Isometries of the unitary group (Q2884433)
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scientific article; zbMATH DE number 6038844
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Isometries of the unitary group |
scientific article; zbMATH DE number 6038844 |
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Isometries of the unitary group (English)
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29 May 2012
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unitary group
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Hilbert space
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isometry
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Mazur-Ulam theorem
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\(C^{*}\)-algebra
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Thompson metric
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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.
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