Action of projections on Banach algebras

Shakir Ali; Amal S. Alali; Naira Noor Rafiquee; Vaishali Varshney; Shakir Ali; Amal S. Alali; Naira Noor Rafiquee; Vaishali Varshney

doi:10.3934/math.2023894

AIMS Mathematics

2023, Volume 8, Issue 8: 17503-17513. doi: 10.3934/math.2023894

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Research article

Action of projections on Banach algebras

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
3.
Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

Received: 26 December 2022 Revised: 04 May 2023 Accepted: 10 May 2023 Published: 19 May 2023
MSC : 47L10, 47B48, 46J10, 17C65

Let $ \mathcal{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 $ \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.
- commutativity,
- continuity,
- open subset,
- Banach algebra,
- prime Banach algebra
Citation: Shakir Ali, Amal S. Alali, Naira Noor Rafiquee, Vaishali Varshney. Action of projections on Banach algebras[J]. AIMS Mathematics, 2023, 8(8): 17503-17513. doi: 10.3934/math.2023894

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Abstract

Let $ \mathcal{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 $ \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.

References

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