On absolutely invertibles

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.

Keywords

• *-ring
• absolutely invertible
• topological ring
• normal subgroup