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Transactions of the American Mathematical Society

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

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Ergodic equivalence relations, cohomology, and von Neumann algebras. I
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by Jacob Feldman and Calvin C. Moore PDF
Trans. Amer. Math. Soc. 234 (1977), 289-324 Request permission

Abstract:

Let $(X,\mathcal {B})$ be a standard Borel space, $R \subset X \times X$ an equivalence relation $\in \mathcal {B} \times \mathcal {B}$. Assume each equivalence class is countable. Theorem 1: $\exists$ a countable group G of Borel isomorphisms of $(X,\mathcal {B})$ so that $R = \{ (x,gx):g \in G\}$. G is far from unique. However, notions like invariance and quasi-invariance and R-N derivatives of measures depend only on R, not the choice of G. We develop some of the ideas of Dye [1], [2] and Krieger [1]-[5] in a fashion explicitly avoiding any choice of G; we also show the connection with virtual groups. A notion of “module over R” is defined, and we axiomatize and develop a cohomology theory for R with coefficients in such a module. Surprising application (contained in Theorem 7): let $\alpha ,\beta$ be rationally independent irrationals on the circle $\mathbb {T}$, and f Borel: $\mathbb {T} \to \mathbb {T}$. Then $\exists$ Borel $g,h:\mathbb {T} \to \mathbb {T}$ with $f(x) = (g(ax)/g(x))(h(\beta x)/h(x))$ a.e. The notion of “skew product action” is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the R-N derivative of an ergodic transformation: we define, for a cocycle c on R with values in the group A, a subgroup of A depending only on the cohomology class of c, and in Theorem 8 identify this with another subgroup, the “normalized proper range” of c, defined in terms of the skew action. See also Schmidt [1].
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 234 (1977), 289-324
  • MSC: Primary 22D40; Secondary 28A65, 46L10
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0578656-4
  • MathSciNet review: 0578656