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 ℵn-bounded topological group is either precompact or ambitable. In the familiar semigroups constructed over ambitable groups, topological centres have an effective characterization.