Enveloping Ellis semigroups as compactifications of transformations groups

TL;DR

This paper introduces proper Ellis semigroup compactifications using equiuniformities, establishing their correspondence with topological group compactifications and applying the framework to permutation and automorphism groups, as well as the unitary group.

Contribution

It develops a new approach to Ellis semigroup compactifications via equiuniformities, linking them to topological group compactifications and applying to various groups.

Findings

Proper Ellis semigroup compactifications correspond to special equiuniformities.

Comparison between Ellis equiuniformity and Roelcke uniformity on groups.

Descriptions of compactifications for permutation and automorphism groups.

Abstract

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.