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 \(\mathbb {C}\)-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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Acknowledgements
The authors would like to thank the reviewers for their valuable comments and remarks which have contributed to improve the presentation and quality of the manuscript. The first author has been supported by Research Grant PGC-101514-B-I00 awarded by the Ministry of Science, Innovation and Universities of Spain, and by the 2014-2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia with Project reference: FEDER-UCA18-105867 and Ministerio de Educación y Ciencia (Grant number MTM2016- 75963-P). The second author has been supported by Project PGC2018-094431-B-100 (MICINN. Spain) and Project 8059/2019 (Universitat Jaume I). The third author is supported by MCIN/AEI/10.13039/501100011033, Project PID2019-105011GB-I00, and by Generalitat Valenciana, Project PROMETEU/2021/070.
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García-Pacheco, F.J., Miralles, A. & Murillo-Arcila, M. Invertibles in topological rings: a new approach. RACSAM 116, 38 (2022). https://doi.org/10.1007/s13398-021-01183-4
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DOI: https://doi.org/10.1007/s13398-021-01183-4