An algebraic property of an isometry between the groups of invertible elements in Banach algebras

2009/04/20 by Osamu Hatori, Hatori, Osamu
Mathematics · #46B04 #47B48 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.FA #msc:46B04 #msc:47B48

paper · pdf · doi:10.48550/arxiv.0904.2940

13pages

arxiv created 2009/04/20 · openalex publication_date 2009/04/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T(e_A))^-1T is extended to a surjective real algebra isomorphism; if T is a surjective isometry from the invertible group of a unital commutative Banach algebra onto that of a unital semisimple Banach algebra, then (T(e_A))^-1T is extended to a surjective isometrical real algebra isomorphism between the two underling algebras.

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