Normed versus topological groups: Dichotomy and duality

Normed groups 21so‖αβ‖ ≤ ‖α‖ + ‖β‖.We say that a function α : S → T is multiplicative if α is bounded andα(ss ′ ) = α(s)α(s ′ ).A function γ : S → T is asymptotically multiplicative if γ = αβ, where α is multiplicativeand bounded and β is bounded. In the commutative situation with S, T normed vectorspaces, the norm here reduces to the operator norm. This group-norm is studied extensivelyin [CSC] in relation to Ulam’s problem. We consider in Section 3.2 the case S = Tand functions α which are inner automorphisms. In Proposition 3.42 we shall see thatthe oscillation of a group X is a bounded function from X to R in the sense above.Proposition 2.16 (Magnification metric). Let T = H(X) with group-norm ‖t‖= d T (t, e T ) = d H (t, e T ) and A a subgroup (under composition) of Auth(T ) (so, fort ∈ T and α ∈ A, α(t) ∈ H(X) is a homeomorphism of X). For any ε ≥ 0, putd ε A(α, β) := sup ‖t‖≤ε ˆdT (α(t), β(t)).Suppose further that X distinguishes the maps {α(e H(X) ) : α ∈ A}, i.e., for α, β ∈ A,there is z = z α,β ∈ X with α(e H(X) )(z) ≠ β(e H(X) )(z).Then d ε A (α, β) is a metric; furthermore, dε A is right-invariant for translations by γ ∈ Asuch that γ −1 maps the ε-ball of X to the ε-ball.Proof. To see that this is a metric, note that for t = e H(X) = id T we have ‖t‖ = 0 andˆd T (α(e H(X) ), β(e H(X) )) = sup z d X (α(e H(X) )(z), β(e H(X) )(z))≥ d X (α(e H(X) )(z α,β ), β(e H(X) )(z α,β )) > 0.Symmetry is clear. Finally the triangle inequality follows as usual:d ε A(α, β) = sup ‖t‖≤1 ˆdT (α(t), β(t)) ≤ sup ‖t‖≤1 [ ˆd T (α(t), γ(t)) + ˆd T (γ(t), β(t))]≤ sup ‖t‖≤1 ˆdT (α(t), γ(t)) + sup ‖t‖≤1 ˆdT (γ(t), β(t))= d ε A(α, γ) + d ε A(γ, β).One cannot hope for the metric to be right-invariant in general, but if γ −1 maps theε-ball to the ε-ball, one hasd ε A(αγ, βγ) = sup ‖t‖≤ε ˆdT (α(γ(t)), β(γ(t))= sup ‖γ −1 (s)‖≤ε ˆd T (α(s), β(s)).In this connection we note the following.Proposition 2.17. In the setting of Proposition 2.16, denote by ‖.‖ ε the norm inducedby d ε A ; then sup ‖t‖≤ε ‖γ(t)‖ T − ε ≤ ‖γ‖ ε ≤ sup ‖t‖≤ε ‖γ(t)‖ T + ε.