4 N. H. Bingham <strong>and</strong> A. J. Ostaszewski2. Metric <strong>versus</strong> normed <strong>groups</strong>This section is devoted to group-norms <strong>and</strong> their associated metrics. We collect heresome pertinent information (some of which is scattered in the literature). A central toolfor applications is the introduction of the subgroup of bounded homeomorphisms of agiven group G of self-homeomorphisms of a <strong>topological</strong> group X; the subgroup possessesa guaranteed right-invariant metric. This is the archetypal example of the symbiosis ofnorms <strong>and</strong> metrics, <strong>and</strong> it bears repetition that, in applications just as here, it is helpfulto work simultaneously with a right-invariant metric <strong>and</strong> its associated group-norm.We say that the group X is normed if it has a group-norm as defined below (cf.[DDD]).Definition. We say that ‖ · ‖ : X → R + is a group-norm if the following propertieshold:(i) Subadditivity (Triangle inequality): ‖xy‖ ≤ ‖x‖ + ‖y‖;(ii) Positivity: ‖x‖ > 0 for x ≠ e <strong>and</strong> ||e|| = 0;(iii) Inversion (Symmetry): ‖x −1 ‖ = ‖x‖.If (i) holds we speak of a group semi-norm; if (i) <strong>and</strong> (iii) <strong>and</strong> ‖e‖ = 0 holds onespeaks of a pseudo-norm (cf. [Pet1]); if (i) <strong>and</strong> (ii) hold we speak of a group pre-norm(see [Low] for a full vocabulary).We say that a group pre-norm, <strong>and</strong> so also a group-norm, is abelian, or more preciselycyclically permutable, if(iv) Abelian norm (cyclic permutation): ‖xy‖ = ‖yx‖ for all x, y.Other properties we wish to refer to are :(i) K for all x, y : ‖xy‖ ≤ K(‖x‖ + ‖y‖),(i) ult for all x, y : ‖xy‖ ≤ max{‖x‖, ‖y‖}.Remarks 1. 1. Mutatis mut<strong>and</strong>is this is just the usual vector norm, but with scalingrestricted to the units ±1. The notation <strong>and</strong> language thus mimick the vector spacecounterparts, with sub<strong>groups</strong> playing the role of subspaces; for example, for a symmetric,subbadditive p : X → R + , the set {x : p(x) = 0} is a subgroup. Indeed the analysis ofBaire subadditive functions (see Section 4) is naturally connected with norms, via regularvariation. That is why normed <strong>groups</strong> occur naturally in regular variation theory.2. When (i) K , for some constant K, replaces (i), one speaks of quasi-norms (see [CSC],cf. ‘distance spaces’ [Rach] for a metric analogue). When (i) ult replaces (i) one speaks ofan ultra-norm, or non-Archimedean norm. For an example of the latter, in connectionwith the p-adic topology of a group, see [Fu, I.7.2].3. Note that (i) implies joint continuity of multiplication at the identity e X , while (iii)implies continuity of inversion at e X , a matter we return to in Th. 2.19 ′ <strong>and</strong> in Section3. (Montgomery [Mon1] shows that joint continuity is implied by separate continuitywhen the group is locally complete – cf. Th. 3.47; Ellis [Ell1] considers when one-sidedcontinuity implies joint continuity in the case of locally compact abelian <strong>groups</strong>.) In arelated theme Żelazko [Zel] considers a locally complete metric structure under which an
<strong>Normed</strong> <strong>groups</strong> 5abelian group has separately continuous multiplication <strong>and</strong> shows this to be a <strong>topological</strong>group. See below for the stronger notion of uniform continuity invoked in the UniformityTheorem of Conjugacy (Th. 12.4).4. Abelian <strong>groups</strong> with ordered norms may also be considered, cf. [JMW].Remarks 2. Subadditivity implies that ‖e‖ ≥ 0 <strong>and</strong> this together with symmetry impliesthat ‖x‖ ≥ 0, since ‖e‖ = ‖xx −1 ‖ ≤ 2‖x‖; thus a group-norm cannot take negative values.Subadditivity also implies that ‖x n ‖ ≤ n‖x‖, for natural n. The norm is said to be 2-homogeneous if ‖x 2 ‖ = 2‖x‖; see [CSC] Prop. 4.12 (Ch. IV.3 p.38) for a proof that if anormed group is amenable or weakly commutative (defined in [CSC] to mean that, forgiven x, y, there is m of the form 2 n , for some natural number n, with (xy) m = x m y m ),then it is embeddable as a subgroup of a Banach space. In the case of an abelian group2-homogeneity corresponds to sublinearity, <strong>and</strong> here Berz’s Theorem characterizes thenorm (see [Berz] <strong>and</strong> [BOst-GenSub]). The abelian property implies only that ‖xyz‖ =‖zxy‖ = ‖yzx‖, hence the alternative name of ‘cyclically permutable’. Harding [H], inthe context of quantum logics, uses this condition to guarantee that the group operationsare jointly continuous (cf. Theorem 2 below) <strong>and</strong> calls this a strong norm. See [Kel, Ch.6 Problem O ] (which notes that a locally compact group with abelian norm has a biinvariantHaar measure). We note Ellis’ Theorem that, for X a locally compact group,continuity of the inverse follows from the separate continuity of multiplication (see [Ell2],or [HS, Section 2.5]). The more recent literature concerning when joint continuity of(x, y) → xy follows from separate continuity reaches back to Namioka [Nam] (see e.g.[Bou1], [Bou2], [HT], [CaMo]).Convention. For a variety of purposes <strong>and</strong> for the sake of clarity, when we deal with ametrizable group X if we assume a metric d X on X is right/left invariant we will writed X R or dX L , omitting the superscript <strong>and</strong> perhaps the subscript if context permits.Remarks 3. For X a metrizable group with right-invariant metric d X <strong>and</strong> identity e X ,the canonical example of a group-norm is identified in Proposition 2.3 below as‖x‖ := d X (x, e X ).It is convenient to use the above notation irrespective of whether the metric d X is invariant.Remarks 4. If f : R + → R + is increasing, subadditive with f(0) = 0, <strong>and</strong> ‖x‖ 1 is agroup-norm, then‖x‖ 2 := f(‖x‖ 1 )is also a group-norm. See [BOst-GenSub] for recent work on Baire (i.e., having the Baireproperty) subadditive functions. These will appear in Sections 3 <strong>and</strong> 4.We begin with two key definitions.
- 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: Normed groups 3Topological complete
- 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 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
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Normed groups 55equipped with an in
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Normed groups 57Proof. To apply Th.
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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,