D(ab)=a^3D(b)+D(a)b^3 . The mapping D:A--->X is called a cubic homogenous map if we have D(lambda a)=lambda^3 D(a) for all ain A and lambdain C. In this paper, we define linear-cubic map and linear-cubic derivation as follows. We say the cubic homogenous map D:A--->X is a linear-cubic map if we have D(lambda a+b)=lambda^3D(a)+D(b) , for all a,bin A and lambdain C and moreover if D:A--->X is a cubic derivation, then we call it a linear-cubic derivation. In this paper, we characterize linear-cubic derivations on various class of Banach algebras such as Banach algebras obtained by theta-Lau product, module extensions of Banach algebras and amalgamated Banach algebras. For characterizing, we define theta-cubic derivation and module cubic mappings. For Banach algebra Atimes_theta B , where thetainsigma(A)cup{0} and A is unital, we show that D:A--->X is a linear-cubic derivation if and only if there are theat -cubic derivation D_B,A:B--->A and linear-cubic derivations D_A:A--->A and D_B:B---> B such that for any (a,b)in Atimes_theta B, be as follows D(a,b)=(D_A(a)+D_B,A(b),D(b)) and these mappings satisfy in the given condition. We obtain similar results for module extensions of Banach algebras and amalgamated Banach algebras.

* The formulas are not displayed correctly.