Existence of a maximal G-compactifications

Having had a hard time finding this material in English I decided to make a post on it myself. Here is a proof of the existence of a universal compactification of $G$ -space of various types.

In what follows $G$ will denote a fixed locally compact group, $X$ will denote a Tychonoff space with a continuous $G$ -action (henceforth a $G$ -space), and $K_G(X)$ will denote the family of all its $G$ -compactifications (up to equivariant isomorphisms). We say that a compact $G$ -space $Y$ is a $G$ -compactification of a $G$ -space $X$ if the exists an equivariant topological embedding $\iota: X\to Y$ , usually taken to have dense image.

The collection of all $G$ -spaces which admit a compactification is called $G$ -Tychonoff spaces and are characterized by the following property – for any closed set $A \subset X$ and any point $x \in X \backslash A$ there exists a $G$ -uniform function $f : X \to [0, 1]$ (i.e. a function uniformly continuous on each fixed $G$ -orbit) such that $f(x) = 0$ and $A \subset f^{-1}(1)$ .

For our purpose we will only need to know that if $X$ is a Tychonoff space then $X$ is a $G$ -Tychonoff space in the above sense (a result proved by Vries in [1]), hence $K_G(X)$ is not empty.

There is a partial ordering on $K_G(X)$ determined by $X_i \leq X_j$ if the identity map on $X$ lifts to a surjective equivariant continuous map $X_j\to X_i$ . In this case we say that $X_j$ dominates $X_i$ . Given a linearly ordered subset of $K_G(X)$ (called a chain),

$… \to \overline{X}^{a_n}\xrightarrow[]{f_{n-1}}… \xrightarrow[]{f_0} \overline{X}^{a_0}$

the resulting projective (or inverse) limit space $\beta_GX = \varprojlim (\overline{X}^{a_i})$ is also compact and admits a strongly continuous $G$ -action.

To see this, recall that elements in the limit space $\beta_GX$ can be thought of as sequences $(x_i)$ where

$x_i \in X^{a_i} \qquad \text{and } \qquad f_n(x_{n+1}) = x_{n}$

endowed with the weak topology given by the projections

$p_n: \beta_GX\to X_n \qquad (x_i) \mapsto x_n.$

The surjectivity of the connecting morphisms assures that the limit is non-empty. There is a well defined $G$ action on $\beta_GX$ given by

$g\cdot (x_i) := (gx_i)$

and since (by definition of the weak topology) any map $f: Y \to \beta_GX$ is continuous if and only if $f\circ p_n: Y \to X_n$ is continuous for all $n$ we immediately get that the map

$G\to \beta_GX \qquad g \mapsto g\cdot (x_i)$

is continuous for all $(x_i)\in \beta_GX$ .

Compactness of the limit requires some thought, but follows from the fact that a limit of compact Hausdorff spaces is a compact space (see for instance [2] Theorem 5).

Hence if we only require strong continuity we may employ Zorn’s lemma to create a “maximal” compact (topological) $G$ -space with a strongly continuous $G$ -action.

However, we can say even more – from Ellis theorem any action

$F: G\times Z \to Z \qquad (g, z)\mapsto gz$

of a locally compact group $G$ on a compact space $Z$ which is continuous in each variable is automatically continuous. So the maximal $G$ -compactification is actually a compact $G$ -space. This universal $G$ -compactification is often denote by $\beta_GX$ to mirror the notation of the Stone-Cech compactification, and is characterized by the following universal property –

If $Y$ is a $G$ -compactification of $X$ such that the identity map on $X$ lifts to a surjective map

$Y\to \beta_GX \quad \text{then} \quad Y\simeq \beta_GX.$

In some special cases we have an actual homeomorphism $\beta X\simeq \beta_GX$ . This happens if for instance $G$ is discrete, or acts trivially on $X$ , or if $X$ is pseudocompact (the latter is proven in [3]).

If one wishes to specialize to subsets of $K_G(X)$ with specific topological attributes, here that are some properties which are preserved by projective limits with morphisms given by surjective equivariant continuous maps –

being compact Hausdorff
being non-empty compact Hausdorff
being connected compact Hausdorff
being compact Hausdorff of covering dimension $\leq n$ .
being topological complete (spaces that admit a complete metric inducing its topology).

That is to say, if we take for instance the collection $K_G^{tc}(X)\subset K_G(X)$ of topologically complete $G$ -compactifications of $X$ (may be empty) the maximal $G$ -compactification $\beta_GX$ will be a topologically complete metric space. In general being a metric space, being paracompact or being compact is not preserved under inverse limits unless one puts some requirements on the connecting morphisms.

The case of metric spaces – Looking more closely at the case of metric $G$ -compactifications, let $\overline{X}$ and $\overline{X}'$ be two such compactifications. Since $X$ imbeds into both these metric spaces as a dense subset, the compactifications can be treated simply as the completions of $X$ with respect to two distinct metrics $d$ and $d'$ , where $d$ and $d'$ induce the same G-compactification if and only if they are uniformly equivalent (not to be confused with strongly equivalent). This follows from existence and uniqueness theorem of completions of metric spaces and the fact that a compact metric space must be complete.

In the above example we can hence assume the two spaces are completions of $X$ with respect to some choice of metrics, say $d$ and $d'$ . We see that $\overline{X}\geq \overline{X}'$ if and only if every Cauchy sequence in $d'$ is also a Cauchy sequence in $d$ . The metric

$\tilde{d} = max\{d, d'\}$

gives a completion of $X$ which hence dominates both $\overline{X}$ and $\overline{X}'$ . It is also quite simply to see that every sequence has a $\tilde{d}$ convergent subsequence, by passing twice to subsequences. The compactification is a $G$ -compactification is also quite trivial to check using Ellis theorem. This is the “smallest” space which dominates both $\overline{X}$ and $\overline{X}'$ among the metric G-compactifications of $X$ .

As an exercise, use this procedure (and the definition of a maximal element) to show that the maximal metric $G$ -compactification must dominate all other metric $G$ -compactifications.

Final remark – The theory of compactification of $G$ -spaces is closely interconnected with the existence of certain invariant proximities/uniformities compatible with the topology, and correspondence with certain subalgebras of $C(X)$ . For more on this see the very good article [4] where much of the above material was taken.

Bibliography

[1] – de Vries, Jan On the existence of G-compactifications. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 3, 275–280.
[2] – Stone, Arthur H. “Inverse limits of compact spaces.” General Topology and its Applications 10.2 (1979): 203-211.
[3] – de Vries, J. G-spaces: Compactifications and pseudocompactness. Stichting Mathematisch Centrum, 1983.
[4] – Kozlov, Konstantin Leonidovich, and Vitalii Al’bertovich Chatyrko. “On G-compactifications.” Mathematical Notes 78.5-6 (2005): 649-661.

Bibliography

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