Transformations of the unitary group on a Hilbert space
type: Item
instance of: scholarly article
title: Transformations of the unitary group on a Hilbert space
zbMATH Open document ID: 1258.47056
author: Lajos Molnár
author: Peter Šemrl
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-02-27T00:00:00Z
review text: This paper contains nine results on the subject of transformations of spaces of operators, all of them dealing with transformations of the unitary group \(\mathcal U\) of a separable complex and infinite dimensional Hilbert space \(\mathcal H\); two of the results deal with symmetries (self-adjoint unitaries), and due to the correspondence \(p\mapsto \epsilon_p=2p-1\) among symmetries and orthogonal projections, these projections play a significant role in the paper. What follows is a typical result of the paper: let \(\phi:\mathcal U\to \mathcal U\) be bijective map with \[ \phi(VW)=\phi(V)\phi(W)\, \text{for all} \, V,W\in\mathcal U, \] then there exists an unitary or antiunitary operator \(U\) in \(\mathcal H\) such that \(\phi(V)=UVU^*\) (\(V\in\mathcal U\)). No additional hypothesis on the continuity of \(\phi\) is required.
reviewed by: Gabriel Larotonda
Mathematics Subject Classification ID: 47B49
zbMATH DE Number: 6009629
zbMATH Keywords: Hilbert space
zbMATH Keywords: unitary group
zbMATH Keywords: automorphisms
zbMATH Keywords: commutativity preserving maps
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.11.007
OpenAlex ID: W2035244287
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DOI: 10.1016/J.JMAA.2011.11.007
description: scientific article; zbMATH DE number 6009629