Ambitable topological groups

A topological group is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the group with its right uniformity is contained in an ambit. For n=0,1,2,…, every locally aleph_n bounded topological group is either precompact or ambitable. In the familiar semigroups constructed over ambitable groups, topological centres have an effective characterization.

💡 Research Summary

The paper introduces the notion of an “ambitable” topological group. Let G be a Hausdorff topological group equipped with its right uniformity rG, and let RP(G) denote the family of continuous right‑invariant pseudometrics on G. For any Δ∈RP(G) the set

 BLip⁺(Δ)= { f:G→ℝ | 0≤f≤1 and |f(x)−f(y)|≤Δ(x,y) for all x,y∈G }

is a compact subset of the product space ℝ^G (pointwise convergence). The right translation action ρₓ(f)(z)=f(zx) is continuous, so (BLip⁺(Δ),ρ) is a compact G‑flow. An “ambit” is a compact G‑flow containing a point with a dense orbit; equivalently, a compact flow of the form orb(f) for some f∈Ub(rG), where Ub(rG) is the space of bounded uniformly continuous functions on rG.

A group G is defined to be ambitable if for every Δ∈RP(G) there exists a function f∈Ub(rG) such that the whole set BLip⁺(Δ) is contained in the orbit orb(f). In other words, every uniformly bounded uniformly equicontinuous family of functions (with respect to the right uniformity) sits inside a single ambit. The paper first shows that no precompact group can be ambitable (Lemma 2.2): the constant functions 0 and 1 cannot simultaneously belong to the same orbit, which contradicts the definition of an ambit.

To analyse when a group is ambitable, three cardinal invariants are introduced for a pseudometric Δ:

• d(Δ) – the Δ‑density of G (smallest size of a Δ‑dense subset);
• η♯(Δ) – the smallest cardinality of a set P⊂G such that G = B P, where B={x | Δ(e,x)≤1};
• η(Δ) – the smallest cardinality of a set P⊂G for which there exists a finite Q⊂G with G = Q B P.

Lemma 3.1 establishes basic inequalities η(Δ) ≤ η♯(Δ) ≤ d(Δ) and shows that d(Δ)=limₖ η(kΔ). Theorem 3.3 proves that finiteness of η(Δ) forces finiteness of η♯(½Δ), while infiniteness of η(Δ) yields η♯(½Δ) ≤ η(Δ). These relations allow the authors to translate “κ‑boundedness” (every neighbourhood of the identity can be covered by ≤κ left translates) and “locally κ‑boundedness” into statements about the three cardinal functions (Lemma 3.5).

The central technical tool is a factorisation lemma derived from the work of Neufang and Ferri‑Neufang. Lemma 4.1 shows that if η(Δ)≥ℵ₀, then for any family {Fα} of non‑empty finite subsets of G indexed by a set A of size η(Δ) one can choose elements xα∈G so that the translated sets Fα xα are pairwise Δ‑separated by more than 1. Using this separation, Lemma 4.2 constructs, for any collection O of ≤η(Δ) non‑empty open subsets of BLip⁺(Δ), a function f∈BLip⁺(Δ) whose orbit meets every member of O. Lemma 4.3 then shows that when d(Δ)=η(Δ)≥ℵ₀, there exists f with BLip⁺(Δ)=orb(f). Consequently, if for every Δ there exists a larger pseudometric Δ′ with d(Δ′)=η(Δ′)≥ℵ₀, the group is ambitable.

Theorem 4.4 applies these ideas: if G is locally κ‑bounded and there is a Δ₀ with η♯(Δ₀)≥κ, then G is ambitable. Corollary 4.5 deduces that any locally κ⁺‑bounded group that is not κ‑bounded must be ambitable. The main classification result, Theorem 4.6, states that for each positive integer n, every locally ℵₙ‑bounded group is either precompact or ambitable. This yields several concrete corollaries: locally compact groups are either compact or ambitable; ℵ₀‑bounded groups are either precompact or ambitable; and the additive group of any infinite‑dimensional normed space (with its norm topology) is ambitable (Theorem 4.9).

Finally, the paper connects ambitable groups to the structure of two important right‑topological semigroups: M(rG), the dual of LUC(G) (the space of bounded right‑uniformly continuous functions), and rG, the uniform compactification of G. In ambitable groups, the space of uniform measures Mᵤ(rG) and the completion c rG serve as the topological centres of M(rG) and rG respectively, extending earlier results of Lau, Pym, and Ferri‑Neufang that were limited to precompact or ℵ₀‑bounded groups.

In summary, the authors develop a new dynamical property (ambitability), relate it to cardinal invariants of the right uniformity, provide sufficient conditions (especially for locally ℵₙ‑bounded groups), and show that this property yields a clean description of the topological centres of the associated semigroups, thereby unifying and extending several known results in topological group theory and abstract harmonic analysis.