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 (α , α) -derivation. In particular, we prove that if A is a unital prime Banach algebra and A has a nonzero continuous linear generalized (α , α) -derivation g associated with a nonzero continuous linear (α , α) -derivation d such that g ((x y) n) - d (x n) d (y n) = 0 or g ((x y) 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. [ABSTRACT FROM AUTHOR]