Isometries of the unitary group
type: Item
instance of: scholarly article
title: Isometries of the unitary group
zbMATH Open document ID: 1244.47032
DOI: 10.1090/S0002-9939-2011-11216-4
author: Osamu Hatori
author: Lajos Molnár
published in: Proceedings of the American Mathematical Society
publication date: 2012-05-29T00:00:00Z
Mathematics Subject Classification ID: 47B49
zbMATH DE Number: 6038844
zbMATH Keywords: unitary group
zbMATH Keywords: Hilbert space
zbMATH Keywords: isometry
zbMATH Keywords: Mazur-Ulam theorem
zbMATH Keywords: \(C^{*}\)-algebra
zbMATH Keywords: Thompson metric
MaRDI profile type: Publication
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full work available at URL: s0002-9939-2011-11216-4
OpenAlex ID: W1970008465
review text: Let \(H\) be a complex Hilbert space and let \(U(H)\) be the group of all unitary operators on \(H\), equipped with the metric that comes from the operator norm. The main result states that if \(\phi:U(H)\to U(H)\) is a surjective isometry, then there exist \(V,W\in U(H)\) such that \(\phi\) takes one of the following forms: (1) \(\phi(A)= VAW\), (2) \(\phi(A)= VA^*W\), (3) \(\phi(A)= VA^TW\), (4) \(\phi(A)= VA^{*^T}W\). A result on Thompson isometries of the space of all invertible positive elements of a \(C^*\)-algebra is also established.
reviewed by: Matej Brešar
description: scientific article; zbMATH DE number 6038844