Isometries of the unitary group
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.
scientific article; zbMATH DE number 6038844
scientific article