Hilbert bundles with ends
Let \(H\) be a Hilbert space equipped with a fixed basis indexed by a countable metric space \(X\), and let \(\mathcal U(H)\) denote the group of unitary operators of finite propagation (with respect to the metric on \(X\)) on \(H\) with the inductive limit topology (with respect to the propagation). A Hilbert bundle with fiber \(H\) is said to \textit{have an end} if its transition functions are continuous maps into \(\mathcal U(H)\). Passing from the group of all unitaries to \(\mathcal U(H)\) allows non-trivial Hilbert bundles. The case considered in the paper is \(X=\mathbb Z\) with the standard metric. In this case, the homotopy type of the classifying space of \(\mathcal U(H)\) is explicitly described. Examples of non-trivial Hilbert bundles are provided, in particular, non-triviality of the Hilbert bundle from spectral decomposition of a family of two-dimensional harmonic oscillators is shown.
scientific article; zbMATH DE number 7870901
scientific article