48 N. H. Bingham <strong>and</strong> A. J. OstaszewskiThus ξ = 0, <strong>and</strong> hence x n → e. So our assumption of (iii) yieldsa final contradiction.‖g n x n gn−1 ‖(1 + ε) ≤ lim n→∞ = 1,‖x n ‖We note the following variant on Theorem 3.30.Theorem 3.31. Let G be a Lipschitz-normed <strong>topological</strong> group. The following are equivalent:(i) the mapping g → M(g) is continuous,(ii) the mapping g → M(g) is continuous at e,(iii) the norm is nearly abelian, i.e. for arbitrary g n → e <strong>and</strong> z n → elim n ‖g n z n g −1n ‖/‖z n ‖ = 1.Proof. Clearly (i)=⇒ (ii). To prove (ii)=⇒ (iii), suppose the mapping is continuous ate, then by the continuity of the maximization operation (cf. [Bor, Ch.12] ) g → M g iscontinuous at e, <strong>and</strong> Theorem 3.30 applies.To prove (iii)=⇒ (ii), assume the condition; it now suffices by Theorem 3.30 to provelower semicontinuity (lsc) at g = e. So suppose that, for some open U, U ∩ M(e) ≠ ∅.Thus U ∩ (1, ∞) ≠ ∅. Choose m ′ < m ′′ with 1 < m such that (m ′ , m ′′ ) ⊂ U ∩ M(e). IfM is not lsc at e, then there is g n → e such(m ′ , m ′′ ) ∩ M(g n ) = ∅.Take, e.g., m := 1 2 (m′ + m ′′ ). As m ′ < m < m ′′ , there is x n ≠ e such thatm‖x n ‖ < ‖g n x n gn−1 ‖.As before, if ‖x n ‖ is unbounded we may assume ‖x n ‖ → ∞, <strong>and</strong> so obtain the contradiction‖g n ‖ + ‖x n ‖ + ‖gn−1 ‖1 < m ≤ lim n→∞ = 1.‖x n ‖Now assume ‖x n ‖ → ξ ≥ 0. If ξ > 0 we have the contradiction‖g n ‖ + ‖x n ‖ + ‖gn−1 ‖m ≤ lim n→∞ = 0 + ξ + 0 = 1.‖x n ‖ξThus ξ = 0. So we obtain x n → 0, <strong>and</strong> now deduce thatagain a contradiction.‖g n x n gn−1 ‖1 < m ≤ lim n→∞ = 1,‖x n ‖
<strong>Normed</strong> <strong>groups</strong> 49Remark. On the matter of continuity a theorem of Mueller ([Mue, Th. 3] , see Th. 4.6below) asserts that in a locally compact group a subadditive p satisfyingis continuous almost everywhere.lim inf x→e (lim sup y→x p(y)) ≤ 03.3. Cauchy <strong>Dichotomy</strong>. In this section we demonstrate the impact on the structureof normed <strong>groups</strong> of the classic Cauchy dichotomy of homomorphisms; conjugacy is ahomomorphism so, in an appropriate setting, it is either continuous or highly discontinuous<strong>and</strong> so pathological, as mentioned in the Introduction. Thus, since conjugacy is atthe heart of normed <strong>groups</strong>, normed <strong>groups</strong> are in turn either <strong>topological</strong> or pathological(see e.g. Theorems 3.39 - 3.41 below, inspired by automatic continuity). The key here isDarboux’s classical result on automatic continuity [Dar], that an additive function on thereals is continuous if it is locally bounded, <strong>and</strong> its later weakening via ‘regularity’ to theBaire property (for which see below) or Haar-measurability in a locally compact context.Our aim here is to develop connections between continuity of automorphisms <strong>and</strong> threeareas: completeness, the Baire property <strong>and</strong> ‘boundedness’ of automorphisms. In respectof completeness, the findings (see e.g. Th. 3.37) are in keeping with tradition as exemplifiedby [Kel, Problem 6.Q]; the Baire property yields normed <strong>groups</strong> as ‘automatically<strong>topological</strong>’ (see the earlier cited theorems); boundedness is far more illuminating – inparticular we see that if the KBD holds in the right norm topology with a left-shift inthe st<strong>and</strong>ard form (i.e., as in Th.1.1, with tz m in lieu of t +z m ), as opposed to the specialform of Th.1.2 (yet to be established in Section 5), then the normed group is <strong>topological</strong>.In view of the importance of the Baire property to subsequent arguments, we recallthat a set is meagre if it is a countable union of nowhere dense sets, a set is Baireif it is open modulo a meagre set, or equivalently if it is closed modulo a meagre set(cf. Engelking [Eng] especially p.198 Section 4.9 <strong>and</strong> Exercises 3.9.J, although we prefer‘meagre’ to ‘of first category’ ). For examples, see Section 11.Definition. Noting that d L (x n , x m ) = d R (x −1n , x −1m ), call a sequence {x n } bi-Cauchy(or two-sided Cauchy) if {x n } is both d R - <strong>and</strong> d L -Cauchy, i.e. both {x n } <strong>and</strong> {x −1n }are d R -Cauchy sequences. Thus a sequence is bi-Cauchy iff it is d S -Cauchy, where d S =max{d R , d L } is the symmetrization metric. Recall that d S induces a norm, the originatingnorm, as ‖x‖ S := d S (x, e) = ‖x‖, but does not in general induce the same topology asd R .Discontinuity of automorphisms may be approached through bi-Cauchy sequences.Indeed, if a normed group is not <strong>topological</strong>, then by Th. 3.4 there are a null sequencez n , a point t, <strong>and</strong> ε > 0 such thatε ≤ lim n ‖tz n t −1 ‖;then d R (tz n , t) ≥ ε, <strong>and</strong> tz n is prevented from converging to t. Thus one asks whether{tz n } has a convergent subsequence {tz m } m∈M (such a sequence would be bi-Cauchy, ast −1 } is Cauchy, since for w n null w n x → R x, for any x).{z −1n
- 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 and 14: Normed groups 9The following result
- Page 15 and 16: Normed groups 11Corollary 2.4. For
- 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: Normed groups 47Theorem 3.30. Let G
- 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 and 100: Normed groups 953. For X a normed g
- Page 101 and 102: Normed groups 97Proof. Note that‖
- 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
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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
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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
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Normed groups 139embeddable, 14enab
- Page 145 and 146:
Normed groups 141Bibliography[AL]J.
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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,