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.


  • 2
    $\begingroup$ 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" (ams.org/mathscinet-getitem?mr=2391387). They also give a number of references at the end of the chapter. $\endgroup$ Apr 15, 2011 at 15:02
  • $\begingroup$ thank you very much for bringing this book to my attention! $\endgroup$
    – user5831
    Apr 18, 2011 at 13:56

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.