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Advances in Noncommutative Geometry pp 245–289Cite as

Lie groupoids, pseudodifferential calculus, and index theory

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Abstract

Alain Connes introduced the use of Lie groupoids in noncommutative geometry in his pioneering work on the index theory of foliations. In the present paper, we recall the basic notion involved: groupoids, their C ∗-algebras, their pseudodifferential calculus, etc. We review several recent and older advances on the involvement of Lie groupoids in noncommutative geometry. We then propose some open questions and possible developments of the subject.

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Notes

  1. 1.

    We may just assume that f satisfies the transversality condition.

  2. 2.

    In [61] this element is just a morphism of K-groups since E-theory of Connes–Higson was defined later.

  3. 3.

    The arrows are in fact E-theory elements or even KK-theory elements.

  4. 4.

    We actually have to assume that the same holds also for the adjoint of Φ.

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Acknowledgements

The authors were partially supported by ANR-14-CE25-0012-01 (SINGSTAR).

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  1. UPMC Paris 06 CNRS, IMJ-PRG UFR de Mathématiques, Université Paris Diderot, Sorbonne Paris Cité Sorbonne Universités, Paris, France

    Claire Debord & Georges Skandalis

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  1. Claire Debord
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Editors and Affiliations

  1. Department of Physics Institute for Mathematics, Astrophysics and Particle Physics, Nijmegen, The Netherlands

    Ali Chamseddine

  2. Department of Mathematics, Johns Hopkins University, Baltimore, MD, USA

    Caterina Consani

  3. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Nigel Higson

  4. Department of Mathematics, The University of Western Ontario, London, ON, Canada

    Masoud Khalkhali

  5. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Henri Moscovici

  6. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Guoliang Yu

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Debord, C., Skandalis, G. (2019). Lie groupoids, pseudodifferential calculus, and index theory. In: Chamseddine, A., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds) Advances in Noncommutative Geometry. Springer, Cham. https://doi.org/10.1007/978-3-030-29597-4_4

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