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On subrelations of ergodic measured type III equivalence relations

100%

Danilenko A. I.

Colloquium Mathematicum

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2000

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tom 84/85

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nr 1

13-22

EN

We discuss the classification up to orbit equivalence of inclusions 𝑆 ⊂ ℛ of measured ergodic discrete hyperfinite equivalence relations. In the case of type III relations, the orbit equivalence classes of such inclusions of finite index are completely classified in terms of triplets consisting of a transitive permutation group G on a finite set (whose cardinality is the index of 𝑆 ⊂ ℛ), an ergodic nonsingular ℝ-flow V and a homomorphism of G to the centralizer of V.

2

Isometric extensions, 2-cocycles and ergodicity of skew products

63%

Danilenko A. I., Lemańczyk M.

Studia Mathematica

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1999

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tom 137

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nr 2

123-142

EN

We establish existence and uniqueness of a canonical form for isometric extensions of an ergodic non-singular transformation T. This is applied to describe the structure of commutors of the isometric extensions. Moreover, for a compact group G, we construct a G-valued T-cocycle α which generates the ergodic skew product extension $T_α$ and admits a prescribed subgroup in the centralizer of $T_α$.

3

On measure theoretical analogues of the Takesaki structure theorem for type III factors

63%

Danilenko A. I., Hamachi T.

Colloquium Mathematicum

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2000

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tom 84/85

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nr 2

485-493

EN

The orbit equivalence of type $III_{0}$ ergodic equivalence relations is considered. We show that it is equivalent to the outer conjugacy problem for the natural trace-scaling action of a countable dense ℝ-subgroup by automorphisms of the Radon-Nikodym skew product extensions of these relations. A similar result holds for the weak equivalence of arbitrary type $III_{0}$ cocycles with values in Abelian groups.

Ograniczanie wyników

2 Colloquium Mathematicum

1 Studia Mathematica

3 Danilenko A. I.

1 Hamachi T.

1 Lemańczyk M.

2 2000

1 1999

On subrelations of ergodic measured type III equivalence relations

Isometric extensions, 2-cocycles and ergodicity of skew products

On measure theoretical analogues of the Takesaki structure theorem for type III factors