2008/10/24 by Paulo Carrillo Rouse, Rouse, Paulo Carrillo
Mathematics · #19K56 #53C10 #58J42 #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.AT #math.KT #msc:19K56 #msc:53C10 #msc:58J42
paper · pdf · doi:10.48550/arxiv.0810.4480
arxiv created 2008/10/24 · arxiv updated 2009/12/01
This paper is concerned with the localization of higher analytic indices for Lie groupoids. Let \gr be a Lie groupoid with Lie algebroid A\gr. Let τ be a (periodic) cyclic cocycle over the convolution algebra \cg. We say that τ can be localized if there is a correspondence K0(A^*\gr)\stackrelIndτ\longrightarrowℂ satisfying Indτ(a)=< ind Da,τ> (Connes pairing). In this case, we call Indτ the higher localized index associated to τ. In Ca4 we use the algebra of functions over the tangent groupoid introduced in Ca2, which is in fact a strict deformation quantization of the Schwartz algebra \sw(A\gr), to prove the following results: \item Every bounded continuous cyclic cocycle can be localized. \item If \gr is étale, every cyclic cocycle can be localized. We will recall this results with the difference that in this paper, a formula for higher localized indices will be given in terms of an asymptotic limit of a pairing at the level of the deformation algebra mentioned above. We will discuss how the higher index formulas of Connes-Moscovici, Gorokhovsky-Lott fit in this unifying setting.