Isometries between groups of invertible elements in Banach algebras

TL;DR

This paper proves that isometries between open subgroups of invertible elements in unital Banach algebras extend to isometric algebra isomorphisms, revealing structural preservation under such mappings.

Contribution

It establishes that isometries between open subgroups of invertible elements extend to full algebra isometries, linking metric and algebraic structures in Banach algebras.

Findings

Isometries induce algebra isomorphisms

Extension of isometries to algebra isomorphisms

Structural preservation in Banach algebra invertible groups

Abstract

We show that if $T$ is an isometry (as metric spaces) from an open subgroup of the group of the invertible elements in a unital semisimple commutative Banach algebra onto an open subgroup of the group of the invertible elements in a unital Banach algebra, then $T(1)^{-1}T$ is an isometrical group isomorphism. In particular, $T(1)^{-1}T$ is extended to an isometrical real algebra isomorphism from $A$ onto $B$.