Normed versus topological groups: Dichotomy and duality

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96 N. H. Bingham and A. J. OstaszewskiWithout loss of generality u n → u ∈ K. Now ‖ux n ‖ → ∞, as ‖x n ‖ − ‖u‖ ≤ ‖ux n ‖, bythe triangle inequality. Thus we may put y = {y n } where y n := ux n ; thenT k (y) := ⋂ n>k {t ∈ T : h(tux n)h(ux n ) −1 < n},and NT ∗ (T k (y)) holds. Now z n := u n u −1 is null. So for some k ∈ ω, t ∈ T k (y) andinfinite M,{t(u m u −1 ) : m ∈ M} ∈ T k (y).Soh(tu m u −1 ux m )h(ux m ) −1 < m and t ∈ T.Now ‖u n x n ‖ → ∞, as ‖x n ‖ − ‖u n ‖ ≤ ‖u n x n ‖ and ‖u n ‖ is bounded. But t ∈ T so, asbefore since h ∗ (t) < ∞, for all n large enoughh(tu n x n )h(u n x n ) −1 < n.Now also u ∈ K ⊂ T. So for all n large enoughButh(ux n )h(x n ) −1 < n.h(u n x n )h(x n ) −1 = h(u n x n )h(tu n x n ) −1 × h(tu n x n )h(ux n ) −1 × h(ux n )h(x n ) −1 .Then for m large enough and in M t we havea contradiction for such m to (**).h(u m x m )h(x m ) −1 < m 3 ,We note a generalization with an almost verbatim proof (requiring, mutatis mutandis,the replacement of h(ux)h(x) −1 by ‖h(ux)h(x) −1 ‖). Note that one cannot deduce Th.6.7A from this variant by referring to the normed group Y = R ∗ +, because the naturalnorm on R ∗ + is ‖x‖ Y = | log x| (cf. Remarks to Corollary 2.9).Theorem 7.9B (Combinatorial Uniform Boundedness Theorem). For h : X → Y amapping between normed topological groups and h ∗ Y (u) as above, suppose that h∗ (t) < ∞on a set T on which h is NT ∗ . Then for compact K ⊂ Tlim sup ‖x‖→∞ sup u∈K ‖h(ux)h(x) −1 ‖ < ∞.We may now deduce the result referred to in the remarks to Corollary 2.9, regardingπ : X → Y a group homomorphism, by reference to the case h(x) = π(x) treated in theLemma below.Theorem 7.10 (NT ∗ property of quasi-isometry). If X is a Baire normed topologicalgroup and π : X → Y a group homomorphism, where ‖.‖ Y is (µ-γ)-quasi-isometric to‖.‖ X under the mapping π, then for any non-meagre Baire set T , π is NT ∗ on T.
Normed groups 97Proof. Note that‖h(tx n )h(x n ) −1 ‖ = ‖π(tx n )π(x n ) −1 ‖ = ‖π(t)‖.Hence, as π(e) = e (see Examples A4 of Section 2),and soNow{t ∈ T : h(tx n )h(x n ) −1 < n} = {t ∈ T : ‖π(t)‖ < n} = B π n(e),⋂n≥k T n(x n ) = {t ∈ T : ‖π(t)‖ < k} = B π k (e).1µ ‖t‖ X − γ ≤ ‖π(t)‖ Y ≤ 1 µ ‖t‖ X + γ,hence B π n(e) is approximated from above and below by the closed sets T ± n :T + n := {t ∈ T : 1 µ ‖t‖ X + γ ≤ n} ⊂ T (x n ) = B π n(e) ⊂ T − n := {t ∈ T : 1 µ ‖t‖ X − γ ≤ n},which yields the equivalent approximation:Hence,¯B µ(k−γ) ∩ T = {t ∈ T : ‖t‖ X ≤ µ(k − γ)} = ⋂ n≥k T + n⊂ T k (x) ⊂ ⋂ n≥k T − n = {t ∈ T : ‖t‖ X ≤ µ(k + γ)} = T ∩ ¯B µ(k+γ) .T = ⋃ k T k(x) = ⋃ k T ∩ ¯B µ(k+γ) .Hence, by the Baire Category Theorem, for some k the set T k (x) contains a Bairenon-meagre set ¯B µ(k−γ) ∩ T and the proof of Th. 7.8 applies. Indeed if T ∩ ¯B µ(k ′ +γ) isnon-meagre for some k ′ , then so is T ∩ ¯B µ(k ′ +γ) for k ≥ k ′ + 2γ and hence also T k (x) isso.Theorem 7.11 (Global bounds at infinity – Global Bounds Theorem). Let X be a locallycompact topological group with with norm having a vanishingly small global word-net.For h : X → R + , if h ∗ is globally bounded, i.e.h ∗ (u) = lim sup ‖x‖→∞ h(ux)h(x) −1 < B (u ∈ X)for some positive constant B, independent of u, then there exist constants K, L, M suchthath(ux)h(x) −1 < ‖u‖ K (u ≥ L, ‖x‖ ≥ M).Hence h is bounded away from ∞ on compact sets sufficiently far from the identity.Proof. As X is locally compact, it is a Baire space (see e.g. [Eng, Section 3.9]). Thus,by Th. 7.8, the Combinatorial Uniform Boundedness Theorem Th. 7.9A may be appliedwith T = X to a compact closed neighbourhood K = ¯B ε (e X ) of the identity e X , wherewithout loss of generality 0 < ε < 1; hence we havelim sup ‖x‖→∞ sup u∈K h(ux)h(x) −1 < ∞.
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N. H. BINGHAM and A. J. OSTASZEWSKI
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Normed groups 3ContentsContents . .
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1. IntroductionGroup-norms, which b
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Normed groups 3Topological complete
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Normed groups 5abelian group has se
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Normed groups 74 (Topological permu
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Normed groups 9The following result
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Normed groups 11Corollary 2.4. For
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Normed groups 13More generally, for
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Normed groups 15definitions, our pr
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Normed groups 17so that fg is in th
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Normed groups 19(iii) The ¯d H -to
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Normed groups 21so‖αβ‖ ≤
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Normed groups 23Remark. Note that,
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Normed groups 25shows that [z n , y
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Normed groups 27Denoting this commo
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Normed groups 29Theorem 3.4 (Equiva
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Normed groups 31argument as again p
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Normed groups 33(ii) For α ∈ H u
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Normed groups 35Definition. A group
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Normed groups 37We now give an expl
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Normed groups 39Theorem 3.19 (Abeli
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Normed groups 412. Further recall t
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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
Page 65 and 66: Normed groups 61Proposition 3.46 (M
Page 67 and 68: Normed groups 63Thus ω δ (s) ≤
Page 69 and 70: Normed groups 65Remark. In the penu
Page 71 and 72: Normed groups 67The result confirms
Page 73 and 74: Normed groups 69Proof. By the Baire
Page 75 and 76: Normed groups 715. Generic Dichotom
Page 77 and 78: Normed groups 73Returning to the cr
Page 79 and 80: Normed groups 75Examples. Here are
Page 81 and 82: Normed groups 77cf. [Eng, 4.3.23].)
Page 83 and 84: Normed groups 79Remarks. 1. See [Fo
Page 85 and 86: Normed groups 81Theorem 6.1 (Catego
Page 87 and 88: Normed groups 83is continuous at th
Page 89 and 90: Normed groups 85compact. Evidently,
Page 91 and 92: Normed groups 87j ∈ ω} which enu
Page 93 and 94: Normed groups 89The result below ge
Page 95 and 96: Normed groups 91left-shift, not in
Page 97 and 98: Normed groups 93As a corollary of t
Page 99: Normed groups 953. For X a normed g
Page 103 and 104: Normed groups 99Taking h(x) := ‖
Page 105 and 106: Normed groups 1019. The Semigroup T
Page 107 and 108: Normed groups 103Theorem 9.5 (Semig
Page 109 and 110: Normed groups 105By the Category Em
Page 111 and 112: Normed groups 107Proof. Say f is bo
Page 113 and 114: Normed groups 109Thus G is locally
Page 115 and 116: Normed groups 111Theorem 10.10 (Bar
Page 117 and 118: Normed groups 113K-analyticity was
Page 119 and 120: Normed groups 115Theorem 11.6 (Disc
Page 121 and 122: Normed groups 117restricted to X\M
Page 123 and 124: Normed groups 119groups need not be
Page 125 and 126: Normed groups 121Proof. In the meas
Page 127 and 128: Normed groups 123Hence, as t i n
Page 129 and 130: Normed groups 125The corresponding
Page 131 and 132: Normed groups 127(t, x) ✛✻Φ T
Page 133 and 134: Normed groups 129Fix s. Since s is
Page 135 and 136: Normed groups 131Hence,‖x‖ −
Page 137 and 138: Normed groups 133converging to the
Page 139 and 140: Normed groups 135Definition. Let {
Page 141 and 142: Normed groups 137However, whilst th
Page 143 and 144: Normed groups 139embeddable, 14enab
Page 145 and 146: Normed groups 141Bibliography[AL]J.
Page 147 and 148: Normed groups 143Series 378, 2010.[
Page 149 and 150: 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,
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Normed versus topological groups: Dichotomy and duality

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