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Compactly supported analytic indices for Lie groupoids

Published online by Cambridge University Press:  12 October 2009

Paulo Carrillo Rouse
Paulo Carrillo Rouse
Affiliation:
Projet d'algèbres d'opérateurs, Université de Paris 7, 175, rue de Chevaleret, Paris, France, carrillo@math.jussieu.fr
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Abstract

For any Lie groupoid we construct an analytic index morphism taking values in a modified K-theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in [CR06]. This allows us in particular to prove a more primitive version of the Connes-Skandalis longitudinal index theorem for foliations, that is, an index theorem taking values in a group which pairs with cyclic cocycles. As another application, for D a -PDO elliptic operator with associated index ind we prove that the pairing

with τ a bounded continuous cyclic cocycle, only depends on the principal symbol class [σ(D)]∈K0. The result is completely general for étale groupoids. We discuss some potential applications to the Novikov conjecture.

Keywords

Lie groupoidsTangent groupoidK-theoryCyclic cohomologyIndex theory
Type
Research Article
Information
Journal of K-Theory , Volume 4 , Issue 2 , October 2009 , pp. 223 - 262
Copyright
Copyright © ISOPP 2009

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