A fixed point theorem for group actions on $L^1$ spaces

This post is a continuation of last week's post. We follow the paper of Bader-Gelander-Monod on fixed point theorems on $L^1$ spaces. I am not a functional analyst so I apologise that some concepts might not seem very motivated: I don't know enough context to tell the full story.
 
Given all the work on fixed points of groups acting on Banach spaces it is natural to ask if there is a fixed point theorem for all isometries of $L^{1}$ that preserve a given bounded set. Unlike many known cases where a geometric argument applies, there is a fundamental obstruction in $L^{1}$: any infinite group $G$ admits a fixed-point-free isometric action on a bounded convex subset of $L^{1}$. 

Example: the $G$-action on the affine subspace of summable functions of sum one on $G$. This action is fixed-point-free and preserves the closed convex bounded set of non-negative functions. The obvious (and only) fixed point, zero, is outside the space. 

Thus, we have to search for fixed points possibly outside the convex set, indeed outside the affine subspace it spans. Before that, however, we will need a bit of technical preparation.

Ryll-Nardzewski Theorem

$\textbf{Lemma 4:}$ Let $(E, \mathfrak{I})$ be a locally convex Hausdorff topological vector space, let $K$ be a nonempty $\mathfrak{I}$-separable, weakly compact, convex subset of $E$, and let $p$ be a continuous pseudo-norm on $E$. Then for each $\epsilon > 0$, there is a closed convex subset $C$ of $K$ such that $C \neq K$ and $p$-diam$(K \setminus C) \leq \epsilon$, where, for any subset $X$ of $E$, $p$-diam$(X) = \sup \{ p(x - y) : x, y \in X \}$.

This seems like a lot of mumbo-jumbo to a non-analyst like me. I think of this as just an approximation lemma: one can approximate weakly compact convex subsets of suitable topological vector spaces by closed convex subsets arbitrarily well.

Let $Q$ be a subset of a locally convex space $E$ and let $S$ be a semigroup of transformations of $Q$ into $Q$. The semigroup $S$ is called $\textit{noncontracting}$ if $0$ does not belong to the closure of $\{Tx-Ty\colon T\in S\}$ whenever $x\neq y$ and $x$, $y\in Q$. Clearly $S$ is noncontracting if and only if, for $x$, $y\in Q$ with $x\neq y$, there is a continuous pseudo-norm $p$ (depending on $x$ and $y$) on $E$ such that $\inf \{p(Tx-Ty)\colon T\in S\}>0$.

$\textbf{Ryll-Nardzewski Theorem:}$ Let $Q$ be a nonempty, weakly compact, convex subset of a locally convex Hausdorff linear topological space $E$, and let $S$ be a noncontracting semigroup of weakly continuous affine maps of $Q$ into itself. Then there is a common fixed point of $S$ in $Q$.Furthermore, when $S$ is finitely generated, the problem of finding a common fixed point of $S$ can be reduced to that of a single operator.

$\textbf{Proof:}$ By a compactness argument, it is sufficient to prove that each finite subset of $S$ has a common fixed point in $Q$. Therefore we may assume that $S$ is generated by $T_{1}$, $T_{2}$, $\cdots$, $T_{r}$. Let $T_{0}=(T_{1}+T_{2}+\cdots+T_{r})/r$. Then $T_{0}$ is a weakly continuous affine map of $Q$ into itself; one can look up the appropriate argument in the Bourbaki volume on topological vector spaces to deduce there is a fixed point $x_{0}$ of $T_{0}$ in $Q$. We will show that $T_{i}x_{0}=x_{0}$ for $i=1$, $\cdots$, $r$. Assume that this is not the case. Then by throwing out those $T_{i}$'s for which $T_{i}x_{0}=x_{0}$, we may assume that $T_{i}x_{0}\neq x_{0}$ for $i=1$, $2$, $\cdots$, $r$. Since $S$ is noncontracting there is a continuous pseudo-norm $p$ on $E$ and $\epsilon>0$ such that

(*) $p(TT_{i}x_{0}-Tx_{0})>\epsilon$ for all $T$ in $S$ and $i=1$, $\cdots$, $r$.

Let $K$ be the closed convex hull of $\{Tx_{0}\colon T\in S\}$. Then $K$ is a weakly compact, convex, separable subset of $E$. Hence, by the lemma, there is a closed convex subset $C$ of $K$ such that $C\neq K$ and $p$-diam $(K \setminus C)\leq \epsilon$. Since $C \neq K$, there is an element $S$ in $S$ such that $Sx_0 \in K \setminus C$. From $T_0 x_0 = x_0$, we see that
\[Sx_0 = (ST_1 x_0 + ST_2 x_0 + \cdots + ST_r x_0)/r.\]
Hence $ST_i x_0 \in K \setminus C$ for at least one $i$, since otherwise $Sx_0 \in C$. It follows that $p(ST_i x_0 - Sx_0) \leq p$-diam $(K \setminus C) \leq \epsilon$, contradicting inequality (*). The proof of the theorem is therefore complete. $\blacksquare$

Hewitt-Yosida decomposition

This subsection is a crash course in some definitions that will be used later but can be skipped if the reader is willing to take a little bit more on faith.

Let $S$ be an abstract space and $Y$ a commutative Hausdorff topological group. We denote by $V$ a base for the neighbourhoods of the identity in $Y$ consisting of closed symmetric sets, and by $H$ a ring of subsets of $S$ (under intersection and symmetric difference) with $S \in H_{\sigma}$, where $H_{\sigma}$ is the set $\{ \cup_n A_n: A \text{is a sequence in} K\}$.

We recall first that a function $\phi$ on $H$ to $Y$ is $s\textit{-bounded}$ if and only if $\phi(A_{n})\to 0$, whenever $A$ is a disjoint sequence in $H$. If $\phi$ is finitely additive, an equivalent condition is that $\phi(A_{n})$ be Cauchy whenever $A$ is an increasing sequence in $H$ (and hence, also, whenever $A$ is a decreasing sequence in $H$).

We assume throughout that $\phi$  is finitely additive and $s$-bounded on $H$ to a complete subset of $Y$. Let $H$ be a field, i.e., that $S\in H$. 

We proceed to describe the Hewitt-Yosida decomposition of the $s$-bounded, finitely additive function $\phi$ on $H$ to $Y$.

We need two definitions.

 $\textbf{Definition:}$ For $\psi$ and $v$, set functions on $H$ to topological groups $Y$ and $Z$, respectively,

1. $\psi$ and $v$ are (topologically) singular, $\psi\perp_{t}v$, if and only if given neighbourhoods $V$ and $W$ of the origins in $Y$ and $Z$, respectively, there exists such an $A$ in $H$ that, for all $E$ in $H$, \[    \psi(E\cap A)\,\in\,V\quad\text{and}\quad v(E\backslash A)\,\in\,W;\]
2. $\psi$ $\textit{is purely finitely additive}$ if and only if $\psi$ is finitely additive and $\psi\perp_{t}v$ for every $\sigma$-additive $s$-bounded $v$ on $H$ to a commutative topological group.

 $\textbf{Hewitt-Yosida Theorem:}$ Every $s$-bounded finitely additive set function $\phi$ on the field $H$ with values in a complete subset of a topological group $Y$ can be uniquely represented in the form $\phi=\phi_{\sigma}+\phi_{p}$, where $\phi_{\sigma}$ is $\sigma$-additive and $s$-bounded and $\phi_{p}$ is purely finitely additive.

Back to the fixed point theorem

Recall that a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. Typical examples include the ring $L^{\infty }(\mathbb {R} )$ of essentially bounded measurable functions on the real line, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space $L^{2}(\mathbb {R} )$ of square-integrable functions, and the algebra ${\mathcal {B}}({\mathcal {H}})$ of all bounded operators on a Hilbert space ${\mathcal {H}}$.

We shall show a fixed point theorem more generally for any $L-\textit{embedded}$ Banach space $V$, that is, a space whose bidual can be decomposed as $V^{**}=V\oplus_{1}V_{0}$ for some $V_{0}\subseteq V^{**}$ (and $\oplus_{1}$ indicates that the norm is the sum of the norms on $V$ and $V_{0}$). $L^{1}$ is L-embedded by the Yosida-Hewitt decomposition and this holds more generally for the predual of any von Neumann algebra; in particular, for the dual of any C*-algebra (recall that Sakai showed that von Neumann algebras can also be defined abstractly as C*-algebras that have a predual; in other words the von Neumann algebra, considered as a Banach space, is the dual of some other Banach space called the predual. The predual of a von Neumann algebra is in fact unique up to isomorphism).

$\textbf{Theorem 5:}$ Let $A$ be a non-empty bounded subset of an $L$-embedded Banach space $V$. Then there is a point in $V$ fixed by every isometry of $V$ preserving $A$. Moreover, one can choose a fixed point which minimises $\sup_{a\in A}\|v-a\|$ over all $v\in V$.

We recall that an isometric action of a group $G$ on a Banach space $V$ is given by a linear part and a cocycle $b:G\to V$. The cocycle is the orbital map of $0\in V$ and a fixed point $v$ corresponds to a trivialisation $b(g)=v-g.v$, where $g.v$ is the linear action. The above norm statement implies that one can arrange $\|v\|\leq\sup_{g}\|b(g)\|$ by considering $A=b(G)\ni 0$.

We first recall the concept of Chebyshev centre. Let $A$ by a non-empty bounded subset of a metric space $V$. The circumradius of $A$ in $V$ is

\[\varrho_{V}(A)=\inf\big{\{}r\geq 0:\exists\,x\in V\text{ with }A\subseteq \overline{B}(x,r)\big{\}},\]

where $\overline{B}(x,r)$ denotes the closed $r$-ball around $x$. The Chebyshev centre of $A$ in $V$ is the (possibly empty) set

\[C_{V}(A)=\big{\{}c\in V:A\subseteq\overline{B}(c,\varrho_{V}(A))\big{\}}.\]

Notice that $C_{V}(A)$ can be written as an intersection of closed balls as follows:

\[C_{V}(A)=\bigcap_{r>\varrho_{V}(A)}C_{V}^{r}(A)\quad\text{where}\quad C_{V}^{r}(A)=\bigcap_{a\in A}\overline{B}(a,r).\]

Thus, when $V$ is a normed space, $C_{V}(A)$ is a bounded closed convex set. More importantly, when $V$ is a dual Banach space, we deduce from Alaoglu's theorem that $C_{V}(A)$ is weak-* compact and that it is non-empty because the non-empty sets $C_{V}^{r}(A)$ are monotone in $r$. 

$\textbf{Proposition 6:}$ Let $A$ be a non-empty bounded subset of an $L$-embedded Banach space $V$. Then the convex set $C_{V}(A)$ is weakly compact and non-empty.

$\textbf{Proof:}$ Consider $A$ as a subset of $V^{**}$ under the canonical embedding $V\subseteq V^{**}$. In view of the above discussion, $C_{V^{**}}(A)$ is a non-empty weak-* compact convex set. We claim that it lies in $V$ and coincides with $C_{V}(A)$; the proposition then follows. Let thus $c\in C_{V^{**}}(A)$ and write $c=c_{V}+c_{V_{0}}$ according to the decomposition $V^{**}=V\oplus_{1}V_{0}$. Then, for any $a\in A$, we have

\[\|a-c\|=\|a-c_{V}\|+\|c_{V_{0}}\|\]

since $A\subseteq V$. Therefore,

\[\varrho_{V^{**}}(A)=\sup_{a\in A}\|a-c\|=\sup_{a\in A}\|a-c_{V}\|+\|c_{V_{0}}\| \geq\varrho_{V}(A)+\|c_{V_{0}}\|.\]

Since $\varrho_{V^{**}}(A)\leq\varrho_{V}(A)$ anyway, we deduce $c_{V_{0}}=0$ and $\varrho_{V^{**}}(A)=\varrho_{V}(A)$, whence the claim. $\square$

$\textbf{Proof of Theorem 5:}$ Since the definition of $C_{V}(A)$ is metric, it is preserved by any isometry preserving $A$. By the proposition, we can apply the Ryll-Nardzewski theorem and deduce that there is a point of $C_{V}(A)$ fixed by all isometries preserving $A$. The norm condition follows from the definition of centres. $\blacksquare$

Many, many corollaries

As a special case (the ``commutative'' case), we recover an important theorem due to Losert, but with an improved (indeed optimal) norm estimate:

$\textbf{Corollary 7:}$ Let $G$ be a group acting by homeomorphisms on a locally compact space $X$. Then any bounded cocycle $b:G\to M(X)$ to the space of (finite Radon) measures on $X$ is trivial. More precisely, there is a measure $\mu$ with $\|\mu\|\leq\sup_{g\in G}\|b(g)\|$ such that $b(g)=\mu-g.\mu$ for all $g\in G$.

Indeed, $M(X)$ is the dual of the (commutative) C*-algebra $C_{0}(X)$ and hence the predual of a von Neumann algebra.

 $\textbf{Corollary 8:}$ If $G$ is a locally compact group, then any derivation from the convolution algebra $L^{1}(G)$ to $M(G)$ is inner. 
 
 This is often phrased in terms of derivations ``of $L^{1}(G)$'' since any derivation $L^{1}(G)\to M(G)$ must range in $L^{1}(G)$ by Paul Cohen's factorisation theorem. It also follows that any derivation of $M(G)$ is inner. The given norm estimate is in fact optimal by Remark 7.2(a) in Losert's paper 'The derivation problem for group algebras. The deduction of Corollary 8 is simply putting together a bunch of results of other people so we omit it. We list some other corollaries of Corollary 7 from Losert's paper.

 $\textbf{Corollary 9:}$ Let $G$ denote a locally compact group, $H$ a closed subgroup. Then any bounded derivation $D:M(H)\to M(G)$ is inner.

Again, the same conclusion applies to bounded derivations $D\colon L^{1}(H)\to M(G)$.

 $\textbf{Corollary 10:}$ For any locally compact group $G$, the first continuous cohomology group $\mathcal{H}^{1}(L^{1}(G),M(G))$ is trivial.

 $\textbf{Corollary 11:}$ Let $G$ be a locally compact group and assume that $T\in\mathrm{VN}(G)$ satisfies $T*u-u*T\in M(G)$ for all $u\in L^{1}(G)$. Then there exists $\mu\in M(G)$ such that $T-\mu$ belongs to the centre of $\mathrm{VN}(G)$.

$\textbf{Proof:}$ With $\mathrm{VN}(G)$ denoting the von Neumann algebra of $G$, $M(G)$ is identified with the corresponding set of left convolution operators on $L^{2}(G)$  and is thus considered as a subalgebra of $\mathrm{VN}(G)$. By analogy, we also use the notation $S*T$ for multiplication in $\mathrm{VN}(G)$. Then $\operatorname{ad}_{T}(u)=T*u-u*T$ defines a derivation from $L^{1}(G)$ to $M(G)$ and (from Corollary 8) $\operatorname{ad}_{T}=\operatorname{ad}_{\mu}$ implies that $T-\mu$ centralizes $L^{1}(G)$. Since $L^{1}(G)$ is dense in $\mathrm{VN}(G)$ for the weak operator topology, it follows that $T-\mu$ is central. $\square$

Now we can go back to listing corollaries of Theorem 5.

 $\textbf{Corollary 12:}$ Let $A$ be a unital C*-algebra. Let $M_{*}$ be the predual of a von Neumann algebra. Assume $M_{*}$ is a Banach bimodule over $A$. Then any arbitrary derivation $D:A\to M_{*}$ is inner. Moreover, we can choose $v\in M_{*}$ with $D(a)=v.a-a.v$ such that $\|v\|\leq\|D\|$.

Haagerup's earlier result on weak amenability of $A$ is given by the special case $M_{*}=A^{*}$. Our definition of Banach bimodule demands $\|a.v.b\|\leq\|a\|\cdot\|v\|\cdot\|b\|$ ($a,b\in A$, $v\in M_{*}$).

$\textbf{Proof sketch:}$ Earlier work of Ringrose implies $D$ is continuous; thus it is bounded (by $\|D\|<\infty$) on the group $G$ of unitaries of $A$. The map $G\to M_{*}$ given by $g\mapsto D(g).g^{-1}$ is a cocycle for the Banach $G$-module structure defined by the rule $v\mapsto g.v.g^{-1}$. Theorem 5 thus yields $v$, with norm bounded by $\|D\|$, such that $D(g)=v.g-g.v$ for all $g\in G$. The statement follows since any element of $A$ is a combination of four unitaries. $\square$

In fact, this shows that any continuous derivation from any normed algebra $A$ to a predual $M_{*}$ of a von Neumann algebra is inner as soon as $A$ is spanned by the elements represented as invertible isometries of $M_{*}$.

Finally, returning to the case $V=L^{1}$ of Theorem 5 we can consider actions without $\textit{a priori}$ boundedness of the orbits and obtain a new characterisation of Kazhdan groups, coming full circle to how we started last post:

\textbf{Corollary 13:} Let $\Omega$ by any measure space. Then any isometric action of a Kazhdan group on $L^{1}(\Omega)$ has a fixed point. Moreover, this fixed point property characterises Kazhdan groups amongst all countable (or locally compact $\sigma$-compact) groups.

We omit the proof of this because, well, there have been quite a few corollaries and the post is long enough already, and for the somewhat less lazy reason that similar arguments to those in the proof of theorem 3 in the previous post go through once one quotes a couple of results.