Normed versus topological groups: Dichotomy and duality

42 N. H. Bingham and A. J. OstaszewskiTheorem 3.20. Let G be a locally compact topological group with a norm having a compactlygenerated, vanishingly small global word-net. For f a continuous automorphism(e.g. f(x) = gxg −1 ), supposeThenβ := lim sup ‖x‖→0+‖f(x)‖‖x‖M = sup x‖f(x)‖‖x‖< ∞.< ∞.We defer the proof to Section 4 as it relies on the development there of the theory ofsubadditive functions.Theorem 3.21. If G is an infinitely divisible group with an N-homogeneous norm, thenits norm has the Lipschitz property, i.e. if f : G → G is a continuous automorphism,then for some M > 0‖f(x)‖ ≤ M‖x‖.Proof. Suppose that δ > 0. Fix x ≠ e. Definep δ (x) := sup{q ∈ Q + : ‖x q ‖ < δ} = δ/‖x‖.Let f be a continuous automorphism. As f(e) = e, there is δ > 0 such that, for ‖z‖ ≤ δ,If ‖x q ‖ < δ, thenThus for each q < p δ (x) we haveTaking limits, we obtain, with M = 1/δ,‖f(z)‖ < 1.‖f(x q )‖ < 1.‖f(x)‖ < 1/q.‖f(x)‖ ≤ 1/p δ (x) = M‖x‖.Definitions. 1. Let G be a Lipschitz-normed topological group. We may now takef(x) = γ g (x) := gxg −1 , since this homomorphism is continuous. The Lipschitz norm isdefined byM g := sup x≠e ‖γ g (x)‖/‖x‖ = sup x≠e ‖x‖ g /‖x‖.(As noted before the introduction of the Lipschitz property this is the Lipschitz-1 norm.)Thus‖x‖ g := ‖gxg −1 ‖ ≤ M g ‖x‖.2. For X a normed group with right-invariant metric d X and g ∈ H u (X) denote thefollowing (inverse) modulus of continuity byδ(g) = δ 1 (g) := sup{δ > 0 : d X (g(z), g(z ′ )) ≤ 1, for all d X (z, z ′ ) ≤ δ}.