[Paper Review] On two problems concerning topological centers

Eli Glasner|arXiv (Cornell University)|Oct 15, 2007

Digital Image Processing Techniques5 references14 citations

TL;DR

This paper resolves two problems concerning topological centers in topological dynamics: first, it proves that left multiplication by any element in the remainder of the Čech-Stone compactification of an infinite discrete group Γ is not Borel measurable, using the nonmeasurability of free ultrafilters; second, for the universal distal compactification D of the integers ℤ, it shows that the topological center of D coincides with both its algebraic center and the canonical image of ℤ in D.

ABSTRACT

Let G be an infinite discrete group and bG its Cech-Stone compactification. Using the well known fact that a free ultrafilter on an infinite set is nonmeasurable, we show that for each element p of the remainder bG G, left multiplication L_p:bG o bG is not Borel measurable. Next assume that G is abelian. Let D \subset \ell^\infty(G)$ denote the subalgebra of distal functions on G and let G^D denote the corresponding universal distal (right topological group) compactification of G. Our second result is that the topological center of G^D (i.e. the set of p in G^D for which L_p:G^D o G^D is a continuous map) is the same as the algebraic center and that for G=Z (the group of integers) this center coincides with the canonical image of G in G^D.

Motivation & Objective

• To resolve Michael Megrelishvili's problem on which elements in the remainder of βΓ have Baire class 1 left multiplication maps.
• To solve Mahmoud Filali's problem on identifying the topological center of the universal distal Ellis group D(Γ) for abelian groups.
• To establish that for Γ = ℤ, the topological center of D coincides with the algebraic center and the canonical image of ℤ in D.
• To provide a rigorous justification for the non-Borel measurability of left multiplication maps on βΓ using measure-theoretic arguments.
• To close a gap in the original proof of Theorem 2.1 by proving a lemma on universally measurable images of Borel sets under continuous surjections to the Cantor set.

Proposed method

• Uses the well-known fact that free ultrafilters on infinite sets are nonmeasurable under the product measure on {0,1}^Γ.
• Constructs a map φ = π₀∘ψ̂∘Lₚ from βΓ to {0,1}, where ψ̂ is the extension of the orbit map to βΓ, and analyzes the preimage of 1 as a Borel set.
• Relies on the identification of βΓ with the enveloping semigroup of the dynamical system (Ω, Γ), where Ω = {0,1}^Γ.
• Applies a duality argument via the involution J(ω)(γ) = ω(γ⁻¹) to relate the preimage set to the ultrafilter p.
• Employs a measure-theoretic lemma to show that if the image of a Borel set under a continuous surjection to the Cantor set were Borel, it would be universally measurable, contradicting the nonmeasurability of free ultrafilters.
• For the distal system case, uses the structure of the universal distal compactification D of ℤ and shows that only elements from ℤ itself induce continuous left multiplication.

Experimental results

Research questions

• RQ1 For which elements p ∈ βΓ ∖ Γ is the left multiplication map Lₚ: βΓ → βΓ Borel measurable?
• RQ2 What is the topological center of the universal distal Ellis group D(Γ) for an abelian discrete group Γ?
• RQ3 Does the topological center of D(ℤ) coincide with the algebraic center and the canonical image of ℤ?
• RQ4 Can the non-Borel measurability of Lₚ for p ∈ βΓ ∖ Γ be rigorously established despite βΓ not being a Polish space?
• RQ5 Is the image of a Borel set under a continuous surjection to the Cantor set universally measurable if the preimage is saturated?

Key findings

• For any infinite discrete group Γ, left multiplication by any p ∈ βΓ ∖ Γ is not Borel measurable, as the corresponding ultrafilter is nonmeasurable under the product measure.
• The topological center of the universal distal compactification D of ℤ is precisely the canonical image of ℤ in D, and this set also coincides with the algebraic center of D.
• The proof of non-Borel measurability is completed via a lemma showing that if a Borel set B ⊂ X satisfies B = f⁻¹(f(B)) for a continuous surjection f: X → C to the Cantor set, then f(B) is universally measurable.
• The nonmeasurability of free ultrafilters on infinite sets implies that their image under the characteristic function χ is not Borel in {0,1}^Γ, hence Lₚ cannot be Borel measurable.
• The construction of the map φ = π₀∘ψ̂∘Lₚ and its preimage Q shows that Lₚ⁻¹({q ∈ βΓ : (qω₀)(e) = 1}) is not Borel, leading to the conclusion.
• The result extends to the case of abelian groups, where the topological center of D(Γ) equals the algebraic center, and for ℤ, it equals the image of ℤ in D.