Normed versus topological groups: Dichotomy and duality

18 N. H. Bingham and A. J. OstaszewskiOne may finesse the left-invariance assumption, using (shift), as we will see in Proposition2.14.Example C. As H(X) is a group and ˆd H is right-invariant, the norm ‖f‖ H gives riseto a conjugacy refinement norm. Working in H(X), suppose that f n → f under thesupremum norm ˆd X = ˆd H . Let g ∈ H(X). Then pointwiselim n f n (g(x)) = f(g(x)).Hence, as f −1 is continuous, we have for any x ∈ X,f −1 (lim n f n (g(x))) = lim n f −1 f n (g(x)) = g(x).Likewise, as g −1 is continuous, we have for any x ∈ X,Thusg −1 (lim n f −1 f n (g(x))) = lim n g −1 f −1 f n (g(x)) = x.g −1 f −1 f n g → id X pointwise.This result is generally weaker than the assertion ‖f −1 f n ‖ g → 0, which requires uniformrather than pointwise convergence.We need the following notion of admissibility (with the norm ‖.‖ ∞ in mind; comparealso Section 3).Definitions. 1. Say that the metric d X satisfies the metric admissibility condition onH ⊂ X if, for any z n → e in H under d X and arbitrary y n ,d X (z n y n , y n ) → 0.2. If d X is left-invariant, the condition may be reformulated as a norm admissibilitycondition on H ⊂ X, since‖yn−1 z n y n ‖ = d X L (yn −1 z n y n , e) = d X L (z n y n , y n ) → 0. (H-adm)3. We will say that the group X satisfies the topological admissibility condition on H ⊂ Xif, for any z n → e in H and arbitrary y ny −1n z n y n → e.The next result extends the usage of ‖ · ‖ H beyond H to X itself (via the left shifts).Proposition 2.14 (Right-invariant sup-norm). For any metric d X on a group X, putH X : = H = {x ∈ X : sup z∈X d X (xz, z) < ∞},‖x‖ H : = sup d X (xz, z), for x ∈ H.For x, y ∈ H, let ¯d H (x, y) := ˆd H (λ x , λ y ) = sup z d X (xz, yz). Then:(i) ¯d H is a right-invariant metric on H, and ¯d H (x, y) = ‖xy −1 ‖ H = ‖λ x λ −1y ‖ H .(ii) If d X is left-invariant, then ¯d H is bi-invariant on H, and so ‖x‖ ∞ = ‖x‖ H and thenorm is abelian on H.