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

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

2009-04-20

arxiv.org/abs/0904.2940v1

Mathematics

Functional Analysis

13pages

Scientific paper

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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