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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Quasinormal subrelations of ergodic equivalence relations
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by Alexandre I. Danilenko PDF
Proc. Amer. Math. Soc. 126 (1998), 3361-3370 Request permission

Abstract:

We introduce a notion of quasinormality for a nested pair $\mathcal {S}\subset \mathcal {R}$ of ergodic discrete hyperfinite equivalence relations of type $II_{1}$. (This is a natural extension of the normality concept due to Feldman-Sutherland-Zimmer.) Such pairs are characterized by an irreducible pair $F\subset Q$ of countable amenable groups or rather (some special) their Polish closure $\overline {F}\subset \overline {Q}$. We show that “most” of the ergodic subrelations of $\mathcal {R}$ are quasinormal and classify them. An example of a nonquasinormal subrelation is given. We prove as an auxiliary statement that two cocycles of $\mathcal {R}$ with dense ranges in a Polish group are weakly equivalent.
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Additional Information
  • Alexandre I. Danilenko
  • Affiliation: Department of Mechanics and Mathematics, Kharkov State University, Freedom square 4, Kharkov, 310077, Ukraine
  • MR Author ID: 265198
  • Email: danilenko@ilt.kharkov.ua
  • Received by editor(s): April 10, 1997
  • Communicated by: Palle E. T. Jorgensen
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 3361-3370
  • MSC (1991): Primary 28D99, 46L55
  • DOI: https://doi.org/10.1090/S0002-9939-98-04909-0
  • MathSciNet review: 1610944