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Roe \(C^*\)-algebra for groupoids and generalized Lichnerowicz vanishing theorem for foliated manifolds

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Abstract

We introduce the concept of Roe \(C^*\)-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes’ tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe \(C^*\)-algebra. We apply our theory to derive a Lichnerowicz type vanishing result for foliations on open manifolds.

Keywords

Index Theory Groupoid Roe \(C^*\)-algebras Coarse Structure 

Mathematics Subject Classification

19K56 58J22 46L80 

Notes

Acknowledgements

Tang’s research and Willett’s research are partially supported by the US NSF (DMS1363250, DMS1401126 and DMS1564281). Yao’s research is partially supported by the NSFC (11522107, 11231002 and 11420101001). We would like to thank Jerome Kaminker, Guoliang Yu, and Weiping Zhang for helpful suggestions and encouragement. The first two authors would like to thank the Shanghai Center for Mathematical Sciences for hosting their visits, where the main part of the work was done.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsWashington UniversitySt. LouisUSA
  2. 2.Department of MathematicsUniversity of Hawai’iHonoluluUSA
  3. 3.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China

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