On absolutely invertibles
In this manuscript, the notion of absolutely invertible was extended consistently from semi-normed rings to the class of general topological rings. Then, the closure of the absolutely invertibles multiplied by a certain element was proved to be contained in the set of topological divisors of the ele...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-12-01
|
| Series: | Electronic Research Archive |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024307 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832590750369447936 |
|---|---|
| author | Francisco Javier García-Pacheco María de los Ángeles Moreno-Frías Marina Murillo-Arcila |
| author_facet | Francisco Javier García-Pacheco María de los Ángeles Moreno-Frías Marina Murillo-Arcila |
| author_sort | Francisco Javier García-Pacheco |
| collection | DOAJ |
| description | In this manuscript, the notion of absolutely invertible was extended consistently from semi-normed rings to the class of general topological rings. Then, the closure of the absolutely invertibles multiplied by a certain element was proved to be contained in the set of topological divisors of the element. Also, a sufficient condition for the closed unit ball of a complete unital normed ring to become a closed unit neighborhood of zero was found. Finally, two applications to classical operator theory were provided, i.e., every Banach space of dimension of at least $ 2 $ could be equivalently re-normed in such a way that the group of surjective linear isometries was not a normal subgroup of the group of isomorphisms, and every infinite-dimensional Banach space, containing a proper complemented subspace isomorphic to it, could be equivalently re-normed so that the set of surjective linear operators was not dense in the Banach algebra of bounded linear operators. |
| format | Article |
| id | doaj-art-6ff28e2c0250475bb89caaa6c952101c |
| institution | Kabale University |
| issn | 2688-1594 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj-art-6ff28e2c0250475bb89caaa6c952101c2025-01-23T07:53:06ZengAIMS PressElectronic Research Archive2688-15942024-12-0132126578659210.3934/era.2024307On absolutely invertiblesFrancisco Javier García-Pacheco0María de los Ángeles Moreno-Frías1Marina Murillo-Arcila2Department of Mathematics, University of Cadiz, Puerto Real 11519, SpainDepartment of Mathematics, University of Cadiz, Puerto Real 11519, SpainDepartment of Mathematics, University of Cadiz, Puerto Real 11519, SpainIn this manuscript, the notion of absolutely invertible was extended consistently from semi-normed rings to the class of general topological rings. Then, the closure of the absolutely invertibles multiplied by a certain element was proved to be contained in the set of topological divisors of the element. Also, a sufficient condition for the closed unit ball of a complete unital normed ring to become a closed unit neighborhood of zero was found. Finally, two applications to classical operator theory were provided, i.e., every Banach space of dimension of at least $ 2 $ could be equivalently re-normed in such a way that the group of surjective linear isometries was not a normal subgroup of the group of isomorphisms, and every infinite-dimensional Banach space, containing a proper complemented subspace isomorphic to it, could be equivalently re-normed so that the set of surjective linear operators was not dense in the Banach algebra of bounded linear operators.https://www.aimspress.com/article/doi/10.3934/era.2024307*-ringabsolutely invertibletopological ringnormal subgroup |
| spellingShingle | Francisco Javier García-Pacheco María de los Ángeles Moreno-Frías Marina Murillo-Arcila On absolutely invertibles Electronic Research Archive *-ring absolutely invertible topological ring normal subgroup |
| title | On absolutely invertibles |
| title_full | On absolutely invertibles |
| title_fullStr | On absolutely invertibles |
| title_full_unstemmed | On absolutely invertibles |
| title_short | On absolutely invertibles |
| title_sort | on absolutely invertibles |
| topic | *-ring absolutely invertible topological ring normal subgroup |
| url | https://www.aimspress.com/article/doi/10.3934/era.2024307 |
| work_keys_str_mv | AT franciscojaviergarciapacheco onabsolutelyinvertibles AT mariadelosangelesmorenofrias onabsolutelyinvertibles AT marinamurilloarcila onabsolutelyinvertibles |