An algebraic property of an isometry between the groups of invertible elements in Banach algebras

TL;DR

This paper demonstrates that isometries between the invertible groups of unital Banach algebras can be extended to real-linear isometries of the entire algebras, revealing algebraic structure preservation.

Contribution

It establishes that such isometries are extendable to real-linear isometries or isomorphisms of the full Banach algebras, depending on algebra types.

Findings

Isometries extend to real-linear isometries of Banach algebras.

In standard operator algebras, extensions are surjective real algebra isomorphisms.

For commutative Banach algebras, isometries extend to real algebra isomorphisms.

Abstract

We show that if $T$ is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then $T$ is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, $(T(e_A))^{-1}T$ is extended to a surjective real algebra isomorphism; if $T$ is a surjective isometry from the invertible group of a unital commutative Banach algebra onto that of a unital semisimple Banach algebra, then $(T(e_A))^{-1}T$ is extended to a surjective isometrical real algebra isomorphism between the two underling algebras.