Normed versus topological groups: Dichotomy and duality

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AbstractThe key vehicle of the recent development of a topological theory of regular variation based ontopological dynamics [BOst-TRI], and embracing its classical univariate counterpart (cf. [BGT])as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), aregroups with a right-invariant metric carrying flows. Following the vector paradigm, they arebest seen as normed groups. That concept only occasionally appears explicitly in the literaturedespite its frequent disguised presence, and despite a respectable lineage traceable back to thePettis closed-graph theorem, to the Birkhoff-Kakutani metrization theorem and further backstill to Banach’s Théorie des opérations linéaires. We collect together known salient features anddevelop their theory including Steinhaus theory unified by the Category Embedding Theorem[?], the associated themes of subadditivity and convexity, and a topological duality inherent totopological dynamics. We study the latter both for its independent interest and as a foundationfor topological regular variation.2010 Mathematics Subject Classification: Primary 26A03; Secondary 22.Key words and phrases: multivariate regular variation, topological dynamics, flows, convexity,subadditivity, quasi-isometry, Souslin-graph theorem, automatic continuity, density topology.[4]
1. IntroductionGroup-norms, which behave like the usual vector norms except that scaling is restrictedto the basic scalars of group theory (the units ±1 in an abelian context and the exponents±1 in the non-commutative context), have played a part in the early development of topologicalgroup theory. They appear naturally in the study of groups of homeomorphisms.Although ubiquitous, they lack a clear and unified exposition. This lack is our motivationhere, since they offer the right context for the recent theory of topological regular variation.This extends the classical theory (for which see, e.g. [BGT]) from the real line tometrizable topological groups. Normed groups are just groups carrying a right-invariantmetric. The basic metrization theorem for groups, the Birkhoff-Kakutani Theorem of1936 ([Bir], [Kak], see [Kel, Ch.6 Problems N-R], [Klee], [Bour, Part 2, Section 4.1], and[ArMa], compare also [Eng, Exercise 8.1.G and Th. 8.1.21]), is usually stated as assertingthat a first-countable Hausdorff group has a right-invariant metric. It is properly speakinga ‘normability’ theorem in the style of Kolmogorov’s Theorem ([Kol], or [Ru, Th. 1.39]; inthis connection see also [Jam], where strong forms of connectedness are used in an abeliansetting to generate norms), as we shall see below. Indeed the metric construction in [Kak]is reminiscent of the more familiar construction of a Minkowski functional (for whichsee [Ru, Sect. 1.33]), but is implicitly a supremum norm – as defined below; in Rudin’sderivation of the metric (for a topological vector space setting, [Ru, Th. 1.24]) this normis explicit. Early use by A. D. Michal and his collaborators was in providing a canonicalsetting for differential calculus (see the review [Michal2] and as instance [JMW]) andincluded the noteworthy generalization of the implicit function theorem by Bartle [Bart](see Th. 10.10). In name the group-norm makes an explicit appearance in 1950 in [Pet1]in the course of his classic closed-graph theorem (in connection with Banach’s closedgraphtheorem and the Banach-Kuratowski category dichotomy for groups). It reappearsin the group context in 1963 under the name ‘length function’, motivated by word length,in the work of R. C. Lyndon [Lyn2] (cf. [LynSch]) on Nielsen’s Subgroup Theorem, thata subgroup of a free group is a free group. (Earlier related usage for function spaces isin [EH].) The latter name is conventional in geometric group theory despite the parallelusage in algebra (cf. [Far]) and the recent work on norm extension (from a normalsubgroup) of Bökamp [Bo].When a group is topologically complete and also abelian, then it admits a metric whichis bi-invariant, i.e. is both right- and left-invariant, as [Klee] first showed (in course ofsolving a problem of Banach). In Section 3 we characterize non-commutative groups thathave a bi-invariant metric, a context of significance for the calculus of regular variation[1]
Page 1 and 2: N. H. BINGHAM and A. J. OSTASZEWSKI
Page 3: Normed groups 3ContentsContents . .
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 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 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
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.
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Normed groups 143Series 378, 2010.[
Page 149 and 150:
Normed groups 145abelian groups, Ma
Page 151 and 152:
Normed groups 147[Kak] S. Kakutani,
Page 153 and 154:
Normed groups 149fields. I. Basic p
Page 155:
Normed groups 151[So]R. M. Solovay,
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Normed versus topological groups: Dichotomy and duality

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