Hilbert bundles with ends
type: Item
instance of: scholarly article
title: Hilbert bundles with ends
author: Tsuyoshi Kato
author: Daisuke Kishimoto
author: Mitsunobu Tsutaya
published in: Journal of Topology and Analysis
publication date: 2024-06-25T00:00:00Z
review text: 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.
reviewed by: Vladimir M. Manuilov
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46C05
Mathematics Subject Classification ID: 34K08
Mathematics Subject Classification ID: 57N20
zbMATH DE Number: 7870901
zbMATH Keywords: Hilbert bundle
zbMATH Keywords: end
zbMATH Keywords: unitary operator of finite propagation
zbMATH Keywords: uniform Roe algebra
zbMATH Keywords: pushforward of a vector bundle
zbMATH Keywords: spectral decomposition
zbMATH Keywords: Fourier transform
zbMATH Keywords: harmonic oscillator
MaRDI profile type: Publication
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DOI: 10.1142/S1793525321500680
description: scientific article; zbMATH DE number 7870901