Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-18T00:16:38.288Z Has data issue: false hasContentIssue false
  • English
  • Français

An amenable equivalence relation is generated by a single transformation

Published online by Cambridge University Press:  19 September 2008

A. Connes ,
J. Feldman  and
B. Weiss
A. Connes
Affiliation:
Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France
J. Feldman
Affiliation:
Department of Mathematics, University of California at Berkeley, California, USA
B. Weiss
Affiliation:
Hebrew University of Jerusalem, Israel
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that for any amenable non-singular countable equivalence relation R⊂X×X, there exists a non-singular transformation T of X such that, up to a null set:

It follows that any two Cartan subalgebras of a hyperfinite factor are conjugate by an automorphism.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 1 , Issue 4 , December 1981 , pp. 431 - 450
Copyright
Copyright © Cambridge University Press 1981

References

REFERENCES

[1]Belinskaya, R. M.. Partitioning of a Lebesgue space into trajectories which are definable by ergodic automorphisms. Funkcional Anal. i. Prilozen 2 (1968), 4–16 (in Russian).Google Scholar
[2]Bowen, R.. Anosov foliations are hyperfinite. Ann. Math. 106 (1977), 549–565.CrossRefGoogle Scholar
[3]Chacon, R. V. & Friedman, N.. Approximation and invariant measures. Z. Wahrscheinlichkeitstheorie verw. Geb. 3 (1965), 286–295.CrossRefGoogle Scholar
[4]Connes, A.. Sur la théorie non commutative de I'intégration. Lecture Notes in Math. No. 725, pp. 19–143. Springer-Verlag: Berlin, 1979.Google Scholar
[5]Connes, A.. Classification of injective factors. Annals of Math. 104 (1976), 73–115.CrossRefGoogle Scholar
[6]Connes, A.. On the cohomology of operator algebras. J. Funct. Analysis 28(2) (1978), 248–253.CrossRefGoogle Scholar
[7]Connes, A. & Krieger, W.. Measure space automorphisms, the normalizer of their full groups and approximate finiteness. J. Funct. Analysis 29 (1977), 336.CrossRefGoogle Scholar
[8]Conze, J. P.. Entropie d'un groupe abélien de transformations. Z. Wahrscheinlichkeitstheorie verw. Geb. 25 (1972), 11–30.CrossRefGoogle Scholar
[9]Dye, H.. On groups of measure preserving transformations. I. Amer. J. Math. 81 (1959), 119–159.CrossRefGoogle Scholar
[10]Dye, H.. On groups of measure preserving transformations. II. Amer. J. Math. 85 (1963), 551–576.CrossRefGoogle Scholar
[11]Feldman, J. & Lind, D.. Hyperfiniteness and the Halmos-Rohlin theorem for non-singular actions. Proc. A.M.S. 55 (1976), 339–344.CrossRefGoogle Scholar
[12]Feldman, J. & Moore, C.. Ergodic equivalence relations, von Neumann algebras and cohomology. I and II. Trans. A.M.S. 234 (1977), 289–324.CrossRefGoogle Scholar
[13]Feldman, J., Moore, C. & Hahn, P.. Orbit structure and countable sections for actions of continuous groups. Adv. Math. 28 (1978), 186–230.CrossRefGoogle Scholar
[14]Golodets, V. Ya.. Crossed products of von Neumann algebras. Uspekhi Math. Nauk. 26(5) (1971), 3–50.Google Scholar
[15]Hopf, E.. Theory of measure and invariant integrals. Trans. A.M.S. 39 (1932), 373–393.CrossRefGoogle Scholar
[16]Tulcea, A. Ionescu. On the category of certain classes of transformations in ergodic theory. Trans. A.M.S. 114 (1965), 261–279.CrossRefGoogle Scholar
[17]Katznelson, Y. & Weiss, B.. Commuting measure preserving transformations. Israel J. Math. 12 (1972), 161–173.CrossRefGoogle Scholar
[18]Krieger, W.. On ergodic flows and the isomorphism of factors. Math. Ann. 223 (1976), 19–70.CrossRefGoogle Scholar
[19]Mackey, G. W.. Ergodic theory and virtual groups. Math. Ann. 166 (1966), 187–207.CrossRefGoogle Scholar
[20]Mokobodski, G.. Limites médiates. Exposé de P. A. Meyer. Sem. Probabilités, Université de Strasbourg. 1971–1972. Lecture Notes in Math. No. 321, pp. 198–204, Springer-Verlag: Berlin, 1973.Google Scholar
[21]Murray, F. J. & Neumann, J. von. On rings of operators. Ann. Math. 37 (1936), 116–229.CrossRefGoogle Scholar
[22]Murray, F. J. & Neumann, J. von. Rings of operators IV. Ann. Math. 44 (1943), 716–804.CrossRefGoogle Scholar
[23]Ornstein, D. & Weiss, B.. The Kakutani–Rohlin theorem for solvable groups. (Preprint, 1976).Google Scholar
[24]Ornstein, D. & Weiss, B.. Ergodic theory of amenable group actions I. The Rohlin lemma. Bull. A.M.S. 2 (1980), 161.CrossRefGoogle Scholar
[25]Ramsay, A.. Measured groupoids and countable equivalence relations. Advances in Math, (to appear).Google Scholar
[26]Schwartz, J.. Two finite, non hyperfinite, non isomorphic factors. Comm. Pure. Appl. Math. 16 (1963), 19–26.CrossRefGoogle Scholar
[27]Series, C.. The Rohlin tower theorem and hyperfiniteness for actions of continuous groups. Israel J. Math. 30 (1978), 99–122.CrossRefGoogle Scholar
[28]Veršik, A. M.. Non measurable decompositions, orbit theory, algebras of operators. Dokl. Acad. Nauk. SSSR 199 (1971), 1004–1007.Google Scholar
[29]Veršik, A. M.. The action of PSL (2, ℤ) on ℝ1 is approximable. Uspekhi Matematicheski Nauk. 33(1) (1978), 209–210 (in Russian).Google Scholar
[30]Zimmer, R.. Hyperfinite factors and amenable ergodic actions. Inv. Math. 41 (1977), 23–31.CrossRefGoogle Scholar
[31]Zimmer, R.. Cocycles and the structure of ergodic group actions. Israel J. Math. 26 (1977), 214–220.CrossRefGoogle Scholar
[32]Zimmer, R.. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Funct. Analysis 27 (1978), 350–372.CrossRefGoogle Scholar