Isometries between groups of invertible elements in Banach algebras

Osamu Hatori

Isometries between groups of invertible elements in Banach algebras

Osamu Hatori

Studia Mathematica (2009)

Volume: 194, Issue: 3, page 293-304
ISSN: 0039-3223

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Abstract

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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.

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Osamu Hatori. "Isometries between groups of invertible elements in Banach algebras." Studia Mathematica 194.3 (2009): 293-304. <http://eudml.org/doc/284709>.

@article{OsamuHatori2009,
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.},
author = {Osamu Hatori},
journal = {Studia Mathematica},
keywords = {commutative Banach algebras; group of invertible elements; isometrical real algebra isomorphism},
language = {eng},
number = {3},
pages = {293-304},
title = {Isometries between groups of invertible elements in Banach algebras},
url = {http://eudml.org/doc/284709},
volume = {194},
year = {2009},
}

TY - JOUR
AU - Osamu Hatori
TI - Isometries between groups of invertible elements in Banach algebras
JO - Studia Mathematica
PY - 2009
VL - 194
IS - 3
SP - 293
EP - 304
AB - 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.
LA - eng
KW - commutative Banach algebras; group of invertible elements; isometrical real algebra isomorphism
UR - http://eudml.org/doc/284709
ER -

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