Derivations on Banach algebras of connected multiplicative linear functionals

Let A and B be Banach algebras with sigma(B) not equal emptyset. Let theta, phi, gamma is an element of sigma( B) and Der( Ax(theta)(phi,gamma) B) be the set of all linear mappings d : AxB -> AxB satisfying d((a, b) center dot theta (x, y)) = d(a, b) center dot phi (x, y) + (a, b) center dot gamma d(x, y) for all a, x is an element of A and b, y is an element of B. In this paper, we characterize elements of Der(A x(theta)(phi,gamma) B) in the case where A has a right identity. We then investigate the concept of centralizing for elements of Der(A x(theta)(phi,gamma) B) and determine dependent elements of Der(A x(theta)(phi,gamma) B). We also apply some results to group algebras.