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**Asymptotically invariant sequences and approximate finiteness.**
*(English)*
Zbl 0638.28014

Let G be a countable group, (X,\({\mathcal S},\mu)\) a nonatomic Lebesgue probability space, and (g,x)\(\to gx\) a nonsingular, ergodic action of G on (X,\({\mathcal S},\mu)\). The action of G is called strongly ergodic if every asymptotically invariant sequence \((B_ n)_{n\in N}\) in \({\mathcal S}\) is trivial, i.e. if \(\mu (B_ n\Delta gB_ n)\to 0\) for every \(g\in G\) implies that \(\mu (B_ n)(1-\mu (B_ n))\to 0\) [see A. Connes and B. Weiss, Israel J. Math. 37, 209-210 (1980; Zbl 0479.28017)].

The first result of the paper reveals the connection between the absence of strong ergodicity and approximate finiteness (the action of G is called approximately finite if there exists a nonsingular automorphism V of (X,\({\mathcal S},\mu)\) with \(Gx=\{V^ nx: n\in {\mathbb{Z}}\}\mu\)-a.e., where \(Gx=\{gx: g\in G\}).\) It turns out that the action of G is not strongly ergodic iff (*) there exists a nonsingular, ergodic automorphism V of a nonatomic Lebesgue probability space (Y,\({\mathcal J},\nu)\) and a nonsingular map \(\psi\) : \(X\to Y\) such that \(Gx=\{V^ n\psi x: n\in {\mathbb{Z}}\}\mu\)- a.e. Moreover, if the action of G is not approximately finite and (*) is satisfied, then \(\psi\) is uncountable-to-one and infinite-to-one on a.e. orbit of G. The authors also solve the problem whether the approximately finite “quotient” in the above theorem can be split off as a direct summand of the orbit structure of G. Namely they show that the action of G is stable (i.e. orbit equivalent to the action \(((g,n),(x,y))\to (gx,V^ ny)\) of \(G\times {\mathbb{Z}}\) on \((X\times Y,\quad {\mathcal S}\times {\mathcal J},\quad \mu \times \nu),\) where V is a measure preserving, ergodic automorphism of a nonatomic Lebesgue probability space (Y,\({\mathcal J},\nu))\) iff it has a suitably defined, stronger asymptotic property than lack of strong ergodicity. This result is closely related to the following problem in the theory of von Neumann algebras: when is a factor isomorphic to its tensor product with the hyperfinite \(II_ 1\)-factor? [see D. McDuff, Proc. Lond. Math. Soc., III. Ser. 21, 443-461 (1970; Zbl 0204.149) and A. Connes, Ann. Math., II. Ser. 104, 73- 115 (1976; Zbl 0343.46042) for solution].

All above results can also be formulated in terms of countable, nonsingular, ergodic relations. The authors give two simple applications of the above theorem on stability and raise some open problems.

The first result of the paper reveals the connection between the absence of strong ergodicity and approximate finiteness (the action of G is called approximately finite if there exists a nonsingular automorphism V of (X,\({\mathcal S},\mu)\) with \(Gx=\{V^ nx: n\in {\mathbb{Z}}\}\mu\)-a.e., where \(Gx=\{gx: g\in G\}).\) It turns out that the action of G is not strongly ergodic iff (*) there exists a nonsingular, ergodic automorphism V of a nonatomic Lebesgue probability space (Y,\({\mathcal J},\nu)\) and a nonsingular map \(\psi\) : \(X\to Y\) such that \(Gx=\{V^ n\psi x: n\in {\mathbb{Z}}\}\mu\)- a.e. Moreover, if the action of G is not approximately finite and (*) is satisfied, then \(\psi\) is uncountable-to-one and infinite-to-one on a.e. orbit of G. The authors also solve the problem whether the approximately finite “quotient” in the above theorem can be split off as a direct summand of the orbit structure of G. Namely they show that the action of G is stable (i.e. orbit equivalent to the action \(((g,n),(x,y))\to (gx,V^ ny)\) of \(G\times {\mathbb{Z}}\) on \((X\times Y,\quad {\mathcal S}\times {\mathcal J},\quad \mu \times \nu),\) where V is a measure preserving, ergodic automorphism of a nonatomic Lebesgue probability space (Y,\({\mathcal J},\nu))\) iff it has a suitably defined, stronger asymptotic property than lack of strong ergodicity. This result is closely related to the following problem in the theory of von Neumann algebras: when is a factor isomorphic to its tensor product with the hyperfinite \(II_ 1\)-factor? [see D. McDuff, Proc. Lond. Math. Soc., III. Ser. 21, 443-461 (1970; Zbl 0204.149) and A. Connes, Ann. Math., II. Ser. 104, 73- 115 (1976; Zbl 0343.46042) for solution].

All above results can also be formulated in terms of countable, nonsingular, ergodic relations. The authors give two simple applications of the above theorem on stability and raise some open problems.

Reviewer: K.Krzyżewski

### MSC:

28D15 | General groups of measure-preserving transformations |

46L35 | Classifications of \(C^*\)-algebras |