Action of projections on Banach algebras

Author:

Ali Shakir1,Alali Amal S.2,Rafiquee Naira Noor1,Varshney Vaishali1

Affiliation:

1. Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

2. Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box-84428, Riyadh-11671, KSA

Abstract

<abstract><p>Let $ \mathcal{A} $ be a Banach algebra and $ n &gt; 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 $ \mathcal{A} $ is a prime Banach algebra admitting a continuous projection $ \mathcal{P} $ with image in $ \mathcal{Z}(\mathcal{A}) $ such that $ \mathcal{P}(a^n) = a^n\; \text{for all} \; a \in \mathcal{G} $, the nonvoid open subset of $ \mathcal{A} $, then $ \mathcal{A} $ is commutative and $ \mathcal{P} $ is the identity mapping on $ \mathcal{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.</p></abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

General Mathematics

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