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Transactions of the American Mathematical Society

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



On extension of cocycles
to normalizer elements, outer conjugacy,
and related problems

Authors: Alexandre I. Danilenko and Valentin Ya. Golodets
Journal: Trans. Amer. Math. Soc. 348 (1996), 4857-4882
MSC (1991): Primary 46L55; Secondary 28D15, 28D99
MathSciNet review: 1376544
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Abstract: Let $T$ be an ergodic automorphism of a Lebesgue space and $\alpha $ a cocycle of $T$ with values in an Abelian locally compact group $G$. An automorphism $\theta $ from the normalizer $N[T]$ of the full group $[T]$ is said to be compatible with $\alpha $ if there is a measurable function $\varphi : X \to G$ such that $\alpha (\theta x, \theta T\theta ^{-1}) = - \varphi (x) + \alpha (x, T) + \varphi (Tx)$ at a.e. $x$. The topology on the set $D(T, \alpha )$ of all automorphisms compatible with $\alpha $ is introduced in such a way that $D(T , \alpha )$ becomes a Polish group. A complete system of invariants for the $\alpha $-outer conjugacy (i.e. the conjugacy in the quotient group $D(T, \alpha )/[T])$ is found. Structure of the cocycles compatible with every element of $N[T]$ is described.

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  • [BG1] S. I. Bezuglyi and V. Ya. Golodets, Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizers of type $III$ full groups, J. Funct. Anal. 60 (1985), 341-369. MR 86e:46052
  • [BG2] -, Outer conjugacy for actions of countable amenable groups on a measure space, Izv. Akad Nauk SSSR Ser. Mat. 50 (1986), 643-660 (Russian); English transl. in. Math. Izv. 29 (1987), 1-18. MR 88d:46115
  • [BG3] -, Weak equivalence and structures of cocycles of an ergodic automorphism, Publ. RIMS, Kyoto Univ. 27 (1991), 577-625. MR 93a:28017
  • [BGD] S. I. Bezuglyi, V. Ya. Golodets, and A. I. Danilenko, Extensions of $1$-cocycles of dynamical systems on normalizer elements, Dokl. Akad. Nauk UkrSSR Ser. A 1988, no. 2, 3-5 (Russian). MR 90h:28016
  • [CK] A. Connes and W. Krieger, Measure space automorphisms, the normalizers of their full groups, and approximate finiteness, J. Funct. Anal. 24 (1977), 336-352. MR 56:3246
  • [CFW] A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergod. Theory and Dyn. Systems 1 (1981), 431-450. MR 84h:46090
  • [Da1] A. I. Danilenko, Ergodic dynamical systems, their cocycles, and automorphisms compatible with cocycles, Ph.D. thesis, Kharkov State University, 1991 (Russian).
  • [Da2] -, On cocycles compatible with normalizers of full groups of measure space transformations, Dokl. Akad. Nauk Ukraine 1994, no. 7, 14-17. MR 95k:28039
  • [Da3] -, The topological structure of Polish groups and groupoids of measure space transformations, Publ. RIMS Kyoto Univ. 31 (1995), 913-940. CMP 96:06
  • [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 58:28261a
  • [FHM] J. Feldman, P. Hahn, and C. C. Moore, Orbit structure and countable sections for actions of continuous groups, Adv. in Math. 28 (1978), 186-230. MR 58:11217
  • [FSZ] J. Feldman, C. E. Sutherland, and R. Zimmer, Subrelations of ergodic equivalence relations, Ergod. Theory and Dyn. Syst. 9 (1989), 239-269. MR 91c:28020
  • [G] V. Ya. Golodets, Description of representations of anticommutation relations, Russian Uspekhi Mat. Nauk 24 (1969), 3-64 (Russian); English transl. in. Russian Math. Surveys 24 (1969).
  • [GD] V. Ya. Golodets and A. I. Danilenko, Ergodic actions of Abelian groups isomorphic to joint actions with themselves, Preprint 4-91, FTINT AN UkrSSR, Kharkov, 1991.
  • [GS1] V. Ya. Golodets and S. D. Sinel'shchikov, Outer conjugacy for actions of continuous amenable groups, Publ. RIMS, Kyoto Univ. 23 (1987), 737-769. MR 89c:46087
  • [GS2] -, Classification and structure of cocycles of amenable ergodic equivalence relations, J. Funct. Anal. 121 (1994), 455-485. MR 95h:28020
  • [H] T. Hamachi, The normalizer group of an ergodic automorphism of type $III$ and the commutant of an ergodic flow, J. Funct. Anal. 40 (1981), 387-403. MR 82m:46070
  • [HO] T. Hamachi and M. Osikawa, Ergodic groups of automorphisms amd Krieger's theorems, Sem. Math. Sci. Keio Univ., 1981. MR 84d:46099
  • [K] A. A. Kirillov, Dynamical systems, factors, and group representations, Uspekhi Matem. Nauk 22, no. 5, (1967), 67-80 (Russian). MR 36:347
  • [Kr] W. Krieger, On ergodic flows and isomorphism of factors, Math. Ann. 24 (1977), 336-352. MR 54:3430
  • [M1] G. W. Mackey, Infinite dimensional group representations, Bull. Amer. Math. Soc. 69 (1963), 628-686. MR 45:2095
  • [M2] -, Ergodic theory and virtual groups, Math. Ann. 166 (1966), 187-207. MR 34:1444
  • [O] D. S. Ornstein, On the root problem in ergodic theory, Proc. 6th Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1967, pp. 347-356. MR 53:3259
  • [R1] A. Ramsay, Virtual groups and group actions, Adv. in Math. 6 (1971), 253-322. MR 43:7590
  • [R2] -, Topologies for measured groupoids, J. Funct. Anal. 47 (1982), 314-343. MR 83k:22014
  • [S] K. Schmidt, Cocycles of ergodic transformation groups, Macmillan Company of India, Ltd, Delhi, 1977. MR 58:28262
  • [Z] R. Zimmer, Extension of ergodic group actions, Illinois J. Math. 20 (1976), 373-409. MR 53:13522

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Additional Information

Alexandre I. Danilenko
Affiliation: Department of Mechanics and Mathematics, Kharkov State University, Freedom Square 4, Kharkov, 310077, Ukraine

Valentin Ya. Golodets
Affiliation: Mathematics Department, Institute for Low Temperature Physics, Lenin Avenue 47, Kharkov, 310164, Ukraine

Keywords: Ergodic dynamical system, cocycle, outer conjugacy
Received by editor(s): January 4, 1995
Additional Notes: The work was supported in part by the International Science Foundation Grant No U2B000.
Article copyright: © Copyright 1996 American Mathematical Society

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