Normed versus topological groups: Dichotomy and duality

84 N. H. Bingham and A. J. OstaszewskiLikewise, for quasi all t ∈ T there is an infinite set M t such that{tz m : m ∈ M t } ⊆ T.Proof. Apply Th. 6.2, taking for d a right-invariant metric, d X R say; the continuous mapsψ n (t) = z n t satisfy d X R (z nt, t) = ‖z n ‖ H → 0, so converge to the identity. Likewise takingfor d a left-invariant metric d X L say, the continuous maps ψ n(t) = tz n satisfy d X R (tz n, t) =‖z n ‖ H → 0, so again converge to the identity.As a corollary of the KBD Theorem of Section 5 (Th. 5.1) we have the followingimportant result known for topological groups (see [RR-TG], Rogers [Jay-Rog, p. 48], and[Kom1] for the topological vector space setting) and here proved in the metric setting.Theorem 6.5 (Piccard-Pettis Theorem – Piccard [Pic1], [Pic2], Pettis [Pet1], [RR-TG] cf.[BOst-TRII]). In a normed topological group whose norm topology is Baire: for A Baireand non-meagre (in the norm topology), the sets AA −1 and A −1 A both have non-emptyinterior.Proof. Suppose otherwise. We work first with the right-invariant metric d R (x, y) =‖xy −1 ‖ and assume A −1 is Baire non-meagre in the right-norm topology. Consider theset A −1 A. Suppose the conclusion fails for A −1 A, for each integer n = 1, 2, ... there isz n ∈ B 1/n (e)\A −1 A; hence z n → z 0 = e. Applying either the KBD Theorem for topologicallycomplete normed groups or its variant for topological groups, there is a ∈ A suchthat for infinitely many naz n ∈ A, or z n ∈ A −1 A,a contradiction. Thus, for some n, the open ball B 1/n (e) is contained in A −1 A. We nextconsider the set AA −1 . As the inversion mapping x → x −1 is a homeomorphism (infact an isometry, see Prop. 2.3) from the right- to the left-norm topology, the set A isBaire non-meagre in the left-norm topology iff A −1 is Baire non-meagre in the right-normtoplogy. But the inversion mapping carries the ball B 1/n (e) into itself, and so we maynow conclude that AA −1 contains an open ball B 1/n (e), as (A −1 ) −1 = A.One says that a set A is thick if e is an interior point of AA −1 (see e.g. [HJ, Section3.4] ). The next result (proved essentially by the same means) applied to the additivegroup R implies the Kestelman-Borwein-Ditor ([BOst-LBII]) theorem on the line. Thename used here refers to a similar (weaker) property studied in Probability Theory (inthe context of probabilities regarded as a semigroup under convolution, for which see[PRV], or [Par, 3.2 and 3.5], [BlHe], [Hey]). We need a definition.Definition (cf.[BOst-StOstr]). In a normed topological group G, say that a set A is(properly) right-shift compact, resp. strongly right-shift compact if, for any sequence ofpoints a n in A, (resp. in G) there is a point t and a subsequence {a n : n ∈ M t } such thata n t lies entirely in A and converges through M t to a point a 0 t in A; similarly for left-shift