LT-equivariant Index from the Viewpoint of KK-theory

Original · EN

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 T, including a Hilbert space H consisting of "L²-sections of a Spinor bundle on the infinite-dimensional manifold", an "LT-equivariant Dirac operator D" acting on H, a "twisted crossed product of the function algebra by LT", and the "twisted group C*-algebra of LT", without the measure on the manifolds, the measure on LT or the function algebra itself. However, we need more sophisticated constructions. In this paper, we study the index problem in terms of KK-theory. Concretely, we focus on the infinite-dimensional version of the latter half of the assembly map defined by Kasparov. Generally speaking, for a Γ-equivariant K-homology class x, the assembly map is defined by μΓ(x):=[c]⊗ jΓ(x), where jΓ is a KK-theoretical homomorphism, [c] is a K-theory class coming from a cut-off function, and ⊗ denotes the Kasparov product with respect to Γ C₀(X). We will define neither the LT-equivariant K-homology nor the cut-off function, but we will indeed define the KK-cycles jLTτ(x) and [c] directly, for a virtual K-homology class x=(H,D) which is mentioned above. As a result, we will get the KK-theoretical index μLTτ(x)∈ KK(C,LTτC). We will also compare μLTτ(x) with the analytic index indLTτC(x) which will be introduced.