@article{Ali2023, 
author = {Shakir Ali and Amal S. Alali and Naira Noor Rafiquee and Vaishali Varshney},
title = {Action of projections on Banach algebras},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {8},
pages = {17503-17513},
keywords = {continuity, Banach algebra, commutativity, open subset, prime Banach algebra},
url = {https://www.sciopen.com/article/10.3934/math.2023894},
doi = {10.3934/math.2023894},
abstract = {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.}
}