Generators and joint spectra for a special class of topological algebras

Sabet, Majid; Morovatpoor, Ali

doi:10.22103/jmmr.2025.24505.1732

Generators and joint spectra for a special class of topological algebras

Document Type : Research Paper

Authors

Majid Sabet ¹

Ali Morovatpoor ²

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

10.22103/jmmr.2025.24505.1732

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.

Keywords

Strongly sequential algebras

Fundamental algebras

Boundedness radius

Carrier space

Joint generators

Subjects

Analysis

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