Ambitable topological groups

A topological group is said to be ambitable if every uniformly bounded uniformly equicontinuous set of functions on the group with its right uniformity is contained in an ambit. For n = 0 , 1 , 2 , …  , every locally ℵ n -bounded topological group is either precompact or ambitable. In the familiar semigroups constructed over ambitable groups, topological centres have an effective characterization.