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

Alexandre Danilenko; Toshihiro Hamachi

Displaying similar documents to “On measure theoretical analogues of the Takesaki structure theorem for type III factors”

On subrelations of ergodic measured type III equivalence relations

Alexandre Danilenko (2000)

Colloquium Mathematicae

Similarity:

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.

Characterizing mildly mixing actions by orbit equivalence of products.

Hawkins, Jane, Silva Cesar, E. (1998)

The New York Journal of Mathematics [electronic only]

Similarity:

Construction of non-constant and ergodic cocycles

Mahesh Nerurkar (2000)

Colloquium Mathematicae

Similarity:

We construct continuous G-valued cocycles that are not cohomologous to any compact constant via a measurable transfer function, provided the underlying dynamical system is rigid and the range group G satisfies a certain general condition. For more general ergodic aperiodic systems, we also show that the set of continuous ergodic cocycles is residual in the class of all continuous cocycles provided the range group G is a compact connected Lie group. The first construction is based on...

Ergodic elements of ergodic actions

Charles Pugh, Michael Shub (1971)

Compositio Mathematica

Similarity:

Induced and amenable ergodic actions of Lie groups

Robert J. Zimmer (1978)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

Ryotaro Sato (1995)

Studia Mathematica

Similarity:

Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{- 1} \sum_{i = 0}^{n - 1} f \circ τ^{i} (x)$ converges almost everywhere to a function f* in $L (p_{1}, q_{1}]$ , where (pq) and $(p_{1}, q_{1}]$ are assumed to be in the set $(r, s) : r = s = 1, o r 1 < r < \infty a n d 1 \leq s \leq \infty, o r r = s = \infty$ . Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized...

Weak almost periodicity of $L_{1}$ contractions and coboundaries of non-singular transformations

Isaac Kornfeld, Michael Lin (2000)

Studia Mathematica

Similarity:

It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on $L_{1}$ is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex $L_{1}$ such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let...

An algebraic group associated to an ergodic diffeomorphism

Robert J. Zimmer (1981)

Compositio Mathematica

Similarity:

Genericity of nonsingular transformations with infinite ergodic index

J. Choksi, M. Nadkarni (2000)

Colloquium Mathematicae

Similarity:

It is shown that in the group of invertible measurable nonsingular transformations on a Lebesgue probability space, endowed with the coarse topology, the transformations with infinite ergodic index are generic; they actually form a dense $G_{δ}$ set. (A transformation has infinite ergodic index if all its finite Cartesian powers are ergodic.) This answers a question asked by C. Silva. A similar result was proved by U. Sachdeva in 1971, for the group of transformations preserving an infinite...

Isometric extensions, 2-cocycles and ergodicity of skew products

Alexandre Danilenko, Mariusz Lemańczyk (1999)

Studia Mathematica

Similarity:

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 $T_{α}$ .