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The Haagerup Property for Discrete Measured Groupoids

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Operator Algebra and Dynamics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 58))

Abstract

We define the Haagerup property in the general context of countable groupoids equipped with a quasi-invariant measure. One of our objectives is to complete an article of Jolissaint devoted to the study of this property for probability measure preserving countable equivalence relations. Our second goal, concerning the general situation, is to provide a definition of this property in purely geometric terms, whereas this notion had been introduced by Ueda in terms of the associated inclusion of von Neumann algebras. Our equivalent definition makes obvious the fact that treeability implies the Haagerup property for such groupoids and that it is not compatible with Kazhdan’s property (T).

Mathematics Subject Classification (2010): 46L55, 37A15, 22A22, 20F65.

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Notes

  1. 1.

    In [8], a countable measured groupoid is called r-discrete. !countable measured !discrete measured Another difference is that we have swapped here the definitions of ν and ν −1.

  2. 2.

    When necessary, we shall write \(\hat{m}\) the element m ∈ M, when viewed in \({L}^{2}(M,\varphi )\), in order to stress this fact.

  3. 3.

    In [25], \(\mathcal{K}(\left \langle M,e_{A}\right \rangle )\) is denoted \(\mathcal{I}_{0}(\left \langle M,e_{A}\right \rangle )\).

  4. 4.

    The reader should not confuse L(f): L 2(G, ν) → L 2(G, ν) with its restriction \(L_{f}: {L}^{2}(A,\tau _{\mu }) \rightarrow {L}^{2}(G,\nu )\).

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Authors and Affiliations

  1. Laboratoire MAPMO – UMR6628, Fédération Denis Poisson (FDP – FR2964), CNRS/Université d’Orléans, 6759, F-45067, Orléans Cedex 2, France

    Claire Anantharaman-Delaroche

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  1. Claire Anantharaman-Delaroche
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Correspondence to Claire Anantharaman-Delaroche .

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Editors and Affiliations

  1. Dept. of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Toke M. Carlsen

  2. Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    Søren Eilers

  3. Dept. of Science and Technology, University of the Faroe Islands, Tórshavn, Faroe Islands

    Gunnar Restorff

  4. Culture and Communication (UKK) Division of Applied Mathematics The School of Education, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

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Anantharaman-Delaroche, C. (2013). The Haagerup Property for Discrete Measured Groupoids. In: Carlsen, T., Eilers, S., Restorff, G., Silvestrov, S. (eds) Operator Algebra and Dynamics. Springer Proceedings in Mathematics & Statistics, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39459-1_1

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