3 The fixed point characterization of amenability

The goal of this chapter is to give a characterization of amenability via a fixed point property. It will allow us to extend the notion of amenability from groups to topological groups.

3.1 Day’s fixed point theorem

Let us recall that a locally convex topological vector space is a topological vector space such that the origin admits a basis of convex neighborhoods.

Definition 3.1 (semi-norm) Let \(E\) be a vector space. A semi-norm is a map \(p\colon E\to\mathbf{R_+}\) such that

\(p(\lambda x)=|\lambda|p(x)\) for all \(x\in E\) and \(\lambda\in \mathbf{R}\)
\(p(x+y)\leq p(x)+p(y)\)

Example 3.1 \(\ \)

Any norm is a semi-norm
If \(X\) is compact space and \(E=C(X)\) with the uniform convergence topology then for any \(x\in X\), \(p_x(f)=|f(x)|\), for \(f\in E\), is a continuous semi-norm on \(E\)

Given a vector space \(E\) and a family of semi-norms \(\mathcal{P}\) on \(E\), the topology given by \(\mathcal{P}\) is the coarsest topology on \(E\) that makes each element of \(\mathcal{P}\) continuous. this a vector space topology.

Proposition 3.1 A topological vector space is locally convex if and only if its topology is given by a collection of semi-norms.

Example 3.2 \(\ \)

Let \(E\) be some Banach space and let \(E^*\) be its dual space with the weak-* topology. This topology on \(E^*\) is given by the semi-norms \(\lambda\mapsto |\lambda(x)|\) for \(x\in E\).
A particular case is the following : Let \(K\) be a compact Hausdorff space. Let \(\mathcal{C}(K)\) be the Banach space of continuous real functions on \(K\) with the norm of uniform convergence. Its dual space is the space of Radon measures on \(K\) by Riesz representation theorem. The topology is given by the semi-norms \(\mu\mapsto|\mu(f)|=\left|\int_Kf(x)d\mu(x)\right|\) for \(f\in\mathcal{K}\).

Proposition 3.2 (Barycenter map) Let \(K\) be a convex compact subset of some locally convex topological vector space \(E\). For any probability measure \(\mu\) on \(K\), there is a unique barycenter \(\textrm{bar}(\mu)\in K\) such that for continuous affine \(f\colon K\to \mathbf{R}\) \(f(\textrm{bar}(\mu))=\mu(f)\).

Morever if \(g\) is an affine continuous transformation of \(K\) then \(\mathrm{bar}(g_*\mu)=g(\mathrm{bar}(\mu))\).

Proof. Uniqueness follows from Hahn-Banach separation theorem: for \(x\neq y\), one can find a linear continuous map \(f\) such that \(f(x)\neq f(y)\). So if \(x,y\) are two barycenters then \(f(x)=\mu(f)=f(y)\) and we have a contradiction.

For the existence, Since \(\{x\in K,\ f(x)=\mu(f)\}\) is compact, by the finite intersection property, it suffices to prove that for any \(f_1,\dots,f_n\) continuous affine maps on \(E\), there is \(x\in K\), such that \(f_i(x)=\mu(f_i)\) for all \(i\). So let us consider the map \[\alpha\colon x\mapsto (f_1(x),\dots,f_n(x)).\]

The space \(\alpha(K)\) is compact convex subspace of \(\mathbf{R}^n\). Let us define \(p=(\mu(f_1),\dots,\mu(f_n))\). We claim that \(p\in \alpha(K)\). If \(\mu\) is a convex combination of Dirac masses \(\mu=\sum_{j=1}^kc_j\delta_{x_j}\) with \(x_j\in K\) and \(c_j\geq0\) with \(\sum_{j=1}^nc_j=1\) then \(p=(f_1(\sum_{j=1}^nc_jx_j),\dots, f_n(\sum_{j=1}^nc_jx_j))\in\alpha(K)\). Now \(\mu\mapsto p\) is continuous and convex combinations of Dirac masses are dense in \(\mathrm{Prob}(K)\) for the weak-* topology. Since \(\alpha(K)\) is closed then \(p\in\alpha(K)\) for any \(\mu\in \mathrm{Prob}(K)\). By construction \(p\cap_{i=1}^n\{x\in K,\ f_i(x)=\mu(f_i)\}\), and thus the finite intersection property holds. Thus there is a unique point \(\mathrm{bar}(\mu)\) in the intersection over all continuous affine functions \(f\) of \(\{x\in K,\ f(x)=\mu(f)\}\).

Finally, let \(g\) be some continuous affine transformations of \(K\) then we have for any continuous affine function \(f\) on \(K\):

\[f(\mathrm{bar}(g_*\mu))=g_*\mu(f)=\mu(f\circ g)=(f\circ g)\mathrm{bar}_\mu=f(g\mathrm{bar}_\mu)\] and thus \(\mathrm{bar}(g_*\mu)=g\mathrm{bar}_\mu\).

Let \(K\) be a convex subset of vector space \(E\). Assume a group \(G\) is acting on K. The action is said to be affine if for any \(g\in G\), \(x,y\in K\) and \(t\in[0,1]\), \(g(tx+(1-t)y)=tg(x)+(1-t)g(y)\).

Example 3.3 If \(G\) acts by affine transformations on \(E\) and \(K\) is \(G\)-invariant then the action on \(K\) is affine. This situation is not the general one.

Theorem 3.1 (Day's fixed point theorem) Let \(G\) be a group. Then \(G\) is amenable if and only if for any affine action on compact convex subset of a locally convex topological vector space, there is a fixed point.

Proof. Let’s assume \(G\) is an amenable group acting affinely on some compact convex set \(K\). Let \(m\) be a left invariant mean on \(G\). For \(f\in\mathcal{C}(K)\) and \(x\in K\), the map \(\theta_f\colon g\mapsto f(gx)\) belongs to \(\ell^\infty(G)\). So let us define

\[\mu(f)=m(\theta_f).\]

One checks that \(\mu\) is a positive linear maps on \(\mathcal{C}(K)\) such that \(\mu(\mathbf{1}_K)=1\), so \(\mu\) is a probability measure on \(K\).

Moreover, \(h_*\mu(f)=\mu(f\circ h)=m(\theta_{f\circ h})=\mu(f)\) since \(m\) is left invariant.

3.2 Amenable topological group

Definition 3.2 A topological group \(G\) is amenable if for any continuous affine actions on a convex compact topological vector space, there is a fixed point.

Remark. Day’s fixed point theorem tells us that a group is amenable if and only if it amenable for the discrete topology.

Remark. If \(\tau_1\) and \(\tau_2\) are two group topologies on \(G\), \(\tau_1\) is finer than \(\tau_2\) then if \(G\) is amenable for \(\tau_1\) then it is also amenable for \(\tau_2\).

In particular, if a group is amenable as an abstract group then it is amenable with respect to any topology. For example, finite groups, Abelian or more generally solvable groups are amenable with respect to any group topology.

Theorem 3.2 Compact groups are amenable.

Proof. We prove that any compact group has an invariant probability measure.

Let \(G\) be a compact group. It suffices to deal with the case where \(G\) is infinite. Let \(\mu\) be some diffuse probability measure on \(G\) with full support. Let \(\lambda\) be some \(\mu\)-stationnary probability measure, i.e. \(\mu\ast \lambda=\lambda\). Such measure exists thanks to Tychonoff fixed theorem.

For \(f\in\mathcal{C}(G)\), let us define

\[\varphi(g)=\int_G f(gh)d\lambda(h).\] The function \(\varphi\) is continuous and thus has a maximum at some \(g_0\). Up to replace \(f\) by \(f'\colon x\mapsto f(g_0x)\), we may assume that \(\varphi\) achieves its maximum at \(e\). By stationnarity

\[\int_G\varphi(g)d\mu(g)=\iint f(gh)d\mu(g)d\lambda(h)=\int f(h)d\lambda(h)=\varphi(e).\]

So, \(\mu\)-a.s. \(\varphi(g)=\varphi(e)\). Since \(\mu\) has full support and \(\varphi\) is continuous, \(\varphi(g)=\varphi(e)\) for all \(g\in G\). That \(\varphi\) is constant and \(\lambda\) is invariant.

Remark. The topological group \(\mathrm{SO}(3)\) is not amenable with the discrete topology because of the paradoxical decompositions we have seen in the first chapter, whereas it is amenable with its usual topology since it is compact.

Definition 3.3 Let \(G\) a topological group. A function \(f\colon G\to \mathbf{R}\) is left uniformly continuous if for any \(\varepsilon>0\), there is an identity neighborhood \(U\) such that for \(g,h\in G\), \(g^{-1}h\in U\implies |f(g)-f(h)|<\varepsilon\).

We denote by \(\mathrm{UCB}(G)\) the space of left uniformly continuous bounded functions on \(G\). With the \(\sup\) norm, this is a Banach space. Moreover, for \(f\in\ell^\infty(G)\), the map \(g\mapsto g\cdot f=f\circ g^{-1}\) is continuous if and \(f\) is left uniformly continuous.

Definition 3.4 Let \(G\) be a topological group. A mean on \(\mathrm{UCB}(G)\) is a linear map \(m\colon \mathrm{UCB}(G)\to\mathbf{R}\) such that

\(m(\mathbf{1}_G)=1\)
if \(f\geq0\) then \(m(f)\geq0\).

Proposition 3.3 Let \(G\) be a topological group. The following are equivalent

\(G\) is amenable
There is a \(G\)-invariant mean on \(\mathrm{UCB}(G)\).

Proof. Let

Proposition 3.4 Let We have the following stability properties

If \(h\colon G\to H\) is a continuous surjective homomorphism between topological groups\(G,H\) then if \(G\) is amenable then \(H\) is amenable.
If \(G\) has a dense amenable subgroup (for the induced topology) then \(G\) is amenable
If \(G\) contains a dense increasing unions of amenable subgroups then \(G\) is amenable.
The topological group \(G\) is amenable if the closure of any finitely generated subgroup is amenable.
If \(H\) is an open subgroup of \(G\) and \(G\) is amenable then \(H\) is amenable.
Let \(N\) be a normal subgroup of \(G\). Then \(G\) is amenable if \(N\) and \(G/N\) are amenable. Moreover, if \(N\) is open this is an equivalence.

Corollary 3.1 We can deduce the following amenability results

The group of all permutation \(S_\infty\) of a countable set with the pointwise convergence is amenable
The group of all unitary operators of a Hilbert space is amenable.

Corollary 3.2 Any solvable group is amenable for any group topology.

Example 3.4 The subgroup of upper triangular matrices is amenable

3.3 Locally compact amenable groups

Locally compact groups are remarkable among topological groups.

Theorem 3.3 Let \(G\) be a locally compact group then there is left invariant positive Radon measure on \(G\). It is unique up to multiplication by some positive number.

Conversely, a topological group with a left invariant Borel measure that is locally finite and inner regular is locally compact.

The main point about locally compact groups for us will be the following point.

Theorem 3.4 Let \(G\) be a locally compact group and \(H\) be a closed subgroup. If \(G\) is amenable then \(H\) is amenable.

Remark. If \(G\) is not locally compact, one can find closed subgroup that are not amenable. For example, by letting \(\mathbf{F}_2\) acting on itself by left multiplications, we can realize \(\mathbf{F}_2\) as a discrete subgroup and thus as a closed subgroup of \(S_\infty\) but \(\mathbf{F}_2\) is not amenable!

Proof. We prove the result in the discrete case. Let \(G\) be an amenable group with mean \(m\) and \(H\) be a subgroup. Let \((g_i)\) a collection of coset representatives in \(H\backslash G\) i.e. \(G=\sqcup_i Hg_i\). For \(f\in\ell^\infty(H)\), we define \(f'(g)=f(h)\) where \(g=hg_i\) for some \(g_i\). Let us observe that for \(h'\in H\), \(f'(h'g)=f(h'h)\) where \(g=hg_i\), i.e. \(h\cdot f'=(h\cdot f)'\). We set

\[m'(f)=m(f').\]

Then \(m'\) is a mean on \(\ell^\infty(H)\) and \[hm'(f)=m'(h^{-1}\cdot f)=m((h^{-1}\cdot f)')=m(h^{-1}\cdot f')=m(f')=m'(f).\]

Thus \(H\) has an invariant mean.

Corollary 3.3 The group \(\mathrm{SL}_n(\mathbf{R})\) is not amenable for \(n\geq 2\).

3.4 Exercices

Exercise 3.1 Prove that the group of transformations that preserves angles in \(\mathbf{R}^n\) is amenable for the compact open topology.

Exercise 3.2 Let \(\mathcal{H}\) be a Hilbert space.

Prove that any isometry of \(\mathcal{H}\) is an affine transformation.
Prove that the linear part of an isometry is an orthogonal transformation.
Prove the isometry group \(\mathrm{Isom}(\mathcal{H})\) is the semi-direct product \((\mathcal{H},+)\rtimes O(\mathcal{H})\) where \(O(\mathcal{H})\) is the orthogonal group of \(\mathcal{H}\).
Let us endow \(\mathrm{Isom}(\mathcal{H})\) with the pointwise convergence topology (i.e. the coarsest topology on \(\mathrm{Isom}(\mathcal{H})\) such that all maps \((g,h)\mapsto d(g(x),h(x))\) and \((g,h)\mapsto d(g^{-1}(x),h^{-1}(x))\) where \(x\in\mathcal{H}\) are continuous). Prove this a group topology.
Prove that \(\mathrm{Isom}(\mathcal{H})\) is amenable.

Exercise 3.3 Let \(G\) be a topological group.

Prove there is a unique maximal (for inclusion) amenable normal subgroup of \(G\). It is called the amenable radical of the group.
Prove that the amenable radical is a closed subgroup.

Exercise 3.4 Let \(G\) be a topological group and \(H\) a closed subgroup such that \(G/H\) is compact for the quotient topology. Prove that if \(H\) is amenable then \(G\) is amenable and if \(G\) is locally compact this is an equivalence.

Exercise 3.5 Prove that any product (finite or not) of amenable groups is amenable for the product topology.