Normed versus topological groups: Dichotomy and duality

Normed groups 5abelian group has separately continuous multiplication and shows this to be a topologicalgroup. See below for the stronger notion of uniform continuity invoked in the UniformityTheorem of Conjugacy (Th. 12.4).4. Abelian groups with ordered norms may also be considered, cf. [JMW].Remarks 2. Subadditivity implies that ‖e‖ ≥ 0 and 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, and here Berz’s Theorem characterizes thenorm (see [Berz] and [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) and 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 and 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 and perhaps the subscript if context permits.Remarks 3. For X a metrizable group with right-invariant metric d X and 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, and ‖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 and 4.We begin with two key definitions.