Higher localized analytic indices and strict deformation quantization

Paulo Carrillo Rouse

A chapter in Deformation Spaces, 2010, pp 91-111 from Springer

Abstract: Abstract This paper is concerned with the localization of higher analytic indices for Lie groupoids. Let G be a Lie groupoid with Lie algebroid AG. Let τ be a (periodic) cyclic cocycle over the convolution algebra $$ C_c^\infty \left( G \right) $$ We say that τ can be localized if there is a morphism $$ K^0 \left( {A^* G} \right)\buildrel {Ind_\tau } \over \longrightarrow C $$ satisfying Ind τ (a)=〈ind D a, τ 〉 (Connes pairing). In this case, we call Ind τ the higher localized index associated to τ. In [CR08a] we use the algebra of functions over the tangent groupoid introduced in [CR08b], which is in fact a strict deformation quantization of the Schwartz algebra S(AG ), to prove the following results: Every bounded continuous cyclic cocycle can be localized. If G 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.

Keywords: Vector Bundle; Elliptic Operator; Hyperbolic Group; Principal Symbol; Analytic Index (search for similar items in EconPapers)
Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-8348-9680-3_4

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DOI: 10.1007/978-3-8348-9680-3_4

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