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- W2754112846 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 cite{T}, including a Hilbert space $mathcal{H}$ consisting of $L^2$-sections of a Spinor bundle on the infinite-dimensional manifold, an Dirac operator $mathcal{D}$ acting on $mathcal{H}$, 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 $Gamma$-equivariant $K$-homology class $x$, the assembly map is defined by $mu^Gamma(x):=[c]otimes j^Gamma(x)$, where $j^Gamma$ is a $KK$-theoretical homomorphism, $[c]$ is a $K$-theory class coming from a cut-off function, and $otimes$ denotes the Kasparov product with respect to $Gammaltimes C_0(X)$. We will define neither the $LT$-equivariant $K$-homology nor the cut-off function, but we will indeed define the $KK$-cycles $j^{LT}_tau(x)$ and $[c]$ directly, for a virtual $K$-homology class $x=(mathcal{H},mathcal{D})$ which is mentioned above. As a result, we will get the $KK$-theoretical index $mu^{LT}_tau(x)in KK(mathbb{C},LTltimes_tau mathbb{C})$. We will also compare $mu^{LT}_tau(x)$ with the analytic index ${rm ind}_{LTltimes_taumathbb{C}}(x)$ which will be introduced." @default.
- W2754112846 created "2017-09-25" @default.
- W2754112846 creator A5016609251 @default.
- W2754112846 date "2017-09-18" @default.
- W2754112846 modified "2023-09-27" @default.
- W2754112846 title "LT-equivariant Index from the Viewpoint of KK-theory" @default.
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