32 N. H. Bingham <strong>and</strong> A. J. OstaszewskiTheorem 3.9 (Ambidextrous Refinement). For X a normed group with norm ‖.‖, putd X S (x, y) := max{‖xy −1 ‖, ‖x −1 y‖} = max{d X R (x, y), d X L (x, y)}.Then X is a <strong>topological</strong> group under the right (or left) norm topology iff X is a <strong>topological</strong>group under the symmetrization refinement metric d X S iff the topologies of dX S <strong>and</strong> of dX Rare identical.Proof. Suppose that under the right-norm topology X is a <strong>topological</strong> group. Then d X L isd X R -continuous, by Th. 3.4 (continuity of inversion), <strong>and</strong> hence dX S is also dX R -continuous.Thus if x n → x under d X R , then also, by continuity of dX L , one has x n → x under d X S .Now if x n → x under d X S , then also x n → x under d X R , as dX R ≤ dX S . Thus dX S generatesthe topology <strong>and</strong> so X is a <strong>topological</strong> group under d X S .Conversely, suppose that X is a <strong>topological</strong> group under d X S . As X is a <strong>topological</strong> group,its topology is generated by the neighbourhoods of the identity. But as already noted,d X S (x, e) := ‖x‖,so the d X S -neighbourhoods of the identity are also generated by the norm; in particularany left-open set aB(ε) is d X S -open (as left shifts are homeomorphisms) <strong>and</strong> so right-open(being a union of right shifts of neighbourhoods of the identity). Hence by Th. 3.4 (orTh. 3.3) X is a <strong>topological</strong> group under either norm topology.As for the final assertion, if the d X S topology is identical with the dX R topology theninversion is d X R -continuous <strong>and</strong> so X is a <strong>topological</strong> group by Th. 3.4. The argument ofthe first paragraph shows that if d X R makes X into a <strong>topological</strong> group then dX R <strong>and</strong> dX Sgenerate the same topology.Thus, according to the Ambidextrous Refinement Theorem, a symmetrization thatcreates a <strong>topological</strong> group structure from a norm structure is in fact redundant. We areabout to see such an example in the next theorem.Given a metric space (X, d), we let H unif (X) denote the subgroup of uniformly continuoushomeomorphisms (relative to d), i.e. homeomorphisms α satisfying the conditionthat, for each ε > 0, there is δ > 0 such thatd(α(x), α(x ′ )) < ε, for d(x, x ′ ) < δ.(u-cont)Lemma 3.10 (Compare [dGMc, Cor. 2.13]). (i) For fixed ξ ∈ H(X), the mapping ρ ξ :α → αξ is continuous.(ii) For fixed α ∈ H unif (X), the mapping λ α : β → αβ is in H unif (X) – i.e. is uniformlycontinuous.(iii) The mapping (α, β) → αβ is continuous from H unif (X) × H unif (X) to H(X) underthe supremum norm.Proof. (i) We haveˆd(αξ, βξ) = sup d(α(ξ(t)), β(ξ(t))) = sup d(α(s), β(s)) = ˆd(α, β).
<strong>Normed</strong> <strong>groups</strong> 33(ii) For α ∈ H unif (X) <strong>and</strong> given ε > 0, choose δ > 0, so that (u-cont) holds. Then, forβ, γ with ˆd(β, γ) < δ, we have d(β(t), γ(t)) < δ for each t, <strong>and</strong> henceˆd(αβ, αγ) = sup d(α(β(t)), α(γ(t))) ≤ ε.(iii) Again, for α ∈ H unif (X) <strong>and</strong> given ε > 0, choose δ > 0, so that (u-cont) holds.Thus, for β, η with ˆd(β, η) < δ, we have d(β(t), η(t)) < δ for each t. Hence for ξ withˆd(α, ξ) < ε we obtainConsequently, we haved(α(β(t)), ξ(η(t))) ≤ d(α(β(t)), α(η(t))) + d(α(η(t)), ξ(η(t)))≤ ε + ˆd(α, ξ) ≤ ε + ε.ˆd(αβ, ξη) = sup d(α(β(t)), ξ(η(t))) ≤ 2ε.Comment. See also [AdC] for a discussion of the connection between choice of metric<strong>and</strong> uniform continuity. The following result is of interest.Proposition 3.11 (deGroot-McDowell Lemma, [dGMc, Lemma 2.2]). Given Φ, a countablefamily of self-homeomorphism of X closed under composition (i.e. a semigroup inAuth(X)), the metric on X may be replaced by a <strong>topological</strong>ly equivalent one such thateach α ∈ Φ is uniformly continuous.Definition. Say that a homeomorphism h is bi-uniformly continuous if both h <strong>and</strong> h −1are uniformly continuous. WriteH u = {h ∈ H unif : h −1 ∈ H unif }.Proposition 3.12 (Group of left-shifts). For a normed <strong>topological</strong> group X with rightinvariantmetric d X , the group T r L (X) of left-shifts is (under composition) a subgroupof H u (X) that is isometric to X.Proof. As X is a <strong>topological</strong> group, we have T r L (X) ⊆ H u (X) by Cor. 3.6; T r L (X) is asubgroup <strong>and</strong> λ : X → T r L (X) is an isomorphism, becauseλ x ◦ λ y (z) = λ x (λ y (z)) = x(λ y (z) = xyz = λ xy (z).Moreover, λ is an isometry, as d X is right-invariant; indeed, we haved T (λ x , λ y ) = sup z d X (xz, yz) = d X (x, y).We now offer a generalization which motivates the <strong>duality</strong> considerations of Section12.
- 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: Normed groups 31argument as again p
- 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 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
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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,