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Ergodic equivalence relations, cohomology, and von Neumann algebras. II

On etend certains des resultats fondamentaux de la theorie des isomorphismes aux actions des groupes unimodulaires moyennables generaux. Les groupes moyennables et leur propre entropie. Entropie des actions mesurables. Theoremes d'isomorphismes

Entropy and isomorphism theorems for actions of amenable groups

Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

We prove that for any amenable non-singular countable equivalence relation R⊂X×X, there exists a non-singular transformation T of X such that, up to a null set:It follows that any two Cartan subalgebras of a hyperfinite factor are conjugate by an automorphism

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An amenable equivalence relation is generated by a single transformation

Harmonic Analysis and the Theory of Probability. By S. Bochner pp. 176. 35s. 1955. (California University Press and Cambridge University Press)

/pdf/ergodic-equivalence-relations-cohomology-and-von-neumann-2b1l9od00g.pdf

Ergodic equivalence relations, cohomology, and von Neumann algebras

A property is introduced, for 1-1 measure-preserving transformations of probability spaces, calledloose Bernoulliness (LB), which is invariant under taking factors, inducing, and tower-building. It amounts to replacing, in Ornstein’s definition ofvery weak Bernoulli, the Hamming distance on strings by a coarser metric. The main result is the construction of a transformationT0 which is ergodic and of entropy 0 butnot LB. On the other hand, any irrational rotationis LB. Consequently, the equivalence relation generated by inducing and tower-building (which I callKakutani equivalence, and the Russians callmonotone equivalence) has at least two distinct equivalence classes among the ergodic entropy zero transformations. A similar situation exists for ergodic positive-entropy transformations: on the one hand, any Bernoulli shift is LB, while on the other hand a non LBK-automorphism\(\hat T_0 \) can be made by skewingT0 over a Bernoulli base.

J. Feldman

Papers

Ergodic equivalence relations, cohomology, and von Neumann algebras. II

An amenable equivalence relation is generated by a single transformation

Ergodic equivalence relations, cohomology, and von Neumann algebras

NewK-automorphisms and a problem of Kakutani

Orbit structure and countable sections for actions of continuous groups