LT-equivariant index from the viewpoint of KK-theory

Abstract

Let T be a circle group, and LT be its loop group. We hope to establish an index theory for infinite-dimensional manifolds which LT acts on, including Hamiltonian LT -spaces, from the viewpoint of KK-theory. We have already constructed several objects in the previous paper [Tak], including a Hilbert space H consisting of "L 2

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

“…The index element [ D] is "assemblable". The value of the "total assembly map" coincides with the analytic index computed in [Tak1,Tak2,Tak3].…”

“…The value of our assembly map coincides with the analytic index in [Tak2,Tak3]. In this sense, our construction is "correct".…”

“…It means that, in order to generalize this procedure to infinite-dimensional manifolds, we need some another trick to find some "nice" subspace. In fact, we have used an algebraic technique in [Tak2] to specify the dense subspace.…”

“…As a first step of this dream, we have been studying the index problem for some special infinite-dimensional manifolds in [Tak1,Tak2,Tak3]. [Tak3] is the PhD thesis of the author, which contains [Tak2], most of [Tak1] and some detailed study on proper LT -spaces which is to be explained soon. The following is a rough version of our problem of the present paper.…”

“…However, the following phrase still makes sense: "An LT -equivariant Kasparov module 3 is a Kasparov product of other two Kasparov modules". Then, we will set the problem precisely, and we will divide it into two parts just like [Tak1,Tak2,Tak3]. Consequently, we will find that we may concentrate on one infinite-dimensional manifold U with the standard U -action.…”