Groups acting on Banach spaces

In this post all groups $G$ are topological groups. By a linear isometric $G$-representation on a Banach space $B$, we shall mean a continuous homomorphism $\rho: G \to O(B)$, where $O(B)$ denotes the “orthogonal” group of all invertible linear isometries $B \to B$. We discuss actions on more general Banach spaces as a prelude to the next post, where we discuss fixed points and applications to derivations of group algebras. 
 
Geometric group theory meets functional analysis most prominently (in my opinion) in the study of groups with Property (T). For a brief introduction to Property (T) see this previous blog post. Property (T) should be thought of as a rigidity result for groups: they do not admit fixed-point free actions on a lot of different spaces.

Recall that $G$ is said to have property $(FH)$ if any continuous action of $G$ on a Hilbert space $H$ by affine isometries has a $G$-fixed point. Delorme and Guichardet proved that properties (T) and $(FH)$ are equivalent for $\sigma$-compact groups. This is an important result in the theory of Property (T) groups as well as being one of the first results in the direction of groups acting on Banach spaces. Naturally, this means there have been generalisations.


$\textbf{Definition:}$ We say that $G$ has property $(F_B)$ if any continuous action of $G$ on $B$ by affine isometries has a $G$-fixed point.

Below we summarize the relations between properties (T) and $(F_B)$ which hold for general groups, but first recall the following:

Frechet spaces

 $\textbf{Definition:}$ A topological vector space \( X \) is a \textbf{Fréchet space} if and only if it satisfies the following three properties:

1. It is a Hausdorff space.
2. Its topology may be induced by a countable family of seminorms \(\{ \| \cdot \|_k \}_{k \in \mathbb{N}_0}\). This means that a subset \( U \subseteq X \) is open if and only if for every \( u \in U \) there exist \( K \geq 0 \) and \( r > 0 \) such that \(\{ v \in X : \| v - u \|_k < r \text{ for all } k \leq K \}\) is a subset of \( U \).
3. It is complete with respect to the family of seminorms.

Some basic facts about Frechet spaces are: 

A family \( P \) of seminorms on \( X \) yields a Hausdorff topology if and only if \[\bigcap_{\| \cdot \| \in P} \{ x \in X : \| x \| = 0 \} = \{ 0 \}.\]

A sequence \((x_n)_{n \in \mathbb{N}}\) in \( X \) converges to \( x \) in the Fréchet space defined by a family of seminorms if and only if it converges to \( x \) with respect to each of the given seminorms.

The relevance to our setting is this:
If \( G \) is a topological group acting by affine transformations on a topological vector space \( V \), continuity of the action

\[G \times V \to V, \quad g \cdot x = \varrho(g)x + c(g),\]

is equivalent to continuity of the linear part \( G \times V \to V \) and the continuity of the cocycle \( c: G \to V \). Indeed \( c(g) = g \cdot 0 \), and \( \varrho(g)x = g \cdot x - c(g) \).

Hence, in the context of topological groups, affine actions should be assumed continuous, and \( Z^1(G, \varrho) \) will include only continuous cocycles \( c: G \to V \) (we assume that the linear part \( \varrho \) is continuous as well). If \( G \) is a locally compact \( \sigma \)-compact group, then \( Z^1(\varrho) \) has a natural structure of a Fréchet space with respect to the family of semi-norms

\[\|c\|_K = \sup_{g \in K} \|c(g)\|_V,\]

where \( K \subseteq G \) runs over a countable family of compact subsets which cover \( G \) and \( \|\cdot\|_V \) is a norm inducing the topology of \( V \). 

Actions on general Banach spaces

$\textbf{Theorem 1:}$ For a locally compact second countable group $G$ we have

1.  $(F_B)$ implies $(T_B)$ for any Banach space $B$;
2. $(T)$ implies $(F_B)$ for closed subspaces $B$ of $L^p(\mu)$, where $1<p<2$; likewise for subspaces of $L^1$ and of the pseudo-normed spaces $L^p(\mu)$, $0<p<1$, except one obtains only bounded orbits instead of fixed points;(*)

$\textbf{Proof of 1:}$ Assume that $G$ does not have property $(T_E)$, where $E$ is a Banach space, and let $\varrho: G \rightarrow O(E)$ be a representation such that $E/E^{\varrho(G)}$ admits almost invariant vectors. In order to show that $H^1(G, \varrho) \neq \{0\}$ it suffices to prove that $B^1(G, \varrho) \subseteq Z^1(G, \varrho)$ is not closed.

As was explained in the previous section, the space of $\varrho$-cocycles $Z^1(G, \varrho)$ is always a Fréchet space (and even a Banach space if $G$ is compactly generated). Note that $B^1(G, \varrho)$ is the image of the bounded linear map

\[\tau: E \longrightarrow Z^1(G, \varrho), \quad (\tau(v))(g) = v - \varrho(g)v.\]

If $\tau(E)$ were closed, and hence a Fréchet space, the open mapping theorem would imply that $\tau^{-1}: B^1(G, \varrho) \rightarrow E/E^{\varrho(G)}$ is a bounded map. That would mean that for some $M < \infty$ and a compact $K \subseteq G$,

\[\|v\| \leq M \|\tau(v)\|_K = M \sup_{g \in K} \|\varrho(g)v - v\|, \quad v \in E/E^{\varrho(G)},\]

contrary to the assumption that $\varrho$ almost contains invariant vectors. $\square$

For the proof of 2 we will need a couple of intermediate facts.

The first is the following. $L^p(\mu)$-spaces, for $p \in (0, 2]$, admit embeddings $j: B:=L^p(\mu) \to \mathcal{H}$ into the unit sphere of a Hilbert space so that

\[\langle j(x), j(y) \rangle = \| x - y \|^p.\]

Having such an embedding is equivalent to the following result.

$\textbf{Proposition 2:}$ For $0<p<2$ and any $s>0$ the function $f \mapsto e^{-s\|f\|^p}$ is positive definite on $L^p(\mu)$, i.e. for any finite collection $f_i \in L^p(\mu)$ and any $\lambda_i \in \mathbb{C}$,

\[\sum_{i,j} e^{-s\|f_i - f_j\|^p} \lambda_i \bar{\lambda}_j \geq 0.\]

In fact, more is known: it was shown by Bretagnolle, Dacunha-Castelle, and Krivine that, for $1 \leq p \leq 2$, a Banach space $X$ is isometric to a closed subspace of $L^p(\mu)$ if and only if $e^{-s\| \cdot \|^p}$ is a positive definite function on $X$ for any $s > 0$.

The second is the following, which I think is fairly believable.

$\textbf{Lemma 3:}$ For a uniform equicontinuous affine action of a group $G$ on a superreflexive space $V$, the following are equivalent:

•  there exists a bounded $G$-orbit;
• all $G$-orbits are bounded;
• $G$ fixes a point in $V$;
• $G$ preserves a (Borel regular) probability measure on $V$.

Note that the notion of a subset $E \subseteq V$ being bounded, means that for any open neighbourhood $U$ of $0 \in V$ there is some $t \in \mathbb{R}$ so that $E \subseteq tU$. This notion agrees with the notion of being bounded with respect to any compatible norm on $V$.

$\textbf{Proof of 2:}$ Let $G$ be a locally compact group with Kazhdan's property $(T)$ acting by affine isometries on a closed subspace $B \subseteq L^p(\mu)$ with $0 < p \leq 2$. Using Proposition 2 and a slight modification of a Delorme–Guichardet argument for $(T) \Rightarrow (FH)$ we shall prove that such an action has bounded orbits. For $1 < p \leq 2$ uniform convexity of $B \subseteq L^p(\mu)$ yields a $G$-fixed point using Lemma 3.

Proposition 2 allows us to define a family, indexed by $s > 0$, of Hilbert spaces $\mathcal{H}_s$, embeddings $U_s: B \rightarrow S(\mathcal{H}_s)$ and unitary representations $\pi_s: G \rightarrow O(\mathcal{H}_s)$ with the following properties:

1.  the image $U_s(B)$ spans a dense subspace of $\mathcal{H}_s$;
2. $\langle U_s(x), U_s(y) \rangle = e^{-s \|x - y\|^p}$ for all $x, y \in B$;
3. $U_s(gx) = \pi_s(g)U_s(x)$ for all $x \in B$ and $g \in G$.

Indeed, one constructs $\mathcal{H}_s$ as the completion of the pre-Hilbert space whose vectors are finite linear combinations $\sum_i a_i x_i$ of points $x_i \in B$, and the inner product is given by

\[\left\langle \sum_i a_i x_i, \sum_j b_j y_j \right\rangle = \sum_{i,j} a_i \bar{b}_j e^{-s \|x_i - y_j\|^p}.\]

The representation $\pi_s$ can be constructed (and is uniquely determined) by property (3).

Since $G$ is assumed to have Kazhdan's property $(T)$, for some compact subset $K \subseteq G$ and $\epsilon > 0$, any unitary $G$-representation with $(K, \epsilon)$-almost invariant vectors has a non-trivial invariant vector.

Let $x_0 \in B$ be fixed. The isometric $G$-action is continuous, so $K x_0$ is a compact and hence bounded subset of $B$, hence

\[R_0 = \sup_{g \in K} \|g x_0 - x_0\| < \infty.\]

For the unit vectors $u_s = U_s(x_0) \in S(\mathcal{H}_s)$ we have

\[\min_{g \in K} |\langle \pi_s(g) u_s, u_s \rangle| \geq e^{-s R_0^p} \to 1, \quad \text{as } s \to 0.\]

In particular, for a sufficiently small $s > 0$, $\max_{g \in K} \|\pi_s(g) u_s - u_s\| < \epsilon$. Let us fix such an $s$, and rely on property $(T)$ to deduce that $\pi_s$ has an invariant vector $v \in S(\mathcal{H}_s)$.

We claim that $G$ must have bounded orbits for its affine isometric action on $B$. Indeed, otherwise there would exist a sequence $g_n \in G$ so that

\[\|g_n x - y\| \to \infty \quad \text{and hence } \langle \pi_s(g_n) U_s(x), U_s(y) \rangle \to 0\]

for all $x, y \in B$. This implies that $\langle \pi_s(g_n) w, u \rangle \to 0$ for any $w, u \in \mathrm{span}(U_s(B))$ and, since $\mathrm{span}(U_s(B))$ is dense in $\mathcal{H}_s$, for any $w, u \in \mathcal{H}_s$. Taking $w = u = v$, we get a contradiction. Therefore the affine isometric $G$-action on $B$ has bounded orbits, and hence fixes a point in case $1 < p \leq 2$. $\square$