On measure theoretical analogues of the Takesaki structure theorem for type III factors

Alexandre Danilenko; Toshihiro Hamachi

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 2, page 485-493
  • ISSN: 0010-1354

Abstract

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The orbit equivalence of type I I I 0 ergodic equivalence relations is considered. We show that it is equivalent to the outer conjugacy problem for the natural trace-scaling action of a countable dense ℝ-subgroup by automorphisms of the Radon-Nikodym skew product extensions of these relations. A similar result holds for the weak equivalence of arbitrary type I I I 0 cocycles with values in Abelian groups.

How to cite

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Danilenko, Alexandre, and Hamachi, Toshihiro. "On measure theoretical analogues of the Takesaki structure theorem for type III factors." Colloquium Mathematicae 84/85.2 (2000): 485-493. <http://eudml.org/doc/210828>.

@article{Danilenko2000,
abstract = {The orbit equivalence of type $III_\{0\}$ ergodic equivalence relations is considered. We show that it is equivalent to the outer conjugacy problem for the natural trace-scaling action of a countable dense ℝ-subgroup by automorphisms of the Radon-Nikodym skew product extensions of these relations. A similar result holds for the weak equivalence of arbitrary type $III_\{0\}$ cocycles with values in Abelian groups.},
author = {Danilenko, Alexandre, Hamachi, Toshihiro},
journal = {Colloquium Mathematicae},
keywords = {orbit equivalence; outer conjugacy; ergodic countable transformation group; ergodic discrete equivalence relation; cocycle},
language = {eng},
number = {2},
pages = {485-493},
title = {On measure theoretical analogues of the Takesaki structure theorem for type III factors},
url = {http://eudml.org/doc/210828},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Danilenko, Alexandre
AU - Hamachi, Toshihiro
TI - On measure theoretical analogues of the Takesaki structure theorem for type III factors
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 2
SP - 485
EP - 493
AB - The orbit equivalence of type $III_{0}$ ergodic equivalence relations is considered. We show that it is equivalent to the outer conjugacy problem for the natural trace-scaling action of a countable dense ℝ-subgroup by automorphisms of the Radon-Nikodym skew product extensions of these relations. A similar result holds for the weak equivalence of arbitrary type $III_{0}$ cocycles with values in Abelian groups.
LA - eng
KW - orbit equivalence; outer conjugacy; ergodic countable transformation group; ergodic discrete equivalence relation; cocycle
UR - http://eudml.org/doc/210828
ER -

References

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  1. [BG] S. I. Bezuglyi and V. Ya. Golodets, Weak equivalence and the structure of cocycles of an ergodic automorphism, Publ. RIMS Kyoto Univ. 27 (1991), 577-635. 
  2. [Da1] A. I. Danilenko, The topological structure of Polish groups and groupoids of measure space transformations, ibid. 31 (1995), 913-940. Zbl0851.22001
  3. [Da2] A. I. Danilenko, Quasinormal subrelations of ergodic equivalence relations, Proc. Amer. Math. Soc. 126 (1998), 3361-3370. Zbl0917.28019
  4. [FHM] J. Feldman, P. Hahn and C. C. Moore, Orbit structure and countable sections for actions of continuous groups, Adv. Math. 28 (1978), 186-230. Zbl0392.28023
  5. [FM] J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras I, Trans. Amer. Math. Soc. 234 (1977), 289-324. Zbl0369.22009
  6. [Go] V. Ya. Golodets, Description of representations of anticommutation relations, Uspekhi Mat. Nauk 24 (1969), no. 4, 3-64 (in Russian); English transl. in Russian Math. Surveys 24 (1969). 
  7. [GS] V. Ya. Golodets and S. D. Sinel'shchikov, Classification and structure of cocycles of amenable ergodic equivalence relations, J. Funct. Anal. 121 (1994), 455-485. Zbl0821.28010
  8. [Kr] W. Krieger, On ergodic flows and isomorphism of factors, Math. Ann. 223 (1976), 19-70. Zbl0332.46045
  9. [Mo] C. C. Moore, Ergodic theory and von Neumann algebras, in: Proc. Symp. Pure Math. 38, Part 2, Amer. Math. Soc., 1982, 179-226. 
  10. [Sc1] K. Schmidt, Cocycles of Ergodic Transformation Groups, Lecture Notes in Math. 1, MacMillan of India, Delhi, 1977. Zbl0421.28017
  11. [Sc2] K. Schmidt, Algebraic Ideas in Ergodic Theory, CBMS Regional Conf. Ser. in Math. 76, Amer. Math. Soc., Providence, RI, 1990. 
  12. [Ta] M. Takesaki, Duality for crossed product and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249-310. Zbl0268.46058

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