Normed versus topological groups: Dichotomy and duality

Normed groups 27Denoting this commonly determined set by B(r), we have seen in Proposition 2.5 thatB R (a, r) = {x : x = ya and d R (a, x) = d R (e, y) < r} = B(r)a,B L (a, r) = {x : x = ay and d L (a, x) = d L (e, y) < r} = aB(r).Thus the open balls are right- or left-shifts of the norm balls at the origin. This is bestviewed in the current context as saying that under d R the right-shift ρ a : x → xa is rightuniformly continuous, sinced R (xa, ya) = d R (x, y),and likewise that under d L the left-shift λ a : x → ax is left uniformly continuous, sinced L (ax, ay) = d L (x, y).In particular, under d R we have y → R b iff yb −1 → R e, as d R (e, yb −1 ) = d R (y, b). Likewise,under d L we have x → L a iff a −1 x → L e, as d L (e, a −1 x) = d L (x, a).Thus either topology is determined by the neighbourhoods of the identity (origin)and according to choice makes the appropriately sided shift continuous; said anotherway, the topology is determined by the neighbourhoods of the identity and the chosenshifts. We noted earlier that the triangle inequality implies that multiplication is jointlycontinuous at the identity e, as a mapping from (X, d R ) to (X, d R ). Likewise inversion isalso continuous at the identity by the symmetry axiom. (See Theorem 2.19 ′ .) To obtainsimilar results elsewhere one needs to have continuous conjugation, and this is linkedto the equivalence of the two norm topologies (see Th. 3.4). The conjugacy map underg ∈ G (inner automorphism) is defined byγ g (x) := gxg −1 .Recall that the inverse of γ g is given by conjugation under g −1 and that γ g is ahomomorphism. Its continuity, as a mapping from (X, d R ) to (X, d R ), is thus determinedby behaviour at the identity, as we verify below. We work with the right topology (underd R ), and sometimes leave unsaid equivalent assertions about the isometric case of (X, d L )replacing (X, d R ).Lemma 3.1. The homomorphism γ gright-to-right continuous at e.is right-to-right continuous at any point iff it isProof. This is immediate since x → R a if and only if xa −1 → R e and γ g (x) → R γ g (a) iffγ g (xa −1 ) → R γ g (e), since‖gxg −1 (gag −1 ) −1 ‖ = ‖gxa −1 g −1 ‖.We note that, by the Generalized Darboux Theorem (Th. 11.22), if γ g is locallynorm-bounded and the norm is N-subhomogeneous (i.e. a Darboux norm – there areconstants κ n → ∞ with κ n ‖z‖ ≤ ‖z n ‖), then γ g is continuous. Working under d R , wewill relate inversion to left-shifts. We begin with the following, a formalization of anearlier observation.