Hyper-instability of Banach algebras

Narjes Alabkary; Narjes Alabkary

doi:10.3934/math.2024681

AIMS Mathematics

2024, Volume 9, Issue 6: 14012-14025. doi: 10.3934/math.2024681

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Research article

Hyper-instability of Banach algebras

Narjes Alabkary ^,

Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa, 31982, Saudi Arabia

Received: 24 December 2023 Revised: 12 March 2024 Accepted: 27 March 2024 Published: 16 April 2024
MSC : 47L10, 46L05, 46J10, 46J99

In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as $ C^* $-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions.
- Banach algebra,
- multiplicative stability,
- hyper-instability,
- Fourier algebra,
- Fourier-Stieltjes algebra,
- $ C^* $-algebra
Citation: Narjes Alabkary. Hyper-instability of Banach algebras[J]. AIMS Mathematics, 2024, 9(6): 14012-14025. doi: 10.3934/math.2024681

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Abstract

In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as $ C^* $-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions.

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