Linear extensions of isometries between groups of invertible elements in Banach algebras

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.