The Bochner-Eberlein-Doss Property for the semigroup algebra $\ell^1({\Bbb Z}^2,\max)$

Document Type : Research Paper

Author

Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, Iran

Abstract

In recent years, and in the continuation of BSE algebras, the concept of BED algebra for a commutative and semisimple Banach algebra $\mathcal A$ has been introduced and studied by some authors. It has also been proved that if $\mathcal A$ is regular and also contains a bounded approximate identity $\{e_\alpha\}_\alpha$ such that for every $\alpha$, the Gelfand transformation $e_\alpha$, i.e. $\widehat{e_\alpha}$ has compact support, then $\mathcal A$ is a BSE algebra if and only if it is a BED algebra. In other words, for such algebras two concepts of BSE and BED coincide. In this paper, let us assume that $S$ is the commutative semigroup $({\Bbb Z}^2,\max)$. In recent years, it has been proven that the semigroup algebra $\ell^1({\Bbb Z}^2,\max)$ is a BSE algebra. In this paper, we show that $\ell^1({\Bbb Z}^2,\max)$ contains a bounded approximate identity with the condition mentioned above. We also show that $\ell^1({\Bbb Z}^2,\max)$ is regular. Thus, as a main result, we obtain that $\ell^1({\Bbb Z}^2,\max)$ is also a BED algebra.

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References

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Mathematics and Society
Volume 10, Issue 3 - Serial Number 3
September 2025
Pages 149-163

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History

  • Receive Date: 09 July 2024
  • Revise Date: 05 December 2024
  • Accept Date: 02 February 2025
  • Publish Date: 23 May 2025

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