Normed versus topological groups: Dichotomy and duality

Normed groups 63Thus ω δ (s) ≤ 3ε. (Since ε > 0 was arbitrary, ω(s) = 0, so γ s is continuous, and sos ∈ Z R .) We use this idea several times over in the next result.We now give a necessary and sufficient criterion for a normed group to be topologicalby refering not to continuity, but to approximation of left-shifts by right-shifts. This turnsout to be equivalent both to a commutator condition and 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 (and 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), and 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 and topologically 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 groups, a sharpening of the Montgomery Theorem(Th. 3.47), in view of Montgomery’s Initial Observation above that, for a semitopologicalgroup, each set Ω(ε) ∩ W is non-meagre for W a non-empty open set (and 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, and 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 and each η > 0, there are y η and 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: