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arXiv · 0904.2940

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

Osamu Hatori
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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, $(T(e_A))^{-1}T$ 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 $(T(e_A))^{-1}T$ is extended to a surjective isometrical real algebra isomorphism between the two underling algebras.

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Osamu Hatori. 2009-04-20. An algebraic property of an isometry between the groups of invertible elements in Banach algebras. https://arxiv.org/abs/0904.2940

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