Normed versus topological groups: Dichotomy and duality

136 N. H. Bingham and A. J. OstaszewskiDefinition. We say that pointwise (resp. uniform) divergence is unconditional divergencein A if, for any (pointwise/uniform) divergent sequence α n ,(i) for any bounded σ, the sequence σα n is (pointwise/uniform) divergent; and,(ii) for any ψ n convergent to the identity, ψ n α n is (pointwise/uniform) divergent.Remark. In clause (ii) each of the functions ψ n has a bound depending on n. The twoclauses could be combined into one by requiring that if the bounded functions ψ n convergeto ψ 0 in the supremum norm, then ψ n α n is (pointwise/uniform) divergent.By Lemma 13.3 uniform divergence in H(X) is unconditional. We move to other formsof this result.Proposition 13.4. If the metric on A is left- or right-invariant, then uniform divergenceis unconditional in A.Proof. If the metric d = d A is left-invariant, then observe that if β n is a bounded sequence,then so is σβ n , sinced(e, σβ n ) = d(σ −1 , β n ) ≤ d(σ −1 , e) + d(e, β n ).Since ‖βn−1 ‖ = ‖β n ‖, the same is true for right-invariance. Further, if ψ n is convergent tothe identity, then also ψ n β n is a bounded sequence, sinced(e, ψ n β n ) = d(ψn−1 , β n ) ≤ d(ψn −1 , e) + d(e, β n ).Here we note that, if ψ n is convergent to the identity, then so is ψn−1 by continuity ofinversion (or by metric invariance). The same is again true for right-invariance.The case where the subgroup A of self-homeomorphisms is the translations Ξ, thoughimmediate, is worth noting.Theorem 13.5. (The case A = Ξ) If the metric on the group X is left- or right-invariant,then uniform divergence is unconditional in A = Ξ.Proof. We have already noted that Ξ is isometrically isomorphic to X.Remarks. 1. If the metric is bounded, there may not be any divergent sequences.2. We already know from Lemma 13.3 that uniform divergence in A = H(X) is unconditional.3. The unconditionality condition (i) corresponds directly to the technical conditionplaced in [BajKar] on their filter F. In our metric setting, we thus employ a strongernotion of limit to infinity than they do. The filter implied by the pointwise setting isgenerated by sets of the form⋂i∈I {α : dX (α n (x i ), x i ) > M ultimately} with I finite.