Normed versus topological groups: Dichotomy and duality

Normed groups 17so that fg is in the ˆd π -ball of radius ε of F G provided f is in the ˆd π -ball of radius ε/2of F and g is in the ˆd πH -ball of radius ε/2 of G.Remark (The compact-open topology). In similar circumstances, we show in Theorem3.17 below that under the strong ∆-refinement topology, so a finer topology, Auth(X)is a normed group and a topological group. Rather than use weak or strong refinementof metrics in Auth(X), one may consider the compact-open topology (the topology ofuniform convergence on compacts, introduced by Fox and studied by Arens in [Ar1],[Ar2]). However, in order to ensure the kind of properties we need (especially in flows),the metric space X would then need to be restricted to a special case. Recall somesalient features of the compact-open topology. For composition to be continuous localcompactness is essential ([Dug, Ch. XII.2], [McCN], [BePe, Section 8.2], or [vM2, Ch.1]).When T is compact the topology is admissible (i.e. Auth(X) is a topological groupunder it), but the issue of admissibility in the non-compact situation is not currentlyfully understood (even in the locally compact case for which counter-examples with noncontinuousinversion exist, and so additional properties such as local connectedness areusually invoked – see [Dij] for the strongest results). In applications the focus of interestmay fall on separable spaces (e.g. function spaces), but, by a theorem of Arens, if X isseparable metric and further the compact-open topology on C (X, R) is metrizable, thenX is necessarily locally compact and σ-compact, and conversely (see e.g [Eng, p.165 and266] ).We will now apply the supremum-norm construction to deduce that right-invariancemay be arranged if for every x ∈ X the left translation λ x has finite sup-norm:‖λ x ‖ H = sup z∈X d X (xz, z) < ∞.We will need to note the connection with conjugate norms.Definition. Recall the g-conjugate norm is defined by‖x‖ g := ‖gxg −1 ‖.The conjugacy refinement norm corresponding to the family of all the g-conjugate normsΓ = {‖.‖ g : g ∈ G} will be denoted byin contexts where this is finite.‖x‖ ∞ := sup g ‖x‖ g ,Clearly, for any g,‖x‖ ∞ = ‖gxg −1 ‖ ∞ ,and so ‖x‖ ∞ is an abelian norm (substitute xg for x). Evidently, if the metric d X L isleft-invariant we have‖x‖ ∞ = sup g ‖x‖ g = sup z∈X d X L (z −1 xz, e) = sup z∈X d X L (xz, z). (shift)