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.

Orthogonality preserving linear maps on group algebras

2015 ◽  
Vol 158 (3) ◽  
pp. 493-504 ◽  
Author(s):  
J. ALAMINOS ◽  
J. EXTREMERA ◽  
A. R. VILLENA
Keyword(s):  
Locally Compact Group ◽  
Group Algebras ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Orthogonal Functions ◽  
Orthogonality Conditions ◽  
Linear Maps ◽  
Jordan Homomorphisms ◽  
The One

AbstractWe consider several types of orthogonality conditions on the group algebra L1(G) of a locally compact group G such as f$\ast $g = 0, f$\ast $g☆ = 0, f☆$\ast $g = 0, f$\ast $g = g$\ast $f = 0 and f$\ast $g☆ = g☆$\ast $f = 0, and we describe the linear maps Φ: L1(G) → L1(H) between the group algebras of locally compact groups G and H that take orthogonal functions of L1(G) into orthogonal functions of L1(H). Roughly speaking, they are weighted homomorphisms in the case where we are concerned with the one-sided orthogonality conditions and weighted Jordan homomorphisms in the case where we treat the two-sided orthogonality conditions.

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.

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.

Locally compact groups with permutable closed subgroups

2021 ◽  
Vol 390 ◽  
pp. 107894
Author(s):  
Wolfgang Herfort ◽  
Karl H. Hofmann ◽  
Francesco G. Russo
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact
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