Enveloping semigroups as compactifications of topological groups
Ellis's "functional approach" allows one to obtain proper compactifications of a topological group $G$ if $G$ can be represented as a subgroup of the homeomorphism group of a space $X$ in the topology of pointwise convergence and $G$-space $X$ is $G$-Tychonoff. These compactifications, called Ellis compactifications, are right topological monoids and $G$-compactifications of the group $G$ with its action by multiplication on the left on itself. A comparison is made between Ellis compactifications of $G$ and the Roelcke compactification of $G$. Uniformity corresponding to the Ellis compactification of $G$ for its representation in a compact space $X$ is established. Proper Ellis semigroup compactifications are described for groups ${\rm S}(X)$ (the permutation group of a discrete space $X$) and ${\rm Aut} (X)$ (automorphism group of an ultrahomogeneous chain $X$) in the permutation topology and ${\rm Aut} (X)$ of LOTS $X$ in the topology of pointwise convergence.