Isometries between groups of invertible elements in Banach algebras

Volume 194 / 2009

Osamu Hatori 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 $B$.

Authors

  • Osamu HatoriDepartment of Mathematics
    Faculty of Science
    Niigata University
    Niigata 950-2181 Japan
    e-mail

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