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Enveloping semigroups as compactifications of topological groups

K. L. Kozlov, B. V. Sorin

TL;DR

The paper develops a general framework to construct Ellis compactifications $e_G$ of a topological group $G$ from a $\tau_p$-representation in a $G$-space $X$ and systematically compares these to Roelcke compactifications. It establishes a functorial correspondence between $G$-compactifications of phase spaces and Ellis compactifications, analyzes when uniformities align (yielding Roelcke precompactness), and applies the theory to concrete groups: permutation groups on discrete spaces and automorphism groups of ultrahomogeneous chains and LOTS. The results yield a detailed taxonomy of $G$-compactifications and their Ellis counterparts, including explicit descriptions of sim-compactifications and Rees quotients, and reveal when Ellis and Roelcke compactifications coincide or stand in a refined hierarchy. Overall, the work clarifies how representations in compact or linearly ordered spaces govern the structure of enveloping semigroups and their compactifications, with implications for topological dynamics and representation theory of large groups.

Abstract

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.

Enveloping semigroups as compactifications of topological groups

TL;DR

The paper develops a general framework to construct Ellis compactifications of a topological group from a -representation in a -space and systematically compares these to Roelcke compactifications. It establishes a functorial correspondence between -compactifications of phase spaces and Ellis compactifications, analyzes when uniformities align (yielding Roelcke precompactness), and applies the theory to concrete groups: permutation groups on discrete spaces and automorphism groups of ultrahomogeneous chains and LOTS. The results yield a detailed taxonomy of -compactifications and their Ellis counterparts, including explicit descriptions of sim-compactifications and Rees quotients, and reveal when Ellis and Roelcke compactifications coincide or stand in a refined hierarchy. Overall, the work clarifies how representations in compact or linearly ordered spaces govern the structure of enveloping semigroups and their compactifications, with implications for topological dynamics and representation theory of large groups.

Abstract

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

Paper Structure

This paper contains 13 sections, 29 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $G$-spaces, admissible group topologies and topology of pointwise convergence
  4. Uniformities on topological groups
  5. Semigroups
  6. Ellis compactification of a group from its $\tau_p$-representation in a compact space
  7. Maps of Ellis compactifications
  8. Comparison of uniformities on topological groups
  9. Permutation group of a discrete space
  10. Automorphism groups of ultrahomogeneous chains and LOTS
  11. $G$-compactifications of ultrahomogeneous discrete chains and LOTS
  12. Ellis compactifications of the automorphism group of ultrahomogeneous LOTS
  13. Ellis compactifications of the automorphism group of ultrahomogeneous discrete chain

Key Result

Lemma 2.2

Let $(G=(G, \tau_p), X, \theta)$ be $G$-Tychonoff and $b X$ is a $G$-compactification of $X$. Then the topology of pointwise convergence $\tau_p^{b X}$ for the extended action $\tilde{\theta}: G\times b X\to b X$ is the least admissible group topology on $G$ and $\tau_p^{b X}=\tau_p$.

Theorems & Definitions (72)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 62 more