@article{García-Pacheco2024, 
author = {Francisco Javier García-Pacheco and María de los Ángeles Moreno-Frías and Marina Murillo-Arcila},
title = {On absolutely invertibles},
year = {2024},
journal = {Electronic Research Archive},
volume = {32},
number = {12},
pages = {6578-6592},
keywords = {*-ring, absolutely invertible, topological ring, normal subgroup},
url = {https://www.sciopen.com/article/10.3934/era.2024307},
doi = {10.3934/era.2024307},
abstract = {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.}
}