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The Fredholm Property for Groupoids is a Local Property

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Abstract

Fredholm Lie groupoids were introduced by Carvalho, Nistor and Qiao as a tool for the study of partial differential equations on open manifolds. This article extends the definition to the setting of locally compact groupoids and proves that “the Fredholm property is local”. Let \({\mathcal {G}}\rightrightarrows X\) be a topological groupoid and \((U_i)_{i\in I}\) be an open cover of X. We show that \({\mathcal {G}}\) is a Fredholm groupoid if, and only if, its reductions \({\mathcal {G}}^{U_i}_{U_i}\) are Fredholm groupoids for all \(i \in I\). We exploit this criterion to show that many groupoids encountered in practical applications are Fredholm. As an important intermediate result, we use an induction argument to show that the primitive spectrum of \(C^*({\mathcal {G}})\) can be written as the union of the primitive spectra of all \(C^*({\mathcal {G}}^{U_i}_{U_i})\), for \(i \in I\).

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Notes

  1. There is an alternate definition of \(C_c({\mathcal {G}})\) that is due to Crainic [11]. Connes’ algebra is smaller: it is a quotient of Crainic’s algebra. Note that both algebras separate points of \({\mathcal {G}}\).

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Acknowledgements

The author would like to thank Victor Nistor for useful discussions and suggestions.

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  1. Institut Élie Cartan de Lorraine, Université de Lorraine, 3 rue Augustin Fresnel, 57000, Metz, France

    Rémi Côme

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Côme, R. The Fredholm Property for Groupoids is a Local Property. Results Math 74, 160 (2019). https://doi.org/10.1007/s00025-019-1084-x

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