Isometries between groups of invertible elements in Banach algebras
Volume 194 / 2009
Studia Mathematica 194 (2009), 293-304
MSC: 47B48, 46B04, 54E10.
DOI: 10.4064/sm194-3-5
Abstract
We show that if $T$ is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra $A$ onto a open subgroup of the group of invertible elements in a unital Banach algebra $B$, then $T(1)^{-1}T$ is an isometrical group isomorphism. In particular, $T(1)^{-1}T$ extends to an isometrical real algebra isomorphism from $A$ onto $B$.
