On topological groups with an approximate fixed point property

Abstract

A topological group $G$ has the Approximate Fixed Point (AFP) property on a bounded convex subset $C$ of a locally convex space if every continuous affine action of $G$ on $C$ admits a net $\left(x_i\right)$, $x_i\in C$, such that $x_i-g{x}_i⟶0$ for all $g\in G$. In this work, we study the relationship between this property and amenability.

Key words
Amenable groups; approximate fixed point property; Følner property; Reiter property

INTRODUCTION

One of the most useful known characterizations of amenability is stated in terms of a fixed point property. A classical theorem of ^{(Day 1961)} says that a topological group $G$ is amenable if and only if every continuous affine action of $G$ on a compact convex subset $C$ of a locally convex space has a fixed point, that is, a point $x\in C$ with $g\cdot x=x$ for all $g\in G$. This result generalizes earlier theorems of (Kakutani ¹⁹³⁸) and (Markov ¹⁹³⁶) obtained for abelian acting groups.

At the same time, an active branch of current research is devoted to the existence of approximate fixed points for single maps. Basically, given a bounded, closed convex set $C$ and a map $f:C⟶C$, one wants to find a sequence $\left(x_n\right)\subset C$ such that $x_n-f\left(x_n\right)⟶0$. A sequence with this property will be called an approximate fixed point sequence.

The main motivation for this topic is purely mathematical and comes from several instances of the failure of the fixed point property in convex sets that are no longer assumed to be compact, cf. (Dobrowolski and Marciszewski ¹⁹⁹⁷, Edelstein and Kok-Keong ¹⁹⁹⁴, Floret ¹⁹⁸⁰, Klee ¹⁹⁵⁵, Nowak ¹⁹⁷⁹) and references therein. One of the most emblematic results on this matter states that if $C$ is a non-compact, bounded, closed convex subset of a normed space, then there exists a Lipschitz map $f:C⟶C$ such that ${inf}_{x\in C}\parallel x-f\left(x\right)\parallel >0$ (Lin and Sternfeld ¹⁹⁸⁵). Notice in this case that there is no approximate fixed point sequence for $f$. Previous results of topological flavour were discovered by many authors, including (Klee ¹⁹⁵⁵) who has characterized the fixed point property in terms of compactness in the framework of metrizable locally convex spaces.

Both results give rise to the natural question whether a given spacewithout the fixed point property might still have the approximate fixed point property. The firstthoughts on this subject were developed in (Barroso ²⁰⁰⁹, Barroso and Lin ²⁰¹⁰, Barroso et al. ²⁰¹², ²⁰¹³) in the context of weak topologies. Another mathematical motivation for the study of theapproximate fixed point property is the following open question.

Question 0.1. Let $X$ be a Hausdorff locally convex space. Assume $C\subset X$ is a sequentially compact convex set and $f:C⟶C$ is a sequentially continuous map. Does $f$ have a fixed point?

So far, the best answers for this question were delivered in (Barroso et al. ²⁰¹², 2013). Let us just summarize the results.

Theorem 0.2 (Theorem 2.1, Proposition 2.5(i) in (Barroso et al. ²⁰¹², 2013)). Let $X$ be a topological vector space, $C\subset X$ a nonempty convex set, and let $f:C⟶\overline{C}$ be a map.

• 1. If $C$ is bounded, $f\left(C\right)\subset C$ and $f$ is affine, then $f$ has an approximate fixed point sequence.

• 2. If $X$ is locally convex and $f$ is sequentially continuous with totally bounded range, then $0\in \overline{\{x-f\left(x\right):x\in C\}}$ . And indeed, $f$ has a fixed point provided that $X$ is metrizable.

• 3. If $C$ is bounded and $f$ is $\tau$ -to- $\sigma (X,Z)$ sequentially continuous, then $0\in {\overline{\{x-f\left(x\right):x\in C\}}}^{\sigma (X,Z)}$ , where $\tau$ is the original topology of $X$ and $Z$ is a subspace of the topological dual $X^{\star }$ . And, moreover, $f$ has a $\sigma (X,Z)$ -approximate fixed point sequence provided that $Z$ is separable under the strong topology induced by $X$ .

The idea of approximate fixed points is an old one. Apparently the first result on this kind was exploited in (Scarf ¹⁹⁶⁷), where a constructive method for computing fixed points of continuous mappings of a simplex into itself was described. Other important works along these lines can be found in Hazewinkel and Vel ( ¹⁹⁷⁸), Hadžić ( ¹⁹⁹⁶), Idzik ( ¹⁹⁸⁸), Park ( ¹⁹⁷²). Approximate fixed point property has a lot of applications in many interesting problems. In (Kalenda ²⁰¹¹) it is proved that a Banach space has the weak-approximate fixed point property if and only if it does not contain any isomorphic copy of ${\ell }^1$. As another instance, it can be used to study the existence of limiting-weak solutions for differential equations in reflexive Banach spaces (Barroso ²⁰⁰⁹).

In this note, we study the existence of common approximate fixed points for a set of transformations forming a topological group. Not surprisingly, the approximate fixed point property for an acting group $G$ is also closely related to amenability of $G$, however, the relationship appears to be more complex.

There is an extremely broad variety of known definitions of amenability of a topological group, which are typically equivalent in the context of locally compact groups yet may diverge beyond this class. One would expect the approximate fixed point property (or rather ‘‘properties,’’ for they depend on the class of convex sets allowed) to provide a new definition of amenability for some class of groups, and delineate a new class of topological groups in more general contexts. This indeed turns out to be the case.

We show that a discrete group $G$ is amenable if and only if every continuous affine action of $G$ on a bounded convex subset $C$ of a locally convex space (LCS) $X$ admits approximate fixed points. For a locally compact group, a similar result holds if we consider actions on bounded convex sets $C$ which are complete in the additive uniformity, while in general we can only prove that $G$ admits weakly approximate fixed points. This criterion of amenability is no longer true in the more general case of a Polish group, even if amenability of Polish groups can be expressed in terms of the approximate fixed point property on bounded convex subsets of the Hilbert space.

We view our investigation as only the first step in this direction, and so we close the article with a discussion of open problems for further research.

1 - AMENABILITY

Here is a brief reminder of some facts about amenable topological groups. For a more detailed treatment, see e.g. (Paterson ¹⁹⁸⁸). All the topologies considered here are assumed to be Hausdorff.

Let $G$ be a topological group. The right uniform structure on $G$ has as basic entourages of the diagonal the sets of the form $U_V=\{(g,h)\in G\times G\mid h{g}^{-1}\in V\},$ where $V$ is a neighbourhood of the identity $e$ in $G$. This structure is invariant under right translations. Accordingly, a function $f:G⟶ℝ$ is right uniformly continuous if for all $\epsilon >0$, there exists a neighbourhood $V$ of $e$ in $G$ such that $x{y}^{-1}\in V$ implies $\left|f\left(x\right)-f\left(y\right)\right|<\epsilon$ for every $x,y\in G$. Let $RUCB\left(G\right)$ denote the space of all right uniformly continuous functions equipped with the uniform norm. The group $G$ acts on $RUCB\left(G\right)$ on the left continuously by isometries: for all $g\in G$ and $f\in RUCB\left(G\right),g\cdot f{=}^{g}f$ where ${}^{g}f\left(x\right)=f\left(g^{-1}x\right)$ for all $x\in G$.

Definition 1.1. A topological group $G$ is amenable if it admits an invariant mean on $RUCB\left(G\right)$, that is, a positive linear functional $m$ with $m\left(1\right)=1$, invariant under the left translations.

Examples of such groups include finite groups, solvable topological groups (including nilpotent, in particular abelian topological groups) and compact groups. Here are some more examples:

1. The unitary group $𝒰\left({\ell }^2\right)$, equipped with the strong operator topology (Harpe ¹⁹⁷³).

2. The infinite symmetric group $S_{\infty }$ with its unique Polish topology.

3. The group $𝒥\left(k\right)$ of all formal power series in a variable $x$ that have the form $f\left(x\right)=x+{\alpha }_1{x}^2+{\alpha }_2{x}^3+\dots .,{\alpha }_n\in k$, where $k$ is a commutative unital ring (Babenko and Bogatyi ²⁰¹¹).

Let us also mention some examples of non-amenable groups:

1. The free discrete group ${𝔽}_2$ of two generators. More generally, every locally compact group containing ${𝔽}_2$ as a closed subgroup.

2. The unitary group $𝒰\left({\ell }^2\right)$, with the uniform operator topology (Harpe ¹⁹⁷⁹).

3. The group $\mathrm{Au}t(X,\mu )$ of all measure-preserving automorphisms of a standard Borel measure space $(X,\mu )$, with the uniform topology, i.e. the topology determined by the metric $d(\tau ,\sigma )=\mu \{x\in X:\tau \left(x\right)\ne \sigma \left(x\right)\}$ (Giordano and Pestov ²⁰⁰²).

The following is one of the main criteria of amenability in the locally compact case.

Theorem 1.2 (Følner’s condition). Let $G$ be a locally compact group and denote $\lambda$ the left invariant Haar measure. Then $G$ is amenable if and only if $G$ satisfies the Følner condition: for every compact set $F\subseteq G$ and $\epsilon >0$, there is a Borel set $U\subseteq G$ of positive finite Haar measure $\lambda \left(U\right)$ such that $\frac{\lambda \left(xU△U\right)}{\lambda \left(U\right)}<\epsilon$ for each $x\in F$.

Recall that a Polish group is a topological group whose topology is Polish, i.e., separable and completely metrizable.

Proposition 1.3 (See e.g. (Al-Gadid et al. ²⁰¹¹), Proposition 3.7).

A Polish group $G$ is amenable if and only if every continuous affine action of $G$ on a convex, compact and metrizable subset $K$ of a locally convex space $X$ admits a fixed point.

For a most interesting recent survey about the history of amenable groups, see (^{Grigorchuk and de la Harpe in press}.).

2 - GROUPS WITH APPROXIMATE FIXED POINT PROPERTY

Definition 2.1. Let $C$ be a convex bounded subset of a topological vector space $X$. Say that a topological group $G$ has the approximate fixed point (AFP) property on $C$ if every continuous affine action of $G$ on $C$ admits an approximate fixed point net, that is, a net $\left(x_i\right)\subseteq C$ such that for every $g\in G,x_i-g{x}_i⟶0$.

We will analyse the AFP property of various classes of amenable topological groups.

2.1 - CASE OF DISCRETE GROUPS

Theorem 2.2. The following properties are equivalent for a discrete group $G$:

• 1. $G$ is amenable,

• 2. $G$ has the AFP property on every convex bounded subset of a locally convex space.

Proof.(1) $\Rightarrow$ (2). Let $G$ a discrete amenable group acting by continuous affine maps on a bounded convex subset $C$ of a locally convex space $X$. Choose a Følner’s ${\left({\Phi }_i\right)}_{i\in I}$ net, that is, a net of finite subsets of $G$ such that

$\frac{\left|g{\Phi }_i△{\Phi }_i\right|}{\left|{\Phi }_i\right|}⟶0\forall g\in G.$

Now, let $\gamma \in G$, fix $x\in C$ and define $x_i=\frac{1}{\left|{\Phi }_i\right|}\sum _{g\in \Phi i}gx$. Since $C$ is convex, $x_i\in C$ for all $i\in I$. Notice that $\left|{\Phi }_i\setminus \gamma {\Phi }_i\right|=\left|\gamma {\Phi }_i\setminus {\Phi }_i\right|=\frac{1}{2}\left|\gamma {\Phi }_i△{\Phi }_i\right|$ for all $i\in I$. Therefore we have

$\begin{array}{cc}{x}_i-\gamma x_i& =\frac{1}{\left|{\Phi }_i\right|}\left[\sum _{g\in \Phi i}gx-\sum _{g\in \Phi i}\gamma gx\right]\\ \\ =\frac{1}{\left|{\Phi }_i\right|}\left[\sum _{g\in \Phi i}gx-\sum _{h\in \gamma \Phi i}hx\right]\\ \\ =\frac{1}{\left|{\Phi }_i\right|}\left[\sum _{g\in \left({\Phi }_i\setminus \gamma {\Phi }_i\right)}gx-\sum _{g\in \left(\gamma {\Phi }_i\setminus {\Phi }_i\right)}gx\right]\\ \\ =\frac{\left|\gamma {\Phi }_i△{\Phi }_i\right|}{2\left|{\Phi }_i\right|}\left[\frac{1}{\left|{\Phi }_i\setminus \gamma {\Phi }_i\right|}\sum _{g\in \left({\Phi }_i\setminus \gamma {\Phi }_i\right)}gx-\frac{1}{\left|\gamma {\Phi }_i\setminus {\Phi }_i\right|}\sum _{g\in \left(\gamma {\Phi }_i\setminus {\Phi }_i\right)}gx\right]\end{array}$

Thus $x_i-\gamma x_i\in \frac{\left|\gamma {\Phi }_i△{\Phi }_i\right|}{2\left|{\Phi }_i\right|}\left(C-C\right)$ and hence $x_i-\gamma x_i⟶0$ since $C$ is bounded.

(2) $\Rightarrow$ (1). Let $G$ be a discrete group acting continuously and by affine transformations on a nonempty compact and convex set $K$ in a locally convex space $X$. By hypothesis, there is a net $\left(x_i\right)\subseteq K$ such that $\forall g\in G,x_i-g{x}_i⟶0$. By compactness of $K$, this net has accumulation points in $K$. Since $\forall g\in G,x_i-g{x}_i⟶0$, this insures invariance of accumulation points and shows the existence of a fixed point in $K$. Therefore $G$ is an amenable group by Day’s fixed point theorem mentioned in the Introduction. ∎

2.2 - CASE OF LOCALLY COMPACT GROUPS

Recall from (Bourbaki ¹⁹⁶³) the following notion of integration of functions with range in a locally convex space. Let $F$ be a locally convex vector space on $ℝ$ or $ℂ$. $F^{\prime }$ denotes the dual space of $F$ and $F^{\prime \ast }$ the algebraic dual of $F^{\prime }$. We identify as usual $F$ (seen as a vector space without topology) with a subspace of $F^{\prime \ast }$ by associating to any $z\in F$ the linear form $F^{\prime }\ni z^{\prime }⟼⟨z,z^{\prime }⟩\in ℝ$.

Let $T$ be a locally compact space and let $\mu$ a positive measure on $T$. A map $f:T⟶F$ is essentially $\mu$-integrable if for every element $z^{\prime }\in F^{\prime },⟨z^{\prime },f⟩$ is essentially $\mu$-integrable. If $f:T⟶F$ is essentially $\mu$-integrable, then $z^{\prime }⟼{\int }_T⟨z^{\prime },f⟩\text{d}\mu$ is a linear map on $F^{\prime }$, i.e. an element of $F^{\prime \ast }$. The integral of $f$ is the element of $F^{\prime \ast }$ denoted ${\int }_{T}f\text{d}\mu$ and defined by the condition: $⟨z^{\prime },{\int }_{T}f\text{d}\mu ⟩={\int }_T⟨z^{\prime },f⟩\text{d}\mu$ for every $z^{\prime }\in F^{\prime }$.

Note that, in general we don’t have ${\int }_{T}f\text{d}\mu \in F$. But we have the following.

Proposition 2.3 ((Bourbaki ¹⁹⁶³), chap. 3, Proposition 7). Let $T$ be a locally compact space, $E$ a locally convex space and $f:T⟶E$ a function with compact support. If $f\left(T\right)$ is contained in a complete (with regard to the additive uniformity) convex subset of $E$, then ${\int }_{T}f\text{𝑑}\mu \in E$.

Theorem 2.4. The following are equivalent for a locally compact group $G$:

• 1. $G$ is amenable,

• 2. $G$ has the AFP property on every complete, convex, and bounded subset of a locally convex space.

Proof. (1) $\Rightarrow$ (2). Let $G$ be a locally compact amenable group acting continuously by affine maps on a complete, bounded, convex subset $C$ of a locally convex space $X$. Again, select a Følner net ${\left(F_i\right)}_{i\in I}$ of compact subsets of $G$ such that $\frac{\lambda \left(g{F}_i△F_i\right)}{\lambda \left(F_i\right)}⟶0\forall g\in G$. Fix $x\in C$ and let ${\eta }_x:G\ni g⟼gx\in C$ be the corresponding orbit map. Define $x_i=\frac{1}{\lambda \left(F_i\right)}{\int }_{F_i}{\eta }_x\left(g\right)\text{d}\lambda \left(g\right)$. By the above, this is an element of $C$; the barycenter of the push-forward measure ${\left({\eta }_x\right)}_{\ast }\left({\lambda |}_{F_i}\right)$ on $X$. We have, just like in the discrete case:

$\begin{array}{cc}{x}_i-\gamma x_i& =\frac{1}{\lambda \left(F_i\right)}\left[{\int }_{F_i}{\eta }_x\left(g\right)\text{d}\lambda \left(g\right)-\gamma {\int }_{F_i}{\eta }_x\left(g\right)\text{d}\lambda \left(g\right)\right]\\ \\ =\frac{1}{\lambda \left(F_i\right)}\left[{\int }_{F_i}{\eta }_x\left(g\right)\text{d}\lambda \left(g\right)-{\int }_{F_i}{\eta }_x\left(\gamma g\right)\text{d}\lambda \left(g\right)\right]\\ \\ =\frac{1}{\lambda \left(F_i\right)}\left[{\int }_{F_i}\left[{\eta }_x\left(g\right)-{\eta }_x\left(\gamma g\right)\right]\text{d}\lambda \left(g\right)\right]\\ \\ =\frac{1}{\lambda \left(F_i\right)}\left[{\int }_{\gamma F_i△F_i}{\eta }_x\left(v\right)\text{d}\lambda \left(v\right)\right].\end{array}$

Now, let $q$ be any continuous seminorm on $C$. We have:

$q\left(x_i-\gamma x_i\right)\le \frac{\lambda \left(\gamma F_i△F_i\right)}{\lambda \left(F_i\right)}K$

where $K=\underset{v\in G}{sup}q\circ {\eta }_x\left(v\right)<\infty$ since $C$ is bounded. Thus $x_i-\gamma x_i⟶0$.

(2) $\Rightarrow$ (1). Same argument as in the case of discrete groups. ∎

The assumption of completeness of $C$ does not look natural in the context of approximate fixed points, but we do not know if it can be removed. It depends on the answer to the following.

Question 2.5. Let $f$ be an affine homeomorphism of a bounded convex subset $C$ of a locally convex space $X$. Can $f$ be extended to a continuous map (hence, a homeomorphism) of the closure of $C$ in $X$?

Nevertheless, we can prove the following.

Theorem 2.6. Every amenable locally compact group $G$ has a weak approximate fixed point property on each bounded convex subset $C$ of a locally convex space $X$.

Proof.

In the notation of the proof of Theorem 2.4, let ${\mu }_i={\left({\eta }_x\right)}_{\ast }\left({\lambda |}_{F_i}\right)$ denote the push-forward of the measure ${\lambda }_i=\lambda \upharpoonright F_i$ along the orbit map ${\eta }_x:G\ni g⟼gx\in C$. Let $x_i$ be the barycenter of ${\mu }_i$. This time, $x_i$ need not belong to $C$ itself, but will belong to the completion $\hat{C}$, of $C$ (the closure of $C$ in the locally convex vector space completion $\hat{X}$).

For every $g\in G$, denote $z_i^g$ the barycenter of the measure $g.{\mu }_i={\left({\eta }_x\right)}_{\ast }\left(g{\lambda }_i\right)$. Just like in the proof of Theorem 2.4., for every $g$ we have $x_i-z_i^g\to 0$.

Now select a net ${\nu }_j$ of measures with finite support on $G$, converging to ${\lambda }_i$ in the vague topology (Bourbaki ¹⁹⁶³). Denote $y_j$ the barycenter of the push-forward measure ${\left({\eta }_x\right)}_{\ast }\left({\nu }_j\right)$. Then $y_j\Rightarrow x_i$ in the vague topology on the space of finite measures on the compact space $F_i.x$. Clearly, $y_j\in C$, and so $g\cdot y_j$ is well-defined and $g\cdot y_j\Rightarrow z_i^g$ for every $g\in \Phi$. It follows that $g\cdot y_j-y_j$ weakly converges to $0$ for every $g\in \Phi$. ∎

Remark 2.7. Clearly the weak AFP property on each bounded convex subset $C$ implies amenability of $G$ as well. Recall that a topological group $G$ has the weak AFP property on $C$ if every weakly continuous affine action of $G$ on $C$ admits an approximate fixed point net.

2.3 - CASE OF POLISH GROUPS

The above criteria do not generalize beyond the locally compact case in the ways one might expect: not every amenable non-locally compact Polish group has the AFP property, even on a bounded convex subset of a Banach space.

Proposition 2.8. The infinite symmetric group $S_{\infty }$ equipped with its natural Polish topology does not have the AFP property on closed convex bounded subsets of ${\ell }^1$.

If we think of $S_{\infty }$ as the group of all self-bijections of the natural numbers $\mathbb{N}$, then the natural (and only) Polish topology on $S_{\infty }$ is induced from the embedding of $S_{\infty }$ into the Tychonoff power ${\mathbb{N}}^{\mathbb{N}}$, where $\mathbb{N}$ carried the discrete topology.

We will use the following well-known criterion of amenability for locally compact groups.

Theorem 2.9 (Reiter’s condition). Let $p$ be any real number with $1\le p<\infty$. A locally compact group $G$ is amenable if and only if for any compact set $C\subseteq G$ and $\epsilon >0$, there exists $f\in L^p\left(G\right)$, $f\ge 0$, ${\parallel f\parallel }_p=1$, such that: $\parallel g\cdot f-f\parallel <\epsilon$ for all $g\in C$.

Proof of Proposition 2.8. Denote $\mathrm{prob}\left(\mathbb{N}\right)$ the set of all Borel probability measures on $\mathbb{N}$, in other words, the set of positive functions $b:\mathbb{N}⟶[0,1]$ such that $\sum _{n\in \mathbb{N}}b\left(n\right)=1$. This is the intersection of the unit sphere of ${\ell }^1$ with the cone of positive elements, a closed convex bounded subset of ${\ell }^1$. The Polish group $S_{\infty }$ acts canonically on ${\ell }^1$ by permuting the coordinates:

$S_{\infty }\times {\ell }^1\left(\mathbb{N}\right)\ni (\sigma ,{\left(x_n\right)}_n)⟼\sigma \cdot {\left(x_n\right)}_n={\left(x_{{\sigma }^{-1}\left(n\right)}\right)}_n\in {\ell }^1\left(\mathbb{N}\right).$

Clearly, $\mathrm{prob}\left(\mathbb{N}\right)$ is invariant and the restricted action is affine and continuous. We will show that the action of $S_{\infty }$ on $\mathrm{prob}\left(\mathbb{N}\right)$ admits no approximate fixed point sequence.

Assume the contrary. Make the free group ${𝔽}_2$ act on itself by left multiplication and identify ${𝔽}_2$ with $\mathbb{N}$. In this way we embed ${𝔽}_2$ into $S_{\infty }$ as a closed discrete subgroup. This means that the action of ${𝔽}_2$ by left regular representation on $\mathrm{prob}\left(\mathbb{N}\right)\cong \mathrm{prob}\left({𝔽}_2\right)$ also has almost fixed points, and ${\ell }^1\left({𝔽}_2\right)$, with regard to the left regular representation of ${𝔽}_2$, has almost invariant vectors. But this is the Reiter’s condition ($p=1$) for ${𝔽}_2$, a contradiction with non-amenability of this group. ∎

However, it is still possible to characterize amenability of Polish groups in terms of the AFP property.

Theorem 2.10. The following are equivalent for a Polish group $G$:

• 1. $G$ is amenable,

• 2. $G$ has the AFP property on every bounded, closed and convex subset of the Hilbert space.

Proof. (1) $\Rightarrow$ (2). It is enough to show that a norm-continuous affine action of $G$ on a bounded closed convex subset $C$ of ${\ell }^2$ is continuous with regard to the weak topology, because then there will be a fixed point in $C$ by Day’s theorem.

Let $x\in C$ and $g\in G$ be any, and let $V$ be a weak neighbourhood of $g.x$ in $C$. The weak topology on the weakly compact set $C$ coincides with the $\sigma (\mathrm{span}C,{\ell }^2)$ topology, hence one can choose $x_1,x_2,\dots ,x_n\in C$ and $\epsilon >0$ so that $y\in V$ whenever $\left|⟨x_i,y-gx⟩\right|<\epsilon$ for all $i$.

Denote $K$ the diameter of $C$. Because the action is norm-continuous, we can find $U\ni e$ in $G$ so that $\parallel u^{-1}{x}_i-x_i\parallel <\epsilon /2K$ for all $i$. The set $Ug$ is a neighbourhood of $g$ in $G$.

As a weak neighbourhood of $x$, take the set $W$ formed by all $\zeta \in C$ with $\left|⟨g^{-1}{x}_i,\zeta -x⟩\right|<\epsilon /2$ for all $i$. Equivalently, the condition on $\zeta$ can be stated $\left|⟨x_i,g\zeta -gx⟩\right|<\epsilon /2$ for all $i$.

If now $u\in U$ and $\zeta \in W$, one has

$\left|⟨x_i,\left(ug\right)\cdot \zeta -gx⟩\right|$	$=$	$\left|⟨u^{-1}{x}_i,g\zeta ⟩-⟨x_i,gx⟩\right|$
$=$	$\left|⟨u^{-1}{x}_i,g\zeta ⟩-⟨x_i,g\zeta ⟩\right|+\left|⟨x_i,g\zeta ⟩-⟨x_i,gx⟩\right|$
$\le$	$\parallel u{x}_i-x_i\parallel \cdot K+\frac{\epsilon }{2}$
$=$	$\epsilon .$

This shows that $\left(ug\right)\cdot \zeta \in V$, and so the action of $G$ on $C$ is continuous with regard to the weak topology.

(2) $\Rightarrow$ (1). Suppose that $G$ acts continuously and by affine transformations on a compact convex and metrizable subset $Q$ of a LCS $E$. If $C\left(Q\right)$ is equipped with the usual norm topology, then $G$ acts continuously by affine transformations on the subspace $A\left(Q\right)$ of $C\left(Q\right)$ consisting of affine continuous functions on $Q$. Since $Q$ is a metrizable compact set, the space $C\left(Q\right)$ is separable, so is the space $A\left(Q\right)$. Fix a dense countable subgroup $H$ of $G$, and let $F=\{f_n:n\in \mathbb{N}\}$ be a dense subset of the closed unit ball of $A\left(Q\right)$ which is $H$-invariant. The map $T:Q\ni x⟼\left(\frac{1}{n}{f}_n\left(x\right)\right)\in {\ell }^2$ is an affine homeomorphism of $Q$ onto a convex compact subset of ${\ell }^2$. The subgroup $H$ acts continuously and by affine transformations on the affine topological copy $T\left(Q\right)$ of $Q$ by, the obvious rule $h\cdot T\left(x\right)=T\left(h\cdot x\right)$. The action of $H$ extends by continuity to a continuous affine action of $G$ on $T\left(Q\right)$. By hypothesis, $G$ admits an approximative fixed point sequence in $T\left(Q\right)$, and every accumulation point of this sequence is a fixed point since $Q$ is compact. Therefore $G$ is amenable.

∎

3 - DISCUSSION AND CONCLUSION

3.1 - APPROXIMATELY FIXED SEQUENCES

As we have already noted, we do not know if every locally compact amenable group has the AFP property on all convex bounded subsets of locally convex spaces. Another interesting problem is to determine when does an acting group possess not merely an approximately fixed net, but an approximate fixed sequence.

Recall that a topological group $G$ is $\sigma$-compact if it is a union of countably many compact subsets. It is easy to see that if an amenable locally compact group $G$ is $\sigma$-compact, then it admits an approximate fixed sequence for every continuous action by affine maps on a closed bounded convex set.

Question 3.1. Let $G$ be a metrizable separable group acting continuously and affinely on a convex bounded subset $C$ of a metrizable and locally convex space. If the action has an approximate fixed net, does there necessarily exist an approximate fixed sequence?

This is the case, for example if $G$ is the union of a countable chain of amenable locally compact (in particular, compact) subgroups, and the convex set $C$ is complete.

Recall in this connection that amenability (and thus Day’s fixed point property) is preserved by passing to the completion of a topological group. At the same time, the AFP property is not preserved by completions. Indeed, the group $S_{\infty }^{fin}$ of all permutations of integers with finite support is amenable as a discrete group, and so, equipped with any group topology, will have the AFP property on every bounded convex subset of a locally convex space. However, its completion with regard to the pointwise topology is the Polish group $S_{\infty }$ which, as we have seen, fails the AFP property on a bounded convex subset of ${\ell }^1$.

Question 3.2. Does every amenable group whose left and right uniformities are equivalent (a SIN group) have the AFP property on complete convex sets?

3.2 - DISTAL ACTIONS

Let $G$ be a topological group acting by homeomorphisms on a compact set $Q$. The flow $(G,Q)$ is called distal if whenever ${lim}_{\alpha }{s}_{\alpha }\cdot x={lim}_{\alpha }{s}_{\alpha }\cdot y$ for some net $s_{\alpha }$ in $G$, then $x=y$. A particular class of distal flows is given by equicontinuous flows, for which the collection of all maps $x\mapsto g\cdot x$ forms an equicontinuous family on the compact space $Q$. We have the following fixed point theorem:

Theorem 3.3 (Hahn ( ¹⁹⁶⁷)). If a compact affine flow $(G,Q)$ is distal, then there is a $G$-fixed point.

An earlier result by (Kakutani ¹⁹³⁸) established the same for the class of equicontinuous flows.

Question 3.4. Is there any approximate fixed point analogue of the above results for distal or equicontinuous actions by a topological group on a (non-compact) bounded convex set $Q$?

3.3 - NON-AFFINE MAPS

Historically, Day’s theorem (and before that, the theorem of Markov and Kakutani) was inspired by the classical Brouwer fixed point theorem (Brouwer ¹⁹¹¹) and its later more general versions, first for Banach spaces (Schauder ¹⁹³⁰) andlater for locally convex linear Hausdorff topological spaces (Tychonoff ¹⁹³⁵).(Recently it was extended to topological vector spaces (Cauty ²⁰⁰¹)). The Tychonoff fixed point theorem states the following. Let $C$ be a nonvoid compact convex subset of a locally convex space and let $f:C⟶C$ be a continuous map. Then $f$ has a fixed point in $C$. The map $f$ is not assumed to be affine here.

However, for a common fixed point of more than one function, the situation is completely different. Papers (Boyce ¹⁹⁶⁹) and (Huneke ¹⁹⁶⁹) contain independent examples of two commuting maps $f,g:[0,1]⟶[0,1]$ without a common fixed point. Hence if a common fixed point theorem were to hold, there should be further restrictions on the nature of transformations beyond amenability, and for Day’s theorem, this restriction is that the transformations are affine.

The Tychonoff fixed point theorem is being extended in the context of approximate fixed points. For instance, here is one recent elegant result.

Theorem 3.5 (Kalenda ( ²⁰¹¹)). Let $X$ be a Banach space. Then every nonempty, bounded, closed, convex subset $C\subseteq X$ has the weak AFP property with regard to each continuous map $f:C⟶C$ if and only if $X$ does not contains an isomorphic copy of ${\ell }^1$.

We do not know if a similar program can be pursued for topological groups.

Question 3.6. Does there exist a non-trivial topological group $G$ which has the approximate fixed point property with regard to every continuous action (not necessarily affine) on a bounded, closed convex subset of a locally convex space? of a Banach space?

Question 3.7. The same, for the weak AFP property.

Natural candidates are the extremely amenable groups, see e.g. Pestov ( ²⁰⁰⁶). A topological group is extremely amenable if every continuous action of $G$ on a compact space $K$ has a common fixed point. The action does not have to be affine, and $K$ is not supposed to be convex. This is a very strong nonlinear fixed point property.

Some of the most basic examples of extremely amenable Polish groups are:

1. The unitary group $U\left({\ell }^2\right)$ with the strong operator topology (Gromov and Milman ¹⁹⁸³).

2. The group $\mathrm{Aut}\left(\mathbb{Q}\right)$ of order-preserving bijections of the rational numbers with the topology induced from $S_{\infty }$ (Pestov ¹⁹⁹⁸).

3. The group $\mathrm{Aut}(X,\mu )$ of measure preserving transformations of a standard Lebesgue measure space equipped with the weak topology. (Giordano and Pestov ²⁰⁰²).

However, at least the group $\mathrm{Aut}\left(\mathbb{Q}\right)$ does not have the AFP property with regard to continuous actions on the Hilbert space. To see this, one can use the same construction as in Proposition 2.8, together with the well-known fact that $\mathrm{Aut}\left(\mathbb{Q}\right)$ contains a closed discrete copy of ${𝔽}_2$.

ACKNOWLEDGMENTS

Cleon S. Barroso is currently as a visiting researcher scholar at the Texas A&M University. He takes the opportunity to express his gratitude to Prof. Thomas Schlumprecht for his support and friendship. Also, he acknowledges Financial Support form Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) by the Science Without Bordes Program, PDE 232883/2014-9. Brice R. Mbombo was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) postdoctoral grant, processo 12/20084-1. Vladimir G. Pestov is a Special Visiting Researcher of the program Science Without Borders of CAPES, processo 085/2012.

REFERENCES

• ^r1
  AL-GADID Y, MBOMBO B AND PESTOV V. 2011. Sur les espaces test pour la moyennabilité. CR Math Acad Sci Soc R Can 33: 65-77.
• ^r2
  BABENKO I AND BOGATYI SA. 2011. The amenability of the substitution group of formal power series. Izv Math 75: 239-252.
• ^r3
  BARROSO CS, KALENDA OFK AND REBOUÇAS MP. 2013. Optimal approximate fixed point result in locally convex spaces. J Math Appl 401: 1-8.
• ^r4
  BARROSO CS, KALENDA OFK AND LIN PK. 2012. On the approximate fixed point property in abstract spaces. Math Z 271: 1271-1285.
• ^r5
  BARROSO CS AND LIN PK. 2010. On the weak approximate fixed point property. J Math Anal Appl 365: 171-175.
• ^r6
  BARROSO CS. 2009. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete Cont Dyn Syst 25: 467-479.
• ^r7
  BOURBAKI N. 1963. Intégration. Hermann, Paris.
• ^r8
  BOYCE WM. 1969. Commuting functions with no common fixed point. Trans Amer Math Soc 137: 77-92.
• ^r9
  BROUWER LEJ. 1911. Uber Abbildung von Mannigfaltigkeiten. Math Ann 71: 97-115.
• ^r10
  CAUTY R. 2001. Solution du problème de point fixe de Schauder. Fund Math 170: 231-246.
• ^r11
  DAY M. 1961. Fixed-point theorems for compact sets. Illinois J Math 5: 585-590.
• ^r12
  DOBROWOLSKI T AND MARCISZEWSKI W. 1997. Rays and the fixed point property in noncompact spaces. Tsukuba J Math 21: 97-112.
• ^r13
  EDELSTEIN M AND KOK-KEONG T. 1994. Fixed point theorems for affine operators on vector spaces. J Math Anal Appl 181: 181-187.
• ^r14
  FLORET K. 1980. Weakly Compact Sets. Lecture Notes in Math. 801, Springer-Verlag, Berlin? Heidelberg? New York.
• ^r15
  GIORDANO T AND PESTOV V. 2002. Some extremely amenable groups. CR Acad Sci Paris Sér I 4: 273-278.
• ^r15a
  GRIGORCHUK R AND DE LA HARPE P. IN PRESS. Amenability and Ergodic Properties of Topological groups: From Bogolyubov onwards.
• ^r16
  GROMOV M AND MILMAN VD. 1983. A topological application of the isoperimetric inequality. Amer J Math 105(4): 843-854.
• ^r17
  HADŽIĆ O. 1996. Almost fixed point and best approximations theorems in H-spaces. Bull Austral Math Soc 53: 447-454.
• ^r18
  HAHN F. 1967. A fixed-point theorem. Math Systems Theory 1: 55-57.
• ^r19
  DE LA HARPE P. 1973. Moyennabilité de quelques groupes topologiques de dimension infinie. CR Acad Sci Paris, Sér A 277: 1037-1040.
• ^r20
  DE LA HARPE P. 1979. Moyennabilité du groupe unitaire et propriété de Schwartz des algèbres de von Neumann. Lecture Notes in Math 725: 220-227.
• ^r21
  HAZEWINKEL M AND VAN DE VEL M. 1978. On almost-fixed-point theory. Canad J Math 30: 673-699.
• ^r22
  HUNEKE JP. 1969. On common fixed points of commuting continuous functions on an interval. Trans Amer Math Soc 139: 371-381.
• ^r23
  IDZIK A. 1988. Almost fixed point theorems. Proc Amer Math Soc 104: 779-784.
• ^r24
  KAKUTANI S. 1938. Fixed-point theorems concerning bicompact convex sets. Proc Imperial Acad Japan 14: 27-31.
• ^r25
  KALENDA OFK. 2011. Spaces not containing `1 have weak approximate fixed point property. J Math Anal Appl 373: 134-137.
• ^r26
  KLEE VL. 1955. Some topological properties of convex sets. Trans Amer Math Soc 78: 30-45.
• ^r27
  LIN PK AND STERNFELD Y. 1985. Convex sets with the Lipschitz fixed point property are compact. Proc Amer Math Soc 93: 33-39.
• ^r28
  MARKOV A. 1936. Quelques théorèmes sur les ensembles abéliens. Dokl Akad Nauk SSSR (NS:) 10: 311-314.
• ^r29
  NOWAK B. 1979. On the Lipschitzian retraction of the unit ball in infinite dimensional Banach spaces onto its boundary. Bull Acad Polon Sci Sr Sci Math 27: 861-864.
• ^r30
  PARK S. 1972. Almost fixed points of multimaps having totally bounded ranges. Nonlinear Anal 51: 1-9.
• ^r31
  PATERSON AT. 1988. Amenability. University Math. Surveys and Monographs 29, Amer Math Soc, Providence, RI.
• ^r32
  PESTOV V. 2006. Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman phenomenon. University Lecture Series, vol. 40, Amer Math Soc, Providence, RI.
• ^r33
  PESTOV V. 1998. On free actions, minimal flows, and a problem by Ellis. Trans of the American Mathematical Society 350: 4149-4165.
• ^r34
  SCARF H. 1967. The approximation of fixed points of a continuous mapping. SIAM J Applied Math 15: 1328-1343.
• ^r35
  SCHAUDER J. 1930. Der Fixpunktsatz in Functionalraumen. Studia Math 02: 171-180.
• ^r36
  TYCHONOFF A. 1935. Ein Fixpunktsatz. Math Ann 03: 767-776.

Publication Dates

• Publication in this collection
  Mar 2017

History

• Received
  25 June 2015
• Accepted
  10 Oct 2016