Amenable actions of groups
Authors:
Scot Adams, George A. Elliott and Thierry Giordano
Journal:
Trans. Amer. Math. Soc. 344 (1994), 803822
MSC:
Primary 22D99; Secondary 22D40, 28D15
MathSciNet review:
1250814
Fulltext PDF Free Access
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References 
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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.
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 [A]
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 [AS]
 S. Adams and G. Stuck, Splitting of nonnegatively curved leaves in minimal sets of foliations, Duke Math. J. 71 (1993), 7192. MR 1230286 (95e:53045)
 [C]
 A. Connes, On hyperfinite factors of type and Krieger's factors, J. Funct. Anal. 18 (1975), 318327. 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), 431450. 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), 225243. 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), 186230. 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), 289324. 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), 12961299; transl. in Soviet Math. Dokl. 41 (1990), 523526. MR 1076484 (92c:22014)
 [GS1]
 , Outer conjugacy for actions of continuous amenable groups, Publ. Inst. Res. Math Sci. 23 (1987), 737769. 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), 225241. 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), 327335. 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., SpringerVerlag, 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), 350372. MR 0473096 (57:12775)
 [Z2]
 , Hyperfinite factors and amenable ergodic actions, Invent. Math. 41 (1977), 2331. MR 0470692 (57:10438)
 [Z3]
 , On the von Neumann algebra of an ergodic group action, Proc. Amer. Math. Soc. 41 (1977), 2331. MR 0460599 (57:592)
 [Z4]
 , Ergodic theory and semisimple groups, Birkhäuser, Boston, MA, 1984.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412508145
PII:
S 00029947(1994)12508145
Article copyright:
© Copyright 1994
American Mathematical Society
