Some infinite dimensional groups and bundles
type: Item
instance of: scholarly article
title: Some infinite dimensional groups and bundles
zbMATH Open document ID: 0575.46059
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: The group \({\mathcal O}\) of Bogoliubov automorphisms of the infinite dimensional Clifford algebra, implementable in a Fock representation, the analogous group of automorphisms of the canonical commutation relations and various generalisations thereof (obtained by replacing the Hilbert- Schmidt condition which enters their definition by a condition involving a general symmetric ideal of compact operators) are discussed. Their homotopy type is determined in the topology naturally defined by the spin and metaplectic representations. In particular the existence of a homomorphism of \({\mathcal O}\) onto \({\mathbb{Z}}/2{\mathbb{Z}}\) which separates connected components leads to a generalisation of a theorem of Araki and Evans on a \({\mathbb{Z}}/2{\mathbb{Z}}\) index for projections differing by a Hilbert-Schmidt operator. The index maps of this paper are interpreted in terms of K-theory of certain Banach algebras, the latter leading to a connection with Connes' cyclic cohomology.
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 46L60
zbMATH DE Number: 3919117
zbMATH Keywords: group of Bogoliubov automorphisms of the infinite dimensional Clifford algebra
zbMATH Keywords: Fock representation
zbMATH Keywords: canonical commutation relations
zbMATH Keywords: homotopy type
zbMATH Keywords: spin
zbMATH Keywords: metaplectic representations
zbMATH Keywords: index maps
zbMATH Keywords: K-theory of certain Banach algebras
zbMATH Keywords: Connes' cyclic cohomology
author: A. L. Carey
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195180373
OpenAlex ID: W2129191727
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DOI: 10.2977/PRIMS/1195180373
description: scientific article; zbMATH DE number 3919117