Some infinite dimensional groups and bundles (Q1064525)

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.