Enveloping Ellis semigroups as compactifications of transformations groups.

The notion of a proper Ellis semigroup compactification is introduced. Using Ellis’s functional approach their connection with equiuniformities on a topological group is established. Proper Ellis semigroup compactification of a topological group G from the maximal equiuniformity on a phase space in the case of isometric action (on a discrete space, on a discrete chain, as liner isometries of a Hilbert space) is described. Its connection with Roelcke uniformity on G is established.