Some Infinite Dimensional Groups and Bundles

Abstract

The group \mathcal{O} of Bogoliubov automorphisms of the infinite dimensional Clifford albegra, implementable in a Fock representation, the analogous group of automorphisms of the canonical commutation relations and various generalisations are discussed. Their homotopy type is determined in a topology naturally defined by the spin and metaplectic representations. A theorem of Araki and Evans on a Z₂-index for certain projections is generalised using our “mod 2” index for \mathcal{O}. Connections with K₁ of certain Banach algebras are described.