Action of projections on Banach algebras.

Let A be a Banach algebra and n > 1 , a fixed integer. The main objective of this paper is to talk about the commutativity of Banach algebras via its projections. Precisely, we prove that if A is a prime Banach algebra admitting a continuous projection P with image in Z (A) such that P (a n) = a n for all a ∈ G , the nonvoid open subset of A , then A is commutative and P is the identity mapping on A . Apart from proving some other results, as an application we prove that, a normed algebra is commutative iff the interior of its center is non-empty. Furthermore, we provide some examples to show that the assumed restrictions cannot be relaxed. Finally, we conclude our paper with a direction for further research. [ABSTRACT FROM AUTHOR]