부산대학교 도서관

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.sup.n]) = [a.sup.n] for all a [member of] 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 if f 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. Keywords: commutativity; continuity; open subset; Banach algebra; prime Banach algebra Mathematics Subject Classification: 47L10, 47B48, 46J10, 17C65
1. Introduction This research has been motivated by the work's of Ali-Khan [2] and Khan [9]. All over this paper unless otherwise stated, A denotes a Banach algebra with the [...]