Normed versus topological groups: Dichotomy and duality

Normed groups 15definitions, our previous analysis allows the First Limit Theorem for subadditive functions(cf. Th. 2.8 and [BOst-GenSub]) to be restated in the context of normed groups.Proposition 2.10. Let ξ be infinitely divisible and embeddable in the one-parametersubgroup T (ξ) of X. Suppose that lim n→∞ ‖x n ‖ = ∞ for x ≠ e X . Then for any Bairesubadditive p : X → R + and t ∈ T (ξ),∂ T (ξ) p(t) := lim s∈T, ‖s‖→∞p(ts)‖s‖ = ‖p‖ T ,i.e., treating the subgroup T (ξ) as a direction, the limit function is determined by thedirection.Proof. By subadditivity, p(s) = p(t −1 ts) ≤ p(t −1 ) + p(ts), sop(s) − p(t −1 ) ≤ p(ts) ≤ p(t) + p(s).For s ∈ T with s ≠ e, divide through by ‖s‖ and let ‖s‖ → ∞ (as in Th. 2.8):‖p‖ T ≤ ∂ T p(t) ≤ ‖p‖ T .(We consider this in detail in Section 4.)Definition (Supremum metric, supremum norm). Let X have a metric d X . As beforeG is a fixed subgroup of Auth(X), for example T r L (X) the group of left-translations λ x(cf. Th. 3.12), defined byλ x (z) = xz.For g, h ∈ G, define the possibly infinite numberˆd G (g, h), or ˆd X (g, h) := sup x∈X d X (g(x), h(x)),where the notation identifies either the domain of the metric or the source metric d X .PutH(X) = H(X, G) := {g ∈ G : ˆd G (g, id X ) < ∞},and call these the bounded elements of G. We write ˆd H for the metric ˆd G restricted toH = H(X) and call ˆd H (g, h) the supremum metric on H; the associated norm‖h‖ H = ‖h‖ H(X) := ˆd H (h, id X ) = sup x∈X d X (h(x), x)is the supremum norm. This metric notion may also be handled in the setting of uniformities(cf. the notion of functions limited by a cover U arising in [AnB, Section 2];see also [BePe, Ch. IV Th. 1.2] ); in such a context excursions into invariant measuresrather than use of Haar measure (as in Section 6) would refer to corresponding resultsestablished by Itzkowitz [Itz] (cf. [SeKu, §7.4]).Our next result justifies the terminology of the definition above.