62 N. H. Bingham <strong>and</strong> A. J. OstaszewskiRemarks. 1. A sequential version may be formulated: call t an ε-shifting point for thenull sequence z n if there exists N(ε) such that for m > N(ε)d R (t, tz m ) < ε.Then t is an ε-shifting point iff t is ε-shifting for every null sequence. Indeed, if t isnot an ε-shifting point, then for each δ = 1/n there is z n with ‖z n ‖ < 1/n such thatd R (t, tz n ) ≥ ε, so t is not ε-shifting for this null sequence.2. In the notation associated with oscillation, t is ε-shifting for {z n } ift ∈ H ε ({z n }) := ⋃ n G ε(z n ).3. Evidently, if t is an ε-shifting point for each ε > 0, then γ t is continuous (being continuousat e) <strong>and</strong> so a member of the <strong>topological</strong> centre Z Γ (X) of the normed group.4. The sequential version is motivated by the Kestelman-Borwein-Ditor Theorem of Section1 (Th. 1.2) which, roughly speaking, says that tz n → t generically. (See Cor. 3.50.)5. In referring to this property, the theorem which follows assumes something less thanthat the centre Z Γ is dense, only that the open set H ε ({z n }) is dense for each ε > 0 <strong>and</strong>each {z n }.Theorem 3.48 (Dense Oscillation Theorem). In a normed group X⋂cl [Ω(1/n)] = ⋂ Ω(1/n) = Z Γ.n∈N n∈NHence, if for each ε > 0 the ε-shifting points are dense, equivalently Ω(ε) = {t : ω(t) < ε}is dense for each ε > 0, then the normed group is <strong>topological</strong>.More generally, if for some open W <strong>and</strong> all ε > 0 the set Ω(ε) ∩ W is dense in W, thenω = 0 on W ; in particular,(i) if e X ∈ W <strong>and</strong> X is connected <strong>and</strong> Baire under its norm topology, then X is a<strong>topological</strong> group,(ii) if X is separable, connected <strong>and</strong> <strong>topological</strong>ly complete in its norm topology, then Xis a <strong>topological</strong> group.Proof. The opening assertion follows from the continuity of ω. For ε > 0, if Ω(ε) is denseon W, then clW ⊆clΩ(ε). Hence, if Ω(ε) is dense on W for all ε > 0, clW ⊆ Z Γ . So ifW = X = Z Γ , i.e. γ s is continuous for all s ∈ X, then the conclusion follows from theEquivalence Theorem (Th. 3.4). For a more general W, the conclusion follows from Th.3.45.Remark. It is instructive to see how the density property of the last theorem bestowsthe ε-shifting property to nearby points. Fix s <strong>and</strong> ε > 0. For n > 1/ε, let t ∈ Ω(1/n) ∩B ε (s).Then for some δ = δ(n) we have ω δ (t) ≤ 1/n, equivalently d(tz, t) ≤ 1/n for‖z‖ ≤ δ, <strong>and</strong> so for such zd R (sz, s) ≤ d R (sz, tz) + d R (tz, t) + d R (t, s)≤ 2d R (s, t) + 1/n ≤ 3ε.
<strong>Normed</strong> <strong>groups</strong> 63Thus ω δ (s) ≤ 3ε. (Since ε > 0 was arbitrary, ω(s) = 0, so γ s is continuous, <strong>and</strong> sos ∈ Z R .) We use this idea several times over in the next result.We now give a necessary <strong>and</strong> sufficient criterion for a normed group to be <strong>topological</strong>by refering not to continuity, but to approximation of left-shifts by right-shifts. This turnsout to be equivalent both to a commutator condition <strong>and</strong> to a shifting property condition.An extended comment on the commutator condition is in order, because the conditionfinesses the descriptive character of the relation x = yz. Proposition 3.49 below employsthe following ‘commutator oscillation’ set (<strong>and</strong> its density):C(ε) : = ⋃ n∈N C 1/n(ε), whereC δ (ε) : = ⋂ ‖z‖≤δ {y : ‖zyz−1 y −1 ‖ ≤ ε } = ⋂ ‖z‖≤δ {y : d R(zy, yz) ≤ ε }.This is an ‘oscillation set’, since ‖γ y (z)‖ − ‖z‖ ≤ ‖[z, y] L ‖ ≤ ‖γ y (z)‖ + ‖z‖; indeed onemight refer to ¯ω(y) := lim δ↘0 sup ‖z‖≤δ ‖zyz −1 y −1 ‖, but for the fact that ¯ω(y) = ω(y).Furthermore, ⋂ n∈N C(1/n) = Z Γ, since for δ < ε we have the ‘inner regularity of C’:Λ δ (ε) ⊆ C δ (2ε), <strong>and</strong> the ‘outer regularity of C’: C δ (ε) ⊆ Λ δ (2ε). So, since density is thevehicle of proof, one may carry over the proof of the Montgomery Theorem (Th. 3.47) withcl[C δ (ε)] in lieu of Λ δ (ε). Note that these inclusions permit use of C δ (ε) even if the latterhas poor descriptive character (i.e. we do not need to know anything about the relationx = yz). Of course, for X separable <strong>and</strong> <strong>topological</strong>ly complete, if {(y, z) : d R (yz, zy) ≤ ε}has analytic graph, then the set C δ (ε) is co-analytic (complement of a Souslin-F set, seeSection 11 for background), becausey /∈ C δ (ε) ⇐⇒ (∃z ∈ B δ (e X ))[d R (yz, zy) ≤ ε].Under these circumstances, C δ (ε) is Baire by Nikodym’s Theorem (Th.11.5); but Prop.3.49 does not need this.The next result is, for normed <strong>groups</strong>, a sharpening of the Montgomery Theorem(Th. 3.47), in view of Montgomery’s Initial Observation above that, for a semi<strong>topological</strong>group, each set Ω(ε) ∩ W is non-meagre for W a non-empty open set (<strong>and</strong> in particulareach set Ω(ε) is dense). This arises from our use of d = d R , when Montgomery usesan arbitrary (compatible) metric d in Th. 3.46, <strong>and</strong> so relegates the implementation ofcategory to the last rather than an earlier step.Proposition 3.49 (Left-right Approximation Criterion). For W a non-empty right-opensubset of a normed group X, the following are equivalent:(a) For each t ∈ W <strong>and</strong> each η > 0, there are y η <strong>and</strong> δ > 0 such that d R (tz, zy η ) ≤ ηfor all ‖z‖ ≤ δ, i.e. for each t ∈ W the left-shift λ t may be locally approximated near theidentity by a right-shift ρ y .(b) For each ε > 0, the set C(ε) = {y : (∃δ > 0)[d R (yz, zy) ≤ ε for all all ‖z‖ ≤ δ]} isdense in W – i.e. C(ε) ∩ W is dense in W.(c) For each ε > 0, the set Ω(ε) = {t : (∃δ > 0)[d R (tz, t) < ε for all ‖z‖ ≤ δ]} is dense inW.Suppose that for each t ∈ W the left-shift λ t may be locally approximated near the identityby a right-shift. Then:
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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
- 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 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: Normed groups 61Proposition 3.46 (M
- 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
- 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
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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,