A Note on the Topological Group c₀

Abstract

A well-known result of Ferri and Galindo asserts that the topological group $c_0$ is not reflexively representable and the algebra WAP$\left(c_0\right)$ of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if the same remains true for a larger important algebra Tame$\left(c_0\right)$ of tame functions. Respectively, it is an open question if $c_0$ is representable on a Rosenthal Banach space. In the present work we show that Tame$\left(c_0\right)$ is small in a sense that the unit sphere S and $2S$ cannot be separated by a tame function f ∈ Tame$\left(c_0\right)$. As an application we show that the Gromov’s compactification of $c_0$ is not a semigroup compactification. We discuss some questions.

1. Introduction

Recall that for every Hausdorff topological group G the algebra WAP$\left(G\right)$ of all weakly almost periodic functions on G determines the universal semitopological semigroup compactification $u_w:G\to G^w$ of G. This map is a topological embedding for many groups including the locally compact case. For some basic material about WAP$\left(G\right)$ we refer to [1,2].

The question if $u_w$ always is a topological embedding (i.e., if $\mathrm{WAP}\left(G\right)$ determines the topology of G) was raised by Ruppert [2]. This question was negatively answered in [1] by showing that the Polish topological group $G:=H_{+}[0,1]$ of orientation preserving homeomorphisms of the closed unit interval has only constant WAP functions and that every continuous representation $h:G\to Is\left(V\right)$ (by linear isometries) on a reflexive Banach space V is trivial. The WAP triviality of $H_{+}[0,1]$ was conjectured by Pestov.

Recall also that for $G:=H_{+}[0,1]$ every Asplund (hence also every WAP) function is constant and every continuous representation $G\to$ Iso$\left(V\right)$ on an Asplund (hence also reflexive) space V must be trivial [3]. In contrast one may show (see [4,5]) that $H_{+}[0,1]$ is representable on a (separable) Rosenthal space (a Banach space is Rosenthal if it does not contain a subspace topologically isomorphic to $l_1$).

We have the inclusions of topological G-algebras

$$\mathrm{WAP}\left(G\right)\subset \mathrm{Asp}\left(G\right)\subset \mathrm{Tame}\left(G\right)\subset \mathrm{RUC}\left(G\right).$$

For details about $\mathrm{Tame}\left(G\right)$ and definition of $\mathrm{Asp}\left(G\right)$ see [5,6,7]. We only remark that $f\in \mathrm{Tame}\left(G\right)$ if and only if f is a matrix coefficient of a Rosenthal representation. That is, there exist: a Rosenthal Banach space V; a continuous homomorphism $h:G\to Is\left(V\right)$ into the topological group of all linear isometries $V\to V$ with strong operator topology; two vectors $v\in V$; $\psi \in V^{*}$ (the dual of V) such that $f\left(g\right)=\psi \left(h\right(g\left)v\right)$ for every $g\in G$.

Similarly, it can be characterized $f\in \mathrm{Asp}\left(G\right)$ replacing Rosenthal spaces by the larger class of Asplund spaces. A Banach space is Asplund if the dual of every separable subspace is separable. Every reflexive space is Asplund and every Asplund is Rosenthal. A standard example of an Asplund but nonreflexive space is just $c_0$.

Recall that $c_0$, as an additive abelian topological group, is not representable on a reflexive Banach space by a well-known result of Ferri and Galindo [8]. In fact, $\mathrm{WAP}\left(c_0\right)$ separates the points but not points and closed subsets. The group $c_0$ admits an injective continuous homomorphism $h:c_0\to Is\left(V\right)$ with some reflexive V but such h cannot be a topological embedding.

Presently it is an open question if every topological group (abelian, or not) G is Rosenthal representable and if $\mathrm{Tame}\left(G\right)$ determines the topology of G. Note that the algebra $\mathrm{Tame}\left(G\right)$ appears as an important modern tool in some new research lines in topological dynamics motivating its detailed study [5,7].

One of the good reasons to study $\mathrm{Tame}\left(G\right)$ is a special role of tameness in the dynamical Berglund-Fremlin-Talagrand dichotomy [5]; as well as direct links to Rosenthal’s $l_1$-dychotomy. In a sense $\mathrm{Tame}\left(G\right)$ is a set of all functions which are not dynamically massive.

By these reasons and since $H_{+}[0,1]$ is Rosenthal representable, it seems to be an attractive concrete question if $c_0$ is Rosenthal representable and it is worth studying how large is $\mathrm{Tame}\left(c_0\right)$. In the present work we show that $\mathrm{Tame}\left(c_0\right)$ is quite small (even for the discrete copy of $c_0$, see Theorem 3).

Theorem 1.

$\mathit{Tame}\left(c_0\right)$ does not separate the unit sphere S and $2S$.

So, the closures of S and $2S$ intersect in the universal tame compactification of $c_0$ (a fortiori, the same is true for the universal Asplund (HNS) semigroup compactification).

Another interesting question is if $c_0$ admits an embedding into a metrizable semigroup compactification. Note that any metrizable semigroup compactification of $H_{+}[0,1]$ is trivial.

In Section 3 we show that the Gromov’s compactification $\gamma :c_0\hookrightarrow P$, which is metrizable (and $\gamma$ is a G-embedding), is not a semigroup compactification.

Theorem 2.

Let $\gamma :c_0\hookrightarrow P$ be the Gromov’s compactification of the metric space $(c_0,\frac{d}{1+d})$, where $d(x,y):=\left|\right|x-y\left|\right|$. Then γ is not a semigroup compactification.

This gives an example of a naturally defined separable unital (original topology determining) G-subalgebra of $\mathrm{RUC}\left(G\right)$ (for $G=c_0$) which is not left m-introverted in the sense of [9].

2. Tame Functions on c₀

Recall that a sequence $f_n$ of real-valued functions on a set X is said to be independent if there exist real numbers $a<b$ such that

$$\bigcap _{n\in P}{f}_n^{-1}(-\infty ,a)\cap \bigcap _{n\in M}{f}_n^{-1}(b,\infty )\ne \varnothing$$

for all finite disjoint subsets $P,M$ of $\mathbb{N}$. Every bounded independent sequence is an $l_1$-sequence [10].

As in [6,7] we say that a bounded family F of real-valued (not necessarily continuous) functions on a set X is a tame family if F does not contain an independent sequence.

Let G be a topological group, $f:G\to \mathbb{R}$ be a real-valued function. For every $g\in G$ define $fg:G\to \mathbb{R}$ as $\left(fg\right)\left(x\right)=f\left(gx\right)$ (for multiplicative G). Denote by $\mathrm{RUC}\left(G\right)$ the algebra of all bounded right uniformly continuous functions on G. So, $f\in \mathrm{RUC}\left(G\right)$ means that f is bounded and for every $ϵ>0$ there exists a neighborhood U of the identity e (of the multiplicative group G) such that $\left|f\right(ux)-f(x\left)\right|<ϵ$ for every $x\in G$ and $u\in U$. This algebra $\mathrm{RUC}\left(G\right)$ corresponds to the greatest G-compactification $G\to {\beta }_{G}G$ of G (with respect to the left action), greatest ambit of G.

We say that $f\in \mathrm{RUC}\left(G\right)$ is a tame function if the orbit $fG:={\left\{fg\right\}}_{g\in G}$ is a tame family. That is, $fG$ does not contain an independent sequence; notation $f\in \mathrm{Tame}\left(G\right)$.

2.1. Proof of Theorem 1

We have to show that $\mathrm{Tame}\left(c_0\right)$ does not separate the spheres S and $2S$ (where $S:=\{x\in c_0:|\left|x\right||=1\}$). In fact we show the following stronger result.

Theorem 3.

Let $G=c_0$ be the additive group of the classical Banach space $c_0$. Assume that $f:c_0\to \mathbb{R}$ be any (not necessarily continuous) bounded function such that

$$\left\{\begin{array}{cc}f\left(x\right)\le a\hfill & \forall \left|\right|x\left|\right|=1\hfill \\ b\le f\left(x\right)\hfill & \forall \left|\right|x\left|\right|=2\hfill \end{array}\right.$$

for some pair $a<b$ of real numbers. Then f is not a tame function on the discrete copy of the group $c_0$.

Proof. 

For every $n\in \mathbb{N}$ consider the function

$$f_n:c_0\to \mathbb{R},x\mapsto f(e_n+x),$$

where $e_n$ is a vector of $c_0$ having 1 as its n-th coordinate and all other coordinates are 0. Clearly, $f_n=f{g}_n$ where $g_n=e_n\in c_0$. We have to check that $fG$ is an untame family. It is enough to show that the sequence ${\left\{f_n\right\}}_{n\in \mathbb{N}}$ in $fG$ is an independent family of functions on $c_0$. We have to show that for every finite nonempty disjoint subsets $I,J$ in $\mathbb{N}$ the intersection

$$\bigcap _{n\in I}{f}_n^{-1}(-\infty ,a]\cap \bigcap _{n\in J}{f}_n^{-1}[b,\infty )$$

is nonempty.

Define $v={\left(v_k\right)}_{k\in \mathbb{N}}\in c_0$ as follows: $v_j=1$ for every $j\in J$ and $v_k=0$ for every $k\notin J$. Then

(1)

$v\in c_0$ and $\left|\right|v\left|\right|=1$.

(2)

$\left|\right|e_i+v\left|\right|=1$, $f_i\left(v\right)=f(e_i+v)\le a$ for every $i\in I$.

(3)

$\left|\right|e_j+v\left|\right|=2$, $f_j\left(v\right)=f(e_j+v)\ge b$ for every $j\in J$.

So we found v such that

$$v\in \bigcap _{n\in I}{f}_n^{-1}(-\infty ,a]\cap \bigcap _{n\in J}{f}_n^{-1}[b,\infty ).$$

 □

Corollary 1.

The bounded RUC function

$$f:c_0\to [-1,1],x\mapsto \frac{\left|\right|x\left|\right|}{1+\left|\right|x\left|\right|}$$

is not tame on $c_0$ (even on the discrete copy of the group $c_0$).

Proof. 

Observe that $f\left(S\right)=\frac{1}{2},f\left(2S\right)=\frac{2}{3}$ and apply Theorem 3. □

Theorem 3 remains true for the spheres $rS$ and $2rS$ for every $r>0$. In the case of Polish $c_0$ it is unclear if the same is true for any pair of different spheres around the zero. If, yes then this will imply that $\mathrm{Tame}\left(c_0\right)$ does not separate the zero and closed subsets. The following question remains open even for any topological group [5,7].

Question 1.

Is it true that $\mathit{Tame}\left(c_0\right)$ separates the points and closed subsets ? Is it true that Polish group $c_0$ is Rosenthal representable ?

3. Gromov’s Compactification Need Not Be a Semigroup Compactification

Studying topological groups G and their dynamics we need to deal with various natural closed unital G-subalgebras $\mathcal{A}$ of the algebra $\mathrm{RUC}\left(G\right)$. Such subalgebras lead to G-compactifications of G (so-called G-ambits,11]). That is we have compact G-spaces K with a dense orbit $Gz\subset K$ such that the Gelfand algebra which corresponds to the compactification $G\to K,g\mapsto gz$ is just $\mathcal{A}$. Frequently but not always such compactifications are the so-called semigroup compactifications, which are very useful in topological dynamics and analysis. Compactifications of topological groups already is a fruitful research line. See among others [12,13,14] and references there. In our opinion semigroup compactifications deserve even much more attention and systematic study in the context of general topological group theory.

A semigroup compactification of G is a pair $(\alpha ,K)$ such that K is a compact right topological semigroup (all right translations are continuous), and $\alpha$ is a continuous semigroup homomorphism from G into K, where $\alpha \left(G\right)$ is dense in K and the left translation $K\to K,x\mapsto \alpha \left(g\right)x$ is continuous for every $g\in G$.

One of the most useful references about semigroup compactifications is a book of Berglund, Junghenn and Milnes [9]. For some new directions (regarding topological groups) see also [3,4,15,16].

Question 2.

Which natural compactifications of topological groups G are semigroup compactifications? Equivalently which Banach unital G-subalgebras of RUC$\left(G\right)$ are left m-introverted (in the sense of [9])?

Recall that left m-introversion of a subalgebra $\mathcal{A}$ of $\mathrm{RUC}\left(G\right)$ means that for every $v\in \mathcal{A}$ and every $\psi \in MM\left(\mathcal{A}\right)$ the matrix coefficient $m(v,\psi )$ belongs to $\mathcal{A}$, where

$$m(v,\psi ):G\to \mathbb{R},g\mapsto \psi \left(g^{-1}v\right)$$

and $MM\left(\mathcal{A}\right)\subset {\mathcal{A}}^{*}$ denotes the spectrum (Gelfand space) of $\mathcal{A}$.

It is not always easy to verify left m-introversion directly. Many natural G-compactifications of G are semigroup compactifications. For example, it is true for the compactifications defined by the algebras $\mathrm{RUC}\left(G\right)$, $\mathrm{Tame}\left(G\right),$ $\mathrm{Asp}\left(G\right)$, $\mathrm{WAP}\left(G\right)$. Of course, the 1-point compactification is a semitopological semigroup compactification for any locally compact group G.

As to the counterexamples. As it was proved in [3], the subalgebra $\mathrm{UC}\left(G\right):=\mathrm{RUC}\left(G\right)\cap \mathrm{LUC}\left(G\right)$ of all uniformly continuous functions is not left m-introverted for $G:=H\left(C\right)$, the Polish group of homeomorphisms of the Cantor set.

In this section we show that the Gromov’s compactification of a metrizable topological group G need not be a semigroup compactification.

Let $\rho$ be a bounded metric on a set X. Then the Gromov’s compactification of the metric space $(X,\rho )$ is a compactification $\gamma :X\to P$ induced by the algebra $\mathcal{A}$ which is generated by the bounded set of functions

$${\{{\rho }_z:X\to \mathbb{R},{\rho }_z\left(x\right)=\rho (z,x)\}}_{z\in X}.$$

Then $\gamma$ always is a topological embedding. If X is separable then P is metrizable. Moreover, if $(X,\rho )$ admits a continuous $\rho$-invariant action of a topological group G then $\gamma$ is a G-compactification of X; see [17].

Here we examine the following particular case. Let G be a metrizable topological group. Choose any left invariant metric d on G. Denote by $\gamma :G\to P$ the Gromov’s compactification of the bounded metric space $(G,\rho )$, where $\rho =\frac{d}{1+d}$.

Consider the following natural bounded RUC function

$$f:G\to \mathbb{R},x\mapsto \frac{\left|\right|x\left|\right|}{1+\left|\right|x\left|\right|}$$

where $\left|\right|x\left|\right|:=d(e,x)$. By ${\mathcal{A}}_f$ we denote the smallest closed unital G-subalgebra of $\mathrm{RUC}\left(G\right)$ which contains $fG=\{fg:g\in G\}$. Then ${\mathcal{A}}_f$ is the algebra which corresponds to the compactification $\gamma$. Indeed, ${\rho }_{g^{-1}}\left(x\right)=\rho (g^{-1},x)=\left(fg\right)\left(x\right)$ for every $g,x\in G$.

Proof of Theorem 2

We have to prove Theorem 2.

Proof. 

By the discussion above, the unital G-subalgebra ${\mathcal{A}}_f$ of $\mathrm{RUC}\left(G\right)$ associated with $\gamma$ is generated by the orbit $fG$, where $f:G\to \mathbb{R},f\left(x\right)=\frac{\left|\right|x\left|\right|}{1+\left|\right|x\left|\right|}$. Since $c_0$ is separable the algebra ${\mathcal{A}}_f$ is separable. Hence, P is metrizable. If we assume that $\gamma$ is a semigroup compactification then the separability of ${\mathcal{A}}_f$ guarantees by [4] ( Prop. 6.13) that ${\mathcal{A}}_f\subset \mathrm{Asp}\left(G\right)$. On the other hand, since $\mathrm{Asp}\left(G\right)\subset \mathrm{Tame}\left(G\right)$, and $f\in {\mathcal{A}}_f$ we have $f\in \mathrm{Tame}\left(G\right)$. Now observe that f separates the spheres S and $2S$ and we get a contradiction to Corollary 1. □

Question 3.

Is it true that the Polish group $c_0$ admits a semigroup compactification $\alpha :c_0\hookrightarrow P$ such that P is metrizable and $\alpha$ is an embedding? What if P is first countable?

This question is closely related to the setting of this work. Indeed, by [4] (Prop. 6.13) (resp., by [4] (Cor. 6.20)) the metrizability (first countability) of P guarantees that the corresponding algebra is a subset of $\mathrm{Asp}\left(G\right)$ (resp. of $\mathrm{Tame}\left(G\right)$).