Generators and joint spectra for a special class of topological algebras

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, Payame Noor University (PNU), Tehran, Iran

2 Department of Basic Sciences, Faculty of Basic Sciences, Technical and Vocational University (TVU), Tehran, Iran

Abstract

Let $u$ be a generator for a commutative Banach algebra with unit. It is well known that the spectrum of $u$ is homeomorphic to the carrier of this algebra. In this paper, we extend this result for a broader class of complete metrizable  topological algebras, particularly those  satisfying the properties of fundamental strongly sequential(FSS) and linearly complete algebras. Specifically, we establish that the  homeomorphism between the spectrum Sp(u) and the carrier space holds for FSS-algebras and linearly complete regular algebras. Thus, we generalize the classical result known for Banach algebras. Furthermore, by assuming that the boundedness radius $\beta$ is subadditive, we prove that the spectrum $Sp(u)$ is polynomially convex. This assumption also enables us to derive a more general result on the polynomial convexity of joint spectra in finitely generated algebras. To demonstrate the significance and nontrivial nature of these extensions, we provide illustrative examples that highlight how the introduced conditions substantially broaden the applicability of existing results.

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References

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Journal of Mahani Mathematical Research
Volume 15, Issue 1 - Serial Number 33
January 2026
Pages 167-181

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History

  • Receive Date: 14 December 2024
  • Revise Date: 02 July 2025
  • Accept Date: 05 August 2025

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