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 is an isometrical group isomorphism. In particular, extends to an isometrical real algebra isomorphism from A onto B.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm194-3-5,
author = {Osamu Hatori},
title = {Isometries between groups of invertible elements in Banach algebras},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {293-304},
zbl = {1190.46014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm194-3-5}
}
Osamu Hatori. Isometries between groups of invertible elements in Banach algebras. Studia Mathematica, Tome 192 (2009) pp. 293-304. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm194-3-5/