Some commutativity theorems on Banach algebras

In this article we discuss the commutativity of a prime Banach algebra A with the help of its generalized (alpha, alpha)-derivation. In particular, we prove that if A is a unital prime Banach algebra and A has a nonzero continuous linear generalized (alpha, alpha)-derivation g associated with a nonzero continuous linear (alpha, alpha)-derivation d such that g((xy)(n)) - d(x(n))d(y(n)) = 0 or g((xy)(n)) - d(y(n))d(x(n)) = 0 for sufficiently many x, y and integer n = n(x, y) > 1 then A must be commutative. In this connection some more results has been found. Further examples are given to show that hypotheses are not superfluous.