10 N. H. Bingham <strong>and</strong> 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‖, <strong>and</strong> 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 <strong>and</strong> with the notation d(x, y) = ‖xy −1 ‖ etc., themapping x → x −1 from (X, d X R ) → (X, dX L ) is an isometry <strong>and</strong> so a homeomorphism, asd L (x −1 , y −1 ) = d R (x, y).The two (inversion) conjugate metrics separately define a right <strong>and</strong> 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 <strong>topological</strong> <strong>groups</strong> of being complete – see [Br-1], [Hal-ET, p. 63] (the caseof measure algebras), [Kel, Ch. 6 Problem Q] , <strong>and</strong> 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 <strong>and</strong> 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 ).
<strong>Normed</strong> <strong>groups</strong> 11Corollary 2.4. For π a homomorphism, the normed <strong>groups</strong> X, Y are (µ-γ)-quasiisometricunder π for the corresponding metrics iff the associated norms are (µ-γ)-quasiequivalent,i.e.1µ ‖x‖ X − γ ≤ ‖π(x)‖ Y ≤ µ‖x‖ X + γ (a, b ∈ X),d Y (y, π[X]) ≤ γ (y ∈ Y ).Proof. This follows from π(e X ) = e Y <strong>and</strong> π(xy −1 ) = π(x)π(y) −1 .Remark. Note that p(x) = ‖π(x)‖ Y is subadditive <strong>and</strong> bounded at x = e. It will followthat p is locally bounded at every point when we later prove Lemma 4.3.The following result (which we use in [BOst-TRII]) clarifies the relationship betweenthe conjugate metrics <strong>and</strong> the group structure. We define the ε-swelling of a set K in ametric space X for a given (e.g. right-invariant) metric d X , to beB ε (K) := {z : d X (z, k) < ε for some k ∈ K} = ⋃ k∈K B ε(k)<strong>and</strong> for the conjugate (resp. left-invariant) case we can write similarlyWe write B ε (x 0 ) for B ε ({x 0 }), so that˜B ε (K) := {z : ˜d X (z, k) < ε for some k ∈ K}.B ε (x 0 ) := {z : ‖zx −10 ‖ < ε} = {wx 0 : w = zx −10 , ‖w‖ < ε} = B ε(e)x 0 .When x 0 = e X , the ball B ε (e X ) is the same under either of the conjugate metrics, asB ε (e X ) := {z : ‖z‖ < ε}.Proposition 2.5. (i) In a locally compact group X, for K compact <strong>and</strong> for ε > 0 smallenough so that the closed ε-ball B ε (e X ) is compact, the swelling B ε/2 (K) is pre-compact.(ii) B ε (K) = {wk : k ∈ K, ‖w‖ X < ε} = B ε (e X )K, where the notation refers toswellings for d X a right-invariant metric; similarly, for ˜d X , the conjugate metric, ˜B ε (K)= KB ε (e X ).Proof. (i) If x n ∈ B ε/2 (K), then we may choose k n ∈ K with d(k n , x n ) < ε/2. Withoutloss of generality k n converges to k. Thus there exists N such that, for n > N,d(k n , k) d X (z, k) =d X (zk −1 , e), then, putting w = zk −1 , we have z = wk ∈ B ε (K).
- Page 1 and 2: N. H. BINGHAM and A. J. OSTASZEWSKI
- Page 3 and 4: Normed groups 3ContentsContents . .
- Page 5 and 6: 1. IntroductionGroup-norms, which b
- Page 7 and 8: Normed groups 3Topological complete
- Page 9 and 10: Normed groups 5abelian group has se
- Page 11 and 12: Normed groups 74 (Topological permu
- Page 13: Normed groups 9The following result
- Page 17 and 18: Normed groups 13More generally, for
- Page 19 and 20: Normed groups 15definitions, our pr
- Page 21 and 22: Normed groups 17so that fg is in th
- Page 23 and 24: Normed groups 19(iii) The ¯d H -to
- Page 25 and 26: Normed groups 21so‖αβ‖ ≤
- Page 27 and 28: Normed groups 23Remark. Note that,
- Page 29 and 30: Normed groups 25shows that [z n , y
- Page 31 and 32: Normed groups 27Denoting this commo
- Page 33 and 34: Normed groups 29Theorem 3.4 (Equiva
- Page 35 and 36: Normed groups 31argument as again p
- Page 37 and 38: Normed groups 33(ii) For α ∈ H u
- Page 39 and 40: Normed groups 35Definition. A group
- Page 41 and 42: Normed groups 37We now give an expl
- Page 43 and 44: Normed groups 39Theorem 3.19 (Abeli
- Page 45 and 46: Normed groups 412. Further recall t
- Page 47 and 48: Normed groups 43Theorem 3.22 (Lipsc
- Page 49 and 50: Normed groups 45Proof. Z γ = G (cf
- Page 51 and 52: Normed groups 47Theorem 3.30. Let G
- Page 53 and 54: Normed groups 49Remark. On the matt
- Page 55 and 56: Normed groups 51As for the conclusi
- Page 57 and 58: Normed groups 53By (C-adm), we may
- Page 59 and 60: Normed groups 55equipped with an in
- Page 61 and 62: Normed groups 57Proof. To apply Th.
- Page 63 and 64: Normed groups 59Definition. A point
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Normed groups 61Proposition 3.46 (M
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Normed groups 63Thus ω δ (s) ≤
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Normed groups 65Remark. In the penu
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Normed groups 67The result confirms
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Normed groups 69Proof. By the Baire
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Normed groups 715. Generic Dichotom
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Normed groups 73Returning to the cr
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Normed groups 75Examples. Here are
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Normed groups 77cf. [Eng, 4.3.23].)
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Normed groups 79Remarks. 1. See [Fo
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Normed groups 81Theorem 6.1 (Catego
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Normed groups 83is continuous at th
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Normed groups 85compact. Evidently,
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Normed groups 87j ∈ ω} which enu
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Normed groups 89The result below ge
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Normed groups 91left-shift, not in
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Normed groups 93As a corollary of t
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Normed groups 953. For X a normed g
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Normed groups 97Proof. Note that‖
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Normed groups 99Taking h(x) := ‖
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Normed groups 1019. The Semigroup T
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Normed groups 103Theorem 9.5 (Semig
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Normed groups 105By the Category Em
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Normed groups 107Proof. Say f is bo
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Normed groups 109Thus G is locally
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Normed groups 111Theorem 10.10 (Bar
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Normed groups 113K-analyticity was
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Normed groups 115Theorem 11.6 (Disc
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Normed groups 117restricted to X\M
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Normed groups 119groups need not be
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Normed groups 121Proof. In the meas
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Normed groups 123Hence, as t i n
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Normed groups 125The corresponding
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Normed groups 127(t, x) ✛✻Φ T
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Normed groups 129Fix s. Since s is
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Normed groups 131Hence,‖x‖ −
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Normed groups 133converging to the
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Normed groups 135Definition. Let {
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Normed groups 137However, whilst th
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Normed groups 139embeddable, 14enab
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Normed groups 141Bibliography[AL]J.
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Normed groups 143Series 378, 2010.[
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Normed groups 145abelian groups, Ma
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Normed groups 147[Kak] S. Kakutani,
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Normed groups 149fields. I. Basic p
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Normed groups 151[So]R. M. Solovay,