An infinite-dimensional index theorem and the Higson-Kasparov-Trout algebra

Abstract

We have been studying the index theory for some special infinite-dimensional manifolds with a "proper cocompact" actions of the loop group LT of the circle T, from the viewpoint of the noncommutative geometry. In this paper, we will introduce the LT-equivariant KK-theory and we will construct three KK-elements: the index element, the Clifford symbol element and the Dirac element. These elements satisfy a certain relation, which should be called the (KK-theoretical) index theorem, or the KK-theoretical Poincaré duality for infinite-dimensional manifolds. We will also discuss the assembly maps.

References in corpus (8)

• Conformal field theory: a case study
• Unbounded bivariant $K$-theory and correspondences in noncommutative geometry
• Dirac operators on quasi-Hamiltonian G-spaces
• A fixed point theorem on noncompact manifolds
• Spinor modules for Hamiltonian loop group spaces
• Covering monopole map and higher degree in non commutative geometry
• Quantization of Hamiltonian loop group spaces
• On C*-algebras and K-theory for infinite-dimensional Fredholm Manifolds