## 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