Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 9 (2014), 55 -- 78


Mădălina Roxana Buneci

Abstract. We start with a groupoid G endowed with a family W of subsets mimicking the properties of a neighborhood basis of the unit space (of a topological groupoid with paracompact unit space). Using the family W we endow each G-space with a uniform structure. The uniformities of the G-spaces allow us to define various notions of amenability for the G-equivariant maps. As in [1], the amenability of the groupoid G is defined as the amenability of its range map. If the groupoid G is a group, all notions of amenability that we introduce coincide with the classical notion of amenability for topological (not necessarily locally-compact) groups.

2010 Mathematics Subject Classification: 22A22; 43A07; 54E15.
Keywords: groupoid; uniform structure; equivariant map; amenability.

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  1. C. Anantharaman-Delaroche and J. Renault, Amenable Groupoids. Monographie de L'Enseignement Mathematique No 36, Geneve, 2000. MR1799683(2001m:22005). Zbl 0960.43003.

  2. W. Arveson, An invitation to C* -algebras, Springer-Verlag, New York, 1976. MR0512360(58 #23621). Zbl 0344.46123.

  3. M. Buneci, Haar systems and homomorphism on groupoids, Operator algebras and mathematical physics (Constanta, 2001), 35--49, Theta, Bucharest, 2003. MR2018222(2004j:22006). Zbl 1284.22002.

  4. M. Buneci, Groupoid C* -algebras, Surveys in Mathematics and its Applications, 1 (2006), 71 - 98.. MR2274294(2007h:46067). Zbl 1130.22002.

  5. A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory and Dynamical Systems 1 (1981), 431--450. MR0662736(84h:46090). Zbl 0491.28018.

  6. P. Hahn, Haar measure for measure groupoids, Trans. Amer. Math. Soc. 242 (1978), 1-33. MR0496796(82a:28012). Zbl 0343.43003.

  7. S. Jackson, A. S. Kechris and A. Louveau, Countable Borel equivalence relations, Journal of Mathematical Logic, 2 (1) (2002), 1--80. MR1900547(2003f:03066). Zbl 1008.03031.

  8. A. S. Kechris, Amenable versus hyperfinite Borel equivalence relations, J. Symb. Logic, 58 (3) (1993), 894--304. MR1242044(95f:03081). Zbl 0795.03067.

  9. A. Kechris and B. Miller, Means on equivalence relations, Isr. J. Math. 163 (2008), 241-262. MR2391131(2009d:54061). Zbl 1140.37003.

  10. J. Kelley, General Topology, Graduate Texts in Mathematics 27, Springer, 1975. MR0370454(51 #6681). Zbl 0306.54002.

  11. G. Mackey, Ergodic theory, group theory and differentiall geometry, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 1184-1191. MR0165034(29 #2325). Zbl 0178.38801.

  12. P. A. Meyer, Limites médiales, d'après Mokobodzki, Séminaire de Probabilités, VII (Univ. Strasbourg, 1971–1972), 198–204. Lecture Notes in Math., Vol. 321, Springer, Berlin, 1973.

  13. A. Ramsay, Topologies on measured groupoids, J. Funct. Anal. 47 (1982), 314-343. MR0665021(83k:22014). Zbl 0519.22003. MR0404564(53 #8364). Zbl 0262.28005.

  14. A. Ramsay, The Mackey-Glimm dichotomy for foliations and other Polish groupoids, J. Funct. Anal. 94(1990), 358-374. MR1081649(93a:46124). Zbl 0717.57016.

  15. J. Renault, A groupoid approach to C* - algebras, Lecture Notes in Math., Springer-Verlag, 793, 1980. MR0584266(82h:46075). Zbl 0433.46049.

  16. J. Renault, Topological amenability is a Borel property, to appear Math. Scand., arXiv:1302.0636.

  17. V. Pestov, Amenability for non-locally compact topological groups, BOAS 2011 (Brazilian Operator Algebras Symposium, 31 January --- 4 February, 2011), Electronic Conference Proceedings, 2011.

  18. R. J. Zimmer, Hyperfinite factors and amenable ergodic actions, Inv. Math., 41 (1977), 23--31. MR0470692(57 #10438). Zbl 0361.46061.

Mădălina Buneci
University Constantin Brâncuşi,
Str. Geneva, Nr. 3, 210136 Târgu-Jiu, Romania.