Equicontinuity, uniform continuity and sequences in topological groups

Let G be an almost metrizable topological group (for example, a locally compact group). This paper is concerned with the proof of two principal results. First, the following criterion for equicontinuity is proved: Let X be a union of Gδ-subsets of G, Y a uniform space and H a set of continuous mappings of X into Y; then for H to be equicontinuous, it is sufficient (and obviously also necessary) that the sequence ((hn(x),hn(xn)))n∈N be eventually in every entourage of Y for each sequence (hn)n∈N in H, each x∈X and each sequence (xn)n∈N in X such that limn→+∞xn=x. Čobanʹs theorem on dyadicity is a basic tool in the proof of this result. Let e be the identity element of G; it follows immediately from the above criterion that G has equal left and right uniform structures if (and only if) limn→+∞anbn−1=e for all sequences (an)n∈N and (bn)n∈N in G such that limn→+∞an−1bn=e (a well-known property in the case when G is metrizable). The second principal result is the following: Let us suppose that the left and right uniform structures on the almost metrizable topological group G are distinct; then the Banach space UR(G) contains a linear isometric copy L of l∞ such that inf{‖2l+h‖∣h∈U(G)}≥‖l‖ for all l∈L (and consequently, the quotient Banach space UR(G)/U(G) is nonseparable); moreover, if G is complete, then “≥” can be replaced by “=” (and consequently, UR(G)/U(G) contains a linear isometric copy of l∞).