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I'm interested in the following kind of questions about groupoid $C^*$-algebras.

1) If $G_1 \times_{H} \ G_2$ is a fibre product of (nice) groupoids do we have something like $$C^\star(G_1 \times_{H} \ G_2) \cong C^*(G_1) \otimes_{C^\star(H)} C^*(G_2) ?$$ 2) Of course, in general, there is an ambiguity about the above tensor product. So what is a good notion of amenability for groupoids? (In the sense that the groupoid $C^*$ algebra of an amenable groupoid is nuclear.)

Apart from Renault's classic about groupoid $C^*$-algebras I do not really know any other reference for this subject.


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Amenability of groupoids and nuclearity of their $C^*$ algebras is discussed in Section 5.6 of N. Brown, and N. Ozawa: "$C^*$-algebras and finite-dimensional approximations" ( They also give a number of references at the end of the chapter. – Jesse Peterson Apr 15 '11 at 15:02
thank you very much for bringing this book to my attention! – user5831 Apr 18 '11 at 13:56

I'll post this, although you probably already know about it:

C. Anantharaman-Delaroche et J. Renault, Amenable groupoids (avec un appendice par E. Germain), Monographie de l'Enseignement Mathématique (Genève), 36, 2000.

I hope this is helpful. Note: I'm not claiming that the amenability discussed here is the sort that you need. I only answer since you claim to know no reference for amenability of groupoids.

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thank you very much for the reference! – user5831 Apr 18 '11 at 13:56

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