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Invertibles in topological rings: a new approach

Authors
  • García-Pacheco, Francisco Javier1
  • Miralles, Alejandro2
  • Murillo-Arcila, Marina3
  • 1 Universidad de Cádiz, Puerto Real, 11510, Spain , Puerto Real (Spain)
  • 2 Universitat Jaume I, Castelló de la Plana, 12071, Spain , Castelló de la Plana (Spain)
  • 3 Universitat Politècnica de València, València, 46022, Spain , València (Spain)
Type
Published Article
Journal
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
Publisher
Springer International Publishing
Publication Date
Nov 15, 2021
Volume
116
Issue
1
Identifiers
DOI: 10.1007/s13398-021-01183-4
Source
Springer Nature
Keywords
Disciplines
  • Original Paper
License
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Abstract

Every element in the boundary of the group of invertibles of a Banach algebra is a topological zero divisor. We extend this result to the scope of topological rings. In particular, we define a new class of semi-normed rings, called almost absolutely semi-normed rings, which strictly includes the class of absolutely semi-valued rings, and prove that every element in the boundary of the group of invertibles of a complete almost absolutely semi-normed ring is a topological zero divisor. To achieve all these, we have to previously entail an exhaustive study of topological divisors of zero in topological rings. In addition, it is also well known that the group of invertibles is open and the inversion map is continuous and C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}$$\end{document}-differentiable in a Banach algebra. We also extend these results to the setting of complete normed rings. Finally, this study allows us to generalize the point, continuous and residual spectra to the scope of Banach algebras.

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