Linear extensions of isometries between groups of invertible elements in Banach algebras

2009/05/07 by Osamu Hatori, Hatori, Osamu
Mathematics · #46B04 #47B48 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B04 #msc:47B48

paper · pdf · doi:10.48550/arxiv.0905.1047

15pages, minor changes, minor changes(2)

arxiv created 2009/05/12 · arxiv updated 2009/12/01

Abstract

We show that if T is an isometry (as metric spaces) from an open subgroup of the invertible group A^-1 of a unital Banach algebra A onto an open subgroup of the invertible group B^-1 of a unital Banach algebra B, then T is extended to a real-linear isometry up to translation between these Banach algebras. We consider multiplicativity or unti-multiplicativity of the isometry. Note that a unital linear isometry between unital semisimple commutative Banach algebra need be multiplicative. On the other hand, we show that if A is commutative and A or B are semisimple, then (T(e_A))^-1T is extended to a isometrical real algebra isomorphism from A onto B. In particular, A^-1 is isometric as a metric space to B^-1 if and only if they are isometrically isomorphic to each other as metrizable groups if and only if A is isometrically isomorphic to B as a real Banach algebra; it is compared by the example of Żelazko concerning on non-isomorphic Banach algebras with homeomorphically isomorphic invertible groups. Maps between standard operator algebras are also investigated.

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