Enveloping Ellis semigroups as compactifications of transformations groups

The notion of a proper Ellis semigroup compactification is introduced. Ellis's functional approach shows how to obtain them from totally bounded equiuniformities on a phase space $X$ when the acting group $G$ is with the topology of pointwise convergence and the $G$-space $(G, X, \curvearrowright)$ is $G$-Tychonoff. The correspondence between proper Ellis semigroup compactifications of a topological group and special totally bounded equiuniformities (called Ellis equiuniformities) on a topological group is established. The Ellis equiuniformity on a topological transformation group $G$ from the maximal equiuniformity on a phase space $G/H$ in the case of its uniformly equicontinuous action is compared with Roelcke uniformity on $G$. Proper Ellis semigroup compactifications are described for groups $S\,(X)$ (the permutation group of a discrete space $X$) and $Aut\,(X)$ (automorphism group of an ultrahomogeneous chain $X$) in the permutation topology. It is shown that this approach can be applied to the unitary group of a Hilbert space.