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Quasi-Locality for étale Groupoids

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Abstract

Let \(\mathcal {G}\) be a locally compact étale groupoid and \(\mathscr {L}(L^2(\mathcal {G}))\) be the \(C^*\)-algebra of adjointable operators on the Hilbert \(C^*\)-module \(L^2(\mathcal {G})\). In this paper, we discover a notion called quasi-locality for operators in \(\mathscr {L}(L^2(\mathcal {G}))\), generalising the metric space case introduced by Roe. Our main result shows that when \(\mathcal {G}\) is additionally \(\sigma \)-compact and amenable, an equivariant operator in \(\mathscr {L}(L^2(\mathcal {G}))\) belongs to the reduced groupoid \(C^*\)-algebra \(C^*_r(\mathcal {G})\) if and only if it is quasi-local. This provides a practical approach to describe elements in \(C^*_r(\mathcal {G})\) using coarse geometry. Our main tool is a description for operators in \(\mathscr {L}(L^2(\mathcal {G}))\) via their slices with the same philosophy to the computer tomography. As applications, we recover a result by Špakula and the second-named author in the metric space case, and deduce new characterisations for reduced crossed products and uniform Roe algebras for groupoids.

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Notes

  1. As we shall see in Example 3.25, \(\mathcal {A}\) is smaller than \(\mathscr {B}(\ell ^2(X))\). However this does not cause any trouble when considering quasi-local operators (see Example 4.9).

  2. More precisely, Anantharaman-Delaroche [1] defined the uniform Roe algebra \(C^*_u(\mathcal {G})\) in a different way, while we show in Lemma 7.4 that it coincides with our definition.

  3. Warning: Note that our definition of \(\mathcal {G}\)-equivariant operators is not the same as the standard one considered in the literature, e.g. [23, Definition 4.6].

  4. We remark that these metrics have also been considered in [28] to study amenability for certain groupoids, using the language of length functions on groupoids.

  5. Note that the formula given in [1] is slightly different from the above, where the set \(\mathcal {G}*_{\textrm{r}} \mathcal {G}\) is considered instead of \(\mathcal {G}*_{\textrm{s}} \mathcal {G}\). Here we follow the formula from [4, Section 6.5] which is more convenient and compatible to our setting.

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Acknowledgements

We appreciate the anonymous referee for a number of valuable suggestions to improve the paper, especially for suggesting us to use the notion of interior tensor product of Hilbert \(C^*\)-modules (see Remark 3.4 and 7.5).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Baojie Jiang

  2. School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, 200433, China

    Jiawen Zhang

  3. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, China

    Jianguo Zhang

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Correspondence to Jiawen Zhang.

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Communicated by Y. Kawahigashi

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Baojie Jiang was supported by NSFC12001066 and NSFC12071183. Jiawen Zhang was supported by NSFC11871342. Jianguo Zhang was supported by NSFC12171156, 12271165. Jiawen Zhang is the corresponding author.

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Jiang, B., Zhang, J. & Zhang, J. Quasi-Locality for étale Groupoids. Commun. Math. Phys. 403, 329–379 (2023). https://doi.org/10.1007/s00220-023-04782-x

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