Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Amenable actions of groups

Authors: Scot Adams, George A. Elliott and Thierry Giordano
Journal: Trans. Amer. Math. Soc. 344 (1994), 803-822
MSC: Primary 22D99; Secondary 22D40, 28D15
MathSciNet review: 1250814
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The equivalence between different characterizations of amenable actions of a locally compact group is proved. In particular, this answers a question raised by R. J. Zimmer in 1977.

References [Enhancements On Off] (What's this?)

  • [A] W. Arveson, Invitation to $ C \ast $-algebras, Springer-Verlag, Berlin, 1976. MR 0512360 (58:23621)
  • [AS] S. Adams and G. Stuck, Splitting of non-negatively curved leaves in minimal sets of foliations, Duke Math. J. 71 (1993), 71-92. MR 1230286 (95e:53045)
  • [C] A. Connes, On hyperfinite factors of type $ {\text{III}}_{0}$ and Krieger's factors, J. Funct. Anal. 18 (1975), 318-327. MR 0372635 (51:8842)
  • [CFW] A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynamical Systems 1 (1981), 431-450. MR 662736 (84h:46090)
  • [CW] A. Connes and E. J. Woods, Hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks, Pacific J. Math. 137 (1989), 225-243. MR 990212 (90h:46100)
  • [EG1] G. A. Elliott and T. Giordano, Amenable actions of discrete groups, Ergodic Theory Dynamical Systems (to appear). MR 1235474 (94i:22023)
  • [FHM] J. Feldman, P. Hahn and C. Moore, Orbit structure and countable sections for actions of continuous groups, Adv. in Math. 28 (1978), 186-230. MR 0492061 (58:11217)
  • [FM] J. Feldman and C.C. Moore, Ergodic equivalence relations, cohomology and von Neumann algebras I, Trans. Amer. Math. Soc. 234 (1977), 289-324. MR 0578656 (58:28261a)
  • [F] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, NJ, 1981. MR 603625 (82j:28010)
  • [GS] V.Ya. Golodets and S.D. Sinel'shchikov, Amenable ergodic group actions and images of cocycles, Dokl. Akad. Nauk SSSR 312 (1990), 1296-1299; transl. in Soviet Math. Dokl. 41 (1990), 523-526. MR 1076484 (92c:22014)
  • [GS1] -, Outer conjugacy for actions of continuous amenable groups, Publ. Inst. Res. Math Sci. 23 (1987), 737-769. MR 934670 (89c:46087)
  • [GS2] -, Classification and the structure of cocycles of amenable ergodic equivalence relations, preprint.
  • [J] W. Jaworski, Poisson and Furstenberg boundaries of random walks, Ph.D. thesis, Queen's, University at Kingston, 1991. MR 1145123
  • [Ka] R. Kallman, Certain quotient spaces are countably separated, III, J. Funct. Anal. 22 (1976), 225-241. MR 0417329 (54:5385)
  • [Ke] A. Kechris, Countable sections for locally compact group actions, preprint. MR 1176624 (94b:22003)
  • [M] G. Mackey, Point realizations of transformations groups, Illinois J. Math 6 (1962), 327-335. MR 0143874 (26:1424)
  • [S] C.E. Sutherland, Preliminary report on Bratteli diagrams, private communication.
  • [V] V.S. Varadarajan, Geometry of quantum theory, 2nd ed., Springer-Verlag, Berlin, 1985. MR 805158 (87a:81009)
  • [Z1] R.J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Funct. Anal. 27 (1978), 350-372. MR 0473096 (57:12775)
  • [Z2] -, Hyperfinite factors and amenable ergodic actions, Invent. Math. 41 (1977), 23-31. MR 0470692 (57:10438)
  • [Z3] -, On the von Neumann algebra of an ergodic group action, Proc. Amer. Math. Soc. 41 (1977), 23-31. MR 0460599 (57:592)
  • [Z4] -, Ergodic theory and semisimple groups, Birkhäuser, Boston, MA, 1984.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22D99, 22D40, 28D15

Retrieve articles in all journals with MSC: 22D99, 22D40, 28D15

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society