The generalized Bohr compactification, II

1976 ◽  
pp. 112-126
Author(s):  
Shmuel Glasner
Keyword(s):  
Bohr Compactification

The Bohr compactification, modulo a metrizable subgroup

1993 ◽  
Vol 143 (2) ◽  
pp. 119-136
Author(s):  
W. W. Comfort ◽  
F. Javier Trigos-Arrieta ◽  
Ta-Sun Wu
Keyword(s):  
Bohr Compactification

scholarly journals Quantum Bohr compactification

2005 ◽  
Vol 49 (4) ◽  
pp. 1245-1270 ◽  
Author(s):  
Piotr M. Sołtan
Keyword(s):  
Bohr Compactification

scholarly journals Relating a Locally Compact Abelian Group to Its Bohr Compactification

1996 ◽  
Vol 120 (2) ◽  
pp. 322-344 ◽  
Author(s):  
W.W. Comfort ◽  
Salvador Hernández ◽  
F.Javier Trigos-Arrieta
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Bohr Compactification

A note on the Bohr compactification

1973 ◽  
Vol 6 (1) ◽  
pp. 362-364 ◽  
Author(s):  
M. Friedberg ◽  
J. W. Stepp
Keyword(s):  
Bohr Compactification

scholarly journals On uniformly distributed sequences of integers and Poincaré recurrence III

2003 ◽  
Vol 68 (2) ◽  
pp. 345-350
Author(s):  
R. Nair
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Probability Space ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Bohr Compactification ◽  
Poincare Recurrence ◽  
Uniformly Distributed Sequences ◽  
Image Position ◽  
Measure Preserving Transformations

Let S be a semigroup contained in a locally compact Abelian group G. Let Ĝ denote the Bohr compactification of G. We say that a sequence contained in S is Hartman uniform distributed on G iffor any character χ in Ĝ. Suppose that (Tg)g∈s is a semigroup of measurable measure preserving transformations of a probability space (X, β, μ) and B is an element of the σ-algebra β of positive μ measure. For a map T: X → X and a set A ⊆ X let T−1A denote {x ∈ X: Tx ∈ A}. In an earlier paper, the author showed that if k is Hartman uniform distributed thenIn this paper we show that ≥ cannot be replaced by =. A more detailed discussion of this situation ensues.

scholarly journals Amenability, connected components, and definable actions

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Ehud Hrushovski ◽  
Krzysztof Krupiński ◽  
Anand Pillay
Keyword(s):  
Compact Space ◽  
Invariant Probability Measure ◽  
Connected Components ◽  
Topological Groups ◽  
Bohr Compactification ◽  
Weakly Almost Periodic ◽  
Closed Type ◽  
Definable Groups ◽  
The Relationship ◽  
Saturated Structure

AbstractWe study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version (Massicot and Wagner in J Ec Polytech Math 2:55–63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189–243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain “weak Bohr compactification” introduced in Krupiński and Pillay (Adv Math 345:1253–1299, 2019). In other words, the conclusion says that certain connected components of G coincide: $$G^{00}_{{{\,\mathrm{{top}}\,}}} = G^{000}_{{{\,\mathrm{{top}}\,}}}$$ G top 00 = G top 000 . We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from Krupiński and Pillay (2019). We also introduce $$\bigvee $$ ⋁ -definable group topologies on a given $$\emptyset $$ ∅ -definable group G (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of G implies (under some assumption) that $${{\,\mathrm{{cl}}\,}}(G^{00}_M) = {{\,\mathrm{{cl}}\,}}(G^{000}_M)$$ cl ( G M 00 ) = cl ( G M 000 ) for any model M. Secondly, we study the relationship between (separate) definability of an action of a definable group on a compact space [in the sense of Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions of G [in the sense of Ellis and Nerurkar (Trans Am Math Soc 313:103–119, 1989)], and stability. We conclude that any group G definable in a sufficiently saturated structure is “weakly definably amenable” in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G-invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080–1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper. Thirdly, we give an example of a $$\emptyset $$ ∅ -definable approximate subgroup X in a saturated extension of the group $${{\mathbb {F}}}_2 \times {{\mathbb {Z}}}$$ F 2 × Z in a suitable language (where $${{\mathbb {F}}}_2$$ F 2 is the free group in 2-generators) for which the $$\bigvee $$ ⋁ -definable group $$H:=\langle X \rangle $$ H : = ⟨ X ⟩ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) “model” exists for each approximate subgroup does not work in general (they proved in (Massicot and Wagner 2015) that it works for definably amenable approximate subgroups).

scholarly journals Remarks on the quantum Bohr compactification

2013 ◽  
Vol 57 (4) ◽  
pp. 1131-1171 ◽  
Author(s):  
Matthew Daws
Keyword(s):  
Bohr Compactification

scholarly journals The Bohr compactification, modulo a metrizable subgroup

1993 ◽  
Vol 143 (2) ◽  
pp. 119-136
Author(s):  
W. W. Comfort ◽  
F. Javier Trigos-Arrieta ◽  
Ta-Sun Wu
Keyword(s):  
Bohr Compactification
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