Normed versus topological groups: Dichotomy and duality

10 N. H. Bingham and A. J. OstaszewskiAs for the converse, given a right-invariant metric d, put ‖x‖ := d(e, x). Now ‖x‖ =d(e, x) = 0 iff x = e. Next, ‖x −1 ‖ = d(e, x −1 ) = d(x, e) = ‖x‖, and sod(xy, e) = d(x, y −1 ) ≤ d(x, e) + d(e, y −1 ) = ‖x‖ + ‖y‖.Also d(xa, ya) = ‖xaa −1 y −1 ‖ = d(x, y).If d is bi-invariant iff d(e, yx −1 ) = d(x, y) = d(e, x −1 y) iff ‖yx −1 ‖ = ‖x −1 y‖. Invertingthe first term yields the abelian property of the group-norm.Finally, for (X, ‖ · ‖) a normed group and with the notation d(x, y) = ‖xy −1 ‖ etc., themapping x → x −1 from (X, d X R ) → (X, dX L ) is an isometry and so a homeomorphism, asd L (x −1 , y −1 ) = d R (x, y).The two (inversion) conjugate metrics separately define a right and left uniformity;their common refinement is the symmetrized metricd X S (x, y) := max{d X R (x, y), d X L (x, y)},defining what is known as the ambidextrous uniformity, the only one of the three capablein the case of topological groups of being complete – see [Br-1], [Hal-ET, p. 63] (the caseof measure algebras), [Kel, Ch. 6 Problem Q] , and also [Br-2]. We return to these mattersin Section 3. Note thatd X S (x, e X ) = d X R (x, e X ) = d X L (x, e X ),i.e. the symmetrized metric defines the same norm.Definitions. 1. For d X R a right-invariant metric on a group X, we are justified by Proposition2.2 in defining the g-conjugate norm from the g-conjugate metric by‖x‖ g := d X g (x, e X ) = d X R (gx, g) = d X R (gxg −1 , e X ) = ‖gxg −1 ‖.2. For ∆ a family of right-invariant metrics on X we put Γ = {‖.‖ d : D ∈ ∆}, the set ofcorresponding norms defined by‖x‖ d := d(x, e X ), for d ∈ ∆.The refinement norm is then, as in Proposition 2.1,‖x‖ Γ := sup d∈∆ d(x, e X ) = sup d∈Γ ‖x‖ d .We will be concerned with special cases of the following definition.Definition ([Gr1], [Gr2], [BH, Ch. I.8]). For constants µ ≥ 1, γ ≥ 0, the metric spacesX and Y are said to be ( µ-γ)-quasi-isometry under the mapping π : X → Y if1µ dX (a, b) − γ ≤ d Y (πa, πb) ≤ µd X (a, b) + γ (a, b ∈ X),d Y (y, π[X]) ≤ γ (y ∈ Y ).