Linear extensions of 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 real-linear isometries of the entire algebras, revealing deep structural links between the invertible groups and the algebras themselves.

Contribution

It establishes the extension of isometries from invertible subgroups to the whole Banach algebras and characterizes when invertible groups are isometrically isomorphic to the algebras.

Findings

Isometries extend to real-linear isometries of Banach algebras.

In commutative semisimple cases, isometries induce algebra isomorphisms.

Invertible groups determine the Banach algebra up to isometric isomorphism.

Abstract

We show that if $T$ is an isometry (as metric spaces) from an open subgroup of the invertible group $A^{-1}$ of a unital Banach algebra $A$ onto an open subgroup of the invertible group $B^{-1}$ of a unital Banach algebra $B$, then $T$ is extended to a real-linear isometry up to translation between these Banach algebras. We consider multiplicativity or unti-multiplicativity of the isometry. Note that a unital linear isometry between unital semisimple commutative Banach algebra need be multiplicative. On the other hand, we show that if $A$ is commutative and $A$ or $B$ are semisimple, then $(T(e_A))^{-1}T$ is extended to a isometrical real algebra isomorphism from $A$ onto $B$. In particular, $A^{-1}$ is isometric as a metric space to $B^{-1}$ if and only if they are isometrically isomorphic to each other as metrizable groups if and only if $A$ is isometrically isomorphic to $B$ as a real Banach algebra; it is compared by the example of \.Zelazko concerning on non-isomorphic Banach algebras with homeomorphically isomorphic invertible groups. Maps between standard operator algebras are also investigated.