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Results in Mathematics

, 74:160 | Cite as

The Fredholm Property for Groupoids is a Local Property

  • Rémi CômeEmail author
Article

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 \(i \in I\).

Keywords

Fredholm operator Fredholm groupoid \(C^*\)-algebra pseudodifferential operator primitive spectrum 

Mathematics Subject Classification

58J40 (primary) 58H05 46L05 47L80 

Notes

Acknowledgements

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

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut Élie Cartan de LorraineUniversité de LorraineMetzFrance

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