Endomorphisms of measured equivalence relations, cocycles with values in non-locally compact groups and applications

1999 ◽  
Vol 19 (3) ◽  
pp. 571-590 ◽  
Author(s):  
ALEXANDRE I. DANILENKO
Keyword(s):  
Projective Limit ◽  
Equivalence Relations ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Ergodic Transformation ◽  
Skew Product ◽  
Locally Compact ◽  
Dense Range ◽  
Group Extensions ◽  
Polish Groups

Consider a class of Polish groups arising from the subclass of amenable locally compact ones via operations of countable projective limit and group extensions. We show that for each group from this class there exists a cocycle of an ergodic transformation with dense range in it. This is applied to extend and provide a short (orbital) proof for one of the main results from [ALV] on non-coalescence of some ergodic skew product extensions.

scholarly journals AMENABILITY, FREE SUBGROUPS, AND HAAR NULL SETS IN NON-LOCALLY COMPACT GROUPS

2006 ◽  
Vol 93 (3) ◽  
pp. 693-722 ◽  
Author(s):  
SŁAWOMIR SOLECKI
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Inverse Limits ◽  
Automatic Continuity ◽  
Locally Compact ◽  
Polish Groups ◽  
Continuity Result ◽  
Haar Null Sets ◽  
The One ◽  
Free Subgroups

The paper has two objectives. On the one hand, we study left Haar null sets, a measure-theoretic notion of smallness on Polish, not necessarily locally compact, groups. On the other hand, we introduce and investigate two classes of Polish groups which are closely related to this notion and to amenability. We show that left Haar null sets form a $\sigma$-ideal and have the Steinhaus property on Polish groups which are ‘amenable at the identity’, and that they lose these two properties in the presence of appropriately embedded free subgroups. As an application we prove an automatic continuity result for universally measurable homomorphisms from inverse limits of sequences of amenable, locally compact, second countable groups to second countable groups.

On the Lp-conjecture for locally compact groups

2007 ◽  
Vol 89 (3) ◽  
pp. 237-242 ◽  
Author(s):  
F. Abtahi ◽  
R. Nasr-Isfahani ◽  
A. Rejali
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

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