# Complementation in continuous cohomology with coefficients in Banach modules

Mario Klisse  KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001
Leuven, Belgium<EMAIL_ADDRESS>
(Date: . _MSC2010:_ 20J06, 22D12, 46B99. The author is supported by the FWO
postdoctoral grant 1203924N of the Research Foundation Flanders.)

###### Abstract.

In this article, we introduce the concept of weakly uniquely stationary
representations. This framework enables us to investigate the
complementability of closed subspaces within the context of continuous
cohomology with coefficients in Banach modules. As an application, we extend
and refine several cohomological results from the literature, particularly in
the settings of nilpotent groups, products of groups, and lattices.

## Introduction

The study of group cohomology with coefficients in Banach modules is a rich
area of mathematics that lies at the intersection of several fields, including
algebra, topology, and functional analysis. Its motivations are both
theoretical, in terms of understanding the algebraic and topological
properties of groups and their actions, and practical, in terms of
applications to areas like representation theory, harmonic analysis, and
geometric group theory. In the context of unitary representations on Hilbert
spaces, a notable application of the theory is the characterization of
Kazhdan’s Property (T) through the vanishing of 1-cohomology groups (see
[12]).

While the continuous cohomology of groups with coefficients in unitary
representations on Hilbert spaces is well-studied, attempts of extending
existing results to more general Banach modules often lead to technical
difficulties due to the lack of concepts like orthogonality. One example of
this phenomenon is the complementability of closed subspaces, whose study for
general Banach modules was initiated by Nowak in [31]. Recall that a subspace
$W$ of a topological vector space $V$ is _complemented_ , if there exists a
subspace $W^{\prime}\subseteq V$ such that $V$ decomposes as a topological
direct sum $V=W\oplus W^{\prime}$. Under suitable conditions, particularly
covering all isometric representations without almost invariant vectors on
uniformly convex Banach spaces, the main result of [31] proves the
complementation of the subspace of 1-coboundaries in the space of all
1-cocycles. This result was later extended to higher degrees by Rosendal in a
more restrictive setting (see [37]).

In this paper, we investigate the complementation of closed subspaces within
the context of continuous cohomology with coefficients in Banach modules. To
this end, we introduce the concept of weakly uniquely stationary
representations.

######  Definition.

_Let $G$ be a topological group and $\mu$ a Baire probability measure on $G$.
We call a uniformly bounded linear representation $\rho$ of $G$ on a Banach
space $V$ _weakly uniquely $\mu$-stationary_ if it is $\mu$-integrable (in the
sense of Definition 2.1) and if every non-trivial functional that is invariant
under the convolution with the probability measure $\mu$ admits an element
$v\in V$ with $\phi(v)\neq 0$ that is invariant under the convolution
$\rho_{\mu}$ with $\mu$._

The $\rho_{\mu}$-invariant elements $V^{\mu}$ of a weakly uniquely
$\mu$-stationary Banach $G$-module $V$ form the range of a unique projection
$\widetilde{\mathbb{E}}_{\mu}$ on $V$. Our definition draws inspiration from
the concept of uniquely stationary actions of groups on C∗-algebras, as
introduced by Hartman and Kalantar in [21]. It unifies a number of existing
concepts, such as weakly almost periodic Banach modules (see [3] and also
[7]), strictly convex Banach modules, and representations vanishing at
infinity. Additionally, it naturally relates to the work of Jaworski and
Neufang on the Choquet–Deny equation in dual Banach spaces in [22], and to
results of Bekka and Valette in [4].

Motivated by an important ingredient in the proof of Shalom’s cohomological
rigidity results for unitary representations of locally compact groups in
[38], in [3] Bader, Rosendal and Sauer proved the vanishing of certain
restriction maps in continuous reduced cohomology of weakly almost periodic
Banach modules. We extend their main theorem to the setting of weakly uniquely
stationary representations, enabling us to derive complementation results for
the corresponding higher-order cohomology groups.

######  Theorem A.

Let $G$ be a topological group containing commuting subgroups $N,C\leq G$. Let
furthermore $\mu$ be a Baire probability measure on $C$, $(V,\rho)$ a
uniformly bounded, strongly operator continuous, weakly uniquely
$\mu$-stationary Banach $G$-module, and $b\in Z^{n}(G,\rho)$ a $n$-cocycle.
Then $\widetilde{\mathbb{E}}_{\mu}\circ(b|_{N^{n+1}})$ defines a $n$-cocycle
that is almost cohomologous to the restriction $b|_{N^{n+1}}$.

The study of the complementation of closed subspaces in continuous cohomology
has several intriguing applications, such as in Hodge theory, in the deduction
of the vanishing of certain cohomology groups (see e.g. [18]), and in Ozawa’s
functional analytic proof of Gromov’s polynomial growth theorem presented in
[32]. Additionally, it has been applied to address rigidity questions
concerning unitary representations of irreducible integrable lattices in
products of groups as in [38], [2] (see also [10], [11]). Inspired by this, we
utilize our aforementioned results to examine the decomposition of the
continuous cohomology of products of groups.

######  Theorem B.

Let $G:=G_{1}\times G_{2}$ be a product of topological groups and $(V,\rho)$ a
uniformly bounded strongly operator continuous Banach $G$-module. Assume that
$G_{1}$ and $G_{2}$ admit Baire probability measures $\mu_{1}$ and $\mu_{2}$
and null sets $\mathcal{N}_{1}\subseteq G_{1}$, $\mathcal{N}_{2}\subseteq
G_{2}$ for which both $G_{1}\setminus\mathcal{N}_{1}$ and
$G_{2}\setminus\mathcal{N}_{2}$ are contained in compact sets. Assume
furthermore that $\|\rho_{\mu_{1}}\rho_{\mu_{2}}\|<1$ and that
$V^{\mu_{1}}=V^{G_{1}}$ and $V^{\mu_{2}}=V^{G_{2}}$. Then there exists a
topological isomorphism of continuous cohomology groups $H_{c}^{1}(G_{1}\times
G_{2},V)\cong H_{c}^{1}(G_{1},V^{G_{2}})\oplus H_{c}^{1}(G_{2},V^{G_{1}})$.

######  Theorem C.

Let $G:=G_{1}\times G_{2}$ be a product of topological groups, $\mu_{1}$,
$\mu_{2}$ Baire probability measures on $G_{1}$ and $G_{2}$, and $(V,\rho)$ a
uniformly bounded strongly operator continuous Banach $G$-module. Assume that
$(V,\rho|_{G_{1}})$ is weakly uniquely $\mu_{1}$-stationary without almost
invariant unit vectors, and that $(V,\rho|_{G_{2}})$ is weakly uniquely
$\mu_{2}$-stationary. Then there exists an embedding of topological vector
spaces $H_{c}^{1}(G_{1}\times G_{2},V)\hookrightarrow
H_{c}^{1}(G_{1},V^{\mu_{2}})\oplus H_{c}^{1}(G_{2},V^{\mu_{1}})$. If
furthermore $V^{\mu_{1}}=V^{G_{1}}$ and $V^{\mu_{2}}=V^{G_{2}}$, then the
embedding is surjective, i.e. $H_{c}^{1}(G_{1}\times G_{2},V)\cong
H_{c}^{1}(G_{1},V^{G_{2}})$.

Building on the ideas presented in [38], [2], [10], we use these
decompositions to derive new superrigidity results for suitable
representations of irreducible integrable lattices in products of groups.

_Structure_. The article is organized as follows. In Section 1, we review
fundamental concepts related to the theory of Baire sets and Baire measures,
Banach modules, and group cohomology. In Section 2 we investigate elements in
the bidual of Banach spaces that are stationary with respect to convolution
with a given Baire probability measure. Motivated by our results, in Section 3
we introduce the concept of weakly uniquely stationary representations and
explore various examples, including uniquely stationary actions of discrete
groups on C∗-algebras, weakly almost periodic Banach modules, and linear
representations vanishing at infinity. Finally, in Section 4, we apply the
notion of weakly uniquely stationary representations to the study of group
cohomology with coefficients in Banach modules, obtaining complementation and
rigidity results.

##  1\. Preliminaries and notation

###  1.1. General notation

In this article, all Banach spaces will be assumed to be real or complex.
Given a Banach space $V$, we denote the bounded linear operators on $V$ by
$\mathcal{B}(V)$ and $\text{GL}(V)\subseteq\mathcal{B}(V)$ is the subset of
invertible bounded linear operators on $V$. The dual space and bidual space of
$V$ will be denoted by $V^{\ast}$ and $V^{\ast\ast}$ respectively. We
furthermore write $C_{b}(X,\mathbb{R})$ for the bounded continuous real-valued
functions on a topological space $X$. The topological groups in this article
are not necessarily assumed to be Hausdorff.

###  1.2. Baire sets and Baire measures

In the study of measures in terms of integrals of continuous functions, it is
often convenient or even required to pass to $\sigma$-algebras that are
smaller than the corresponding Borel $\sigma$-algebra. In the general
topological setting considered in this article, the notion of Baire measures
will be suitable. Note that there exist several (inequivalent) definitions of
this concept; we will be following the conventions and notation of [6] and
[17].

Define the _Baire $\sigma$-algebra_ $\mathcal{B}a(X)$ of a topological space
$X$ as the smallest $\sigma$-algebra with respect to which all continuous
real-valued functions on $X$ are measurable. In general $\mathcal{B}a(X)$ does
not need to coincide with the Borel $\sigma$-algebra of $X$; for metrizable
spaces however this is the case. A (signed) measure on $\mathcal{B}a(X)$ is
called a _Baire measure_ and the set of all Baire measures will be denoted by
$\mathcal{M}_{\sigma}(X)$. Its subset of finite Baire measures can be equipped
with the (locally convex, Hausdorff) _weak topology_ induced by the
requirement that a net $(\mu_{\lambda})_{\lambda\in\Lambda}$ _weakly
converges_ to $\mu$ if and only if
$\int_{X}fd\mu_{\lambda}\rightarrow\int_{X}fd\mu$ for all $f\in
C_{b}(X,\mathbb{R})$. (Note that this terminology, despite being standard,
conflicts with the standard one in functional analysis.) We denote the
subspace of _Baire probability measures_ on $X$ by $\mathcal{P}_{\sigma}(X)$
and equip it with the subspace topology.

We will make use of the following well-known statement, see e.g. [6, Example
4.2.5]. We include its proof for the convenience of the reader.

######  Lemma 1.1.

Let $X$ be a topological space. Then the convex hull of the set of all point
measures in $X$ is weakly dense in $\mathcal{P}_{\sigma}(X)$.

###### Proof.

Let $\mu\in\mathcal{P}_{\sigma}(X)$. Sets of the form

$\mathcal{U}_{f_{1},...,f_{n},\varepsilon}(\mu):=\left\\{\nu\in\mathcal{P}_{\sigma}(X)\mid\left|\int_{X}f_{i}d\mu-\int_{X}f_{i}d\nu\right|<\varepsilon\text{
for }i=1,...,n\right\\}$

with $\varepsilon>0$ and $f_{1},...,f_{n}\in C_{b}(X,\mathbb{R})$ define a
neighbourhoods basis of $\mu$. It hence suffices to show that every such set
intersects non-trivially with the convex hull of all point measures
$\delta_{x}$, $x\in X$. Since the functions $f_{1},...,f_{n}$ are measurable
and bounded, we find simple functions $g_{1},...,g_{n}$ for which $\sup_{1\leq
i\leq n}\|f_{i}-g_{i}\|_{\infty}<\varepsilon/2$. The $g_{i}$ are (without loss
of generality) of the form $g_{i}=\sum_{j=1}^{m}c_{i,j}\chi_{M_{j}}$ with
pairwise disjoint Baire sets $M_{1},...,M_{m}\in\mathcal{B}a(X)$ and
$c_{i,1},...,c_{i,m}\in\mathbb{R}$. For every $1\leq j\leq m$ pick $x_{j}\in
M_{j}$, let $y\in X\setminus(\bigcup_{j=1}^{m}M_{j})$ and define
$\nu:=\mu\left(X\setminus(\bigcup_{j=1}^{m}M_{j})\right)\delta_{y}+\sum_{j=1}^{m}\mu(M_{j})\delta_{x_{j}}$.
Then $\nu$ is a Baire probability measure with
$\int_{X}g_{i}d\mu=\sum_{j=1}^{m}c_{i,j}\mu(M_{j})=\int_{X}g_{i}d\nu$ for
$i=1,...,n$ and hence $\nu\in\mathcal{U}_{f_{1},...,f_{n},\varepsilon}(\mu).$
∎

###  1.3. Banach modules

Let $G$ be a topological group and $\rho:G\rightarrow\text{GL}(V)$,
$g\mapsto\rho_{g}$ a uniformly bounded linear representation of $G$ on a
Banach space $V$. For the sake of brevity we also say that $(V,\rho)$ is a
_uniformly bounded Banach $G$-module_ and simply write $g.v:=\rho_{g}(v)$ for
$g\in G$, $v\in V$. The representation $\rho$ naturally induces a uniformly
bounded linear representation $\rho^{\ast}:G\rightarrow\text{GL}(V^{\ast})$ of
$G$ on the dual space $V^{\ast}$ via
$\rho_{g}^{\ast}(\phi):=\phi\circ\rho_{g^{-1}}$ for $g\in G$, $\phi\in
V^{\ast}$, which is called the _dual representation_ or _contragradient
representation_. Recall that there exists an isometric embedding
$\iota_{V}:V\hookrightarrow V^{\ast\ast}$ of $V$ into its bidual via
$(\iota_{V}(v))(\phi):=\phi(v)$ for every $v\in V$, $\phi\in V^{\ast}$. We
will also write $\widehat{v}:=\iota_{V}(v)$. The representation $\rho$ then
extends to a uniformly bounded linear representation
$\rho^{\ast\ast}:G\rightarrow\text{GL}(V^{\ast\ast})$ via
$\rho_{g}^{\ast\ast}(f):=f\circ\rho_{g^{-1}}^{\ast}$ for $g\in G$, $f\in
V^{\ast\ast}$ . With respect to $\rho^{\ast\ast}$ the embedding $\iota_{V}$ is
$G$-equivariant and we have that $\sup_{g\in G}\|\rho_{g}\|=\sup_{g\in
G}\|\rho_{g}^{\ast}\|=\sup_{g\in G}\|\rho_{g}^{\ast\ast}\|$. By Goldstine’s
theorem the image of the closed unit ball in $V$ under $\iota_{V}$ is
weak∗-dense in $V^{\ast\ast}$, so in particular $\text{im}(\iota_{V})\subseteq
V^{\ast\ast}$ is weak∗-dense. If $G$ is a topological group, $\rho$ (or
$(V,\rho)$) is called _strongly operator continuous_ if the map $G\ni
g\mapsto\rho_{g}(v)$ is continuous with respect to the norm on $V$ for every
$v\in V$. Similarly, $\rho$ (or $(V,\rho)$) is called _weakly operator
continuous_ if the map $G\ni g\mapsto\rho_{g}(v)$ is continuous with respect
to the weak topology on $V$ for every $v\in V$.

We denote the space of bounded operators on $V$, $V^{\ast}$ and $V^{\ast\ast}$
by $\mathcal{B}(V)$, $\mathcal{B}(V^{\ast})$ and $\mathcal{B}(V^{\ast\ast})$
respectively and endow these spaces with the corresponding operator norms. We
can furthermore consider the _point-weak ∗ topology_ on
$\mathcal{B}(V^{\ast\ast})$ which is the locally convex Hausdorff topology
induced by the family $\\{p_{\phi,f}\\}_{\phi\in V^{\ast},f\in V^{\ast\ast}}$
of seminorms with $p_{\phi,f}(T):=|\left(T(f)\right)(\phi)|$, i.e.
$T_{\lambda}\rightarrow T$ for a net
$(T_{\lambda})_{\lambda\in\Lambda}\subseteq\mathcal{B}(V^{\ast\ast})$ and
$T\in\mathcal{B}(V^{\ast\ast})$ if and only if
$\left(T_{\lambda}(f)\right)(\phi)\rightarrow\left(T(f)\right)(\phi)$ for
every $f\in V^{\ast\ast}$, $\phi\in V^{\ast}$.

###  1.4. Cohomology of groups

Let $(V,\rho)$ be a uniformly bounded Banach $G$-module and assume that $\rho$
is strongly operator continuous. For $n\in\mathbb{N}$ denote the vector space
of all continuous functions $f:G^{n}\rightarrow V$ by $C^{n}(G,V)$. Here
$G^{n}$ is equipped with the product topology and $V$ with the norm topology.
We endow $C^{n}(G,V)$ with the _compact-open topology_ ; that is, the topology
generated by the subbase of sets of the form $\\{f\in C^{n}(G,V)\mid
f(K)\subseteq U\\}$, where $K\subseteq G^{n}$ is compact and $U\subseteq V$ is
open. Note that in our particular setting, the compact-open topology coincides
with the _topology of uniform convergence on compact subsets_ , see e.g. [41,
Theorem 9.21]. The subspace $C^{n}(G,V)^{G}$ of elements $f\in C^{n}(G,V)$
that are _$G$ -equivariant_ in the sense that
$f(gg_{1},...,gg_{n})=\rho_{g}(f(g_{1},...,g_{n}))$ for all
$g,g_{1},...,g_{n}\in G$ is closed in $C^{n}(G,V)$.

Consider the cochain complex

$C^{0}(G,V)^{G}\overset{\partial^{0}}{\longrightarrow}C^{1}(G,V)^{G}\overset{\partial^{1}}{\longrightarrow}...\overset{\partial^{n-1}}{\longrightarrow}C^{n}(G,V)^{G}\overset{\partial^{n}}{\longrightarrow}...,$

where
$\left(\partial^{n}f\right)(g_{1},...,g_{n+1}):=\sum_{i=1}^{n+1}(-1)^{i+1}f(g_{1},...,\widehat{g}_{i},...,g_{n+1})$,
$n\in\mathbb{N}$ denote the _standard differentials_. The $\partial^{n}$ are
continuous with respect to the compact-open topology and one checks that
$\partial^{n+1}\circ\partial^{n}=0$ for all $n\in\mathbb{N}$. The subspace of
_$n$ -cocycles_ is given by $Z^{n}(G,\rho):=\ker(\partial^{n+1})\subseteq
C^{n+1}(G,V)^{G}$, whereas $B^{n}(G,\rho):=\text{im}(\partial^{n})\subseteq
C^{n+1}(G,V)^{G}$ denotes the subspace of _coboundaries_. The quotient

$H_{c}^{n}(G,\rho):=Z^{n}(G,\rho)/B^{n}(G,\rho)$

is called the _$n$ -th continuous cohomology group_ of the representation. In
general $B^{n}(G,\rho)$ need not be closed in $C^{n+1}(G,V)^{G}$. One thus
also considers the _$n$ -th continuous reduced cohomology group_

$\overline{H}_{c}^{n}(G,\rho):=Z^{n}(G,\rho)/\overline{B^{n}(G,\rho)},$

by replacing $B^{n}(G,\rho)$ by its closure $\overline{B^{n}(G,\rho)}$. If the
representation $\rho$ is clear, we will also write
$Z^{n}(G,V):=Z^{n}(G,\rho)$, $B^{n}(G,V):=B^{n}(G,\rho)$,
$H_{c}^{n}(G,V):=H_{c}^{n}(G,\rho)$ and
$\overline{H}_{c}^{n}(G,V):=\overline{H}_{c}^{n}(G,\rho)$ respectively.

Note that for $n=1$ the space $Z^{1}(G,\rho)$ identifies with $\\{f\in
C(G,V)\mid f(gh)=\rho_{g}(f(h))+f(g)\text{ for all }g,h\in G\\}$ via
$Z^{1}(G,\rho)\ni f\mapsto f(e,\cdot)$. In this picture, $B^{n}(G,\rho)$ is
the subspace of functions of the form $G\ni g\mapsto v-\rho_{g}(v)$ where
$v\in V$and $\overline{B^{n}(G,\rho)}$ is its closure with respect to the
compact-open topology.

##  2\. Stationary elements in the bidual Banach space

Let $G$ be a topological group and $\rho:G\rightarrow\text{GL}(V)$ a uniformly
bounded linear representation of $G$ on a Banach space $V$. If $\rho$ is
weakly operator continuous, the function $g\mapsto\phi(\rho_{g}(v))$ is Baire
measurable and $\mu$-integrable for every Baire probability measure
$\mu\in\mathcal{P}_{\sigma}(G)$ and $v\in V$, $\phi\in V^{\ast}$. One can
hence define an element in $V^{\ast\ast}$ via

$V^{\ast}\rightarrow\mathbb{C},\,\phi\mapsto\int_{G}\phi\left(\rho_{g}(v)\right)d\mu(g),$
(2.1)

where the boundedness of the map follows from the closed graph theorem.

######  Definition 2.1.

Let $G$ be a topological group, $\rho:G\rightarrow\text{GL}(V)$ a uniformly
bounded linear representation of $G$ on a Banach space $V$ and
$\mu\in\mathcal{P}_{\sigma}(G)$ a Baire probability measure. We call $\rho$
_$\mu$ -integrable_ if $\rho$ is weakly operator continuous and if for all
$v\in V$ the element (2.1) is contained in $\text{im}(\iota_{V})\subseteq
V^{\ast\ast}$. We denote its preimage in $V$ by $\rho_{\mu}(v)$ and call the
map $\rho_{\mu}:v\mapsto\rho_{\mu}(v)$ the _Markov operator_ on $V$ (with
respect to $\mu$). For convenience we will often abbreviate by saying that
$(V,\rho)$ is a _$\mu$ -integrable Banach $G$-module._

We call $\rho$ _Baire-Pettis integrable_ (or simply _BP-integrable_) if $\rho$
is $\mu$-integrable for all $\mu\in\mathcal{P}_{\sigma}(G)$. In this case we
also say that $(V,\rho)$ is a _BP-integrable Banach $G$-module._

######  Remark 2.2.

_(i)_ Let $G$ be a topological group, $\rho:G\rightarrow\text{GL}(V)$ a weakly
operator continuous uniformly bounded linear representation of $G$ on a Banach
space $V$ and $\mu\in\mathcal{P}_{\sigma}(G)$. It follows from Pettis’
measurability theorem (see [33, Theorem 1.1]) that $\rho$ is $\mu$-integrable
if for any $v\in V$ there exists a $\mu$-null set $\mathcal{N}\subseteq G$
such that $\\{\rho_{g}(v)\mid g\in G\setminus\mathcal{N}\\}\subseteq V$ is
separable. In particular, $(V,\rho)$ will be BP-integrable if $V$ is separable
or if $G$ is countable. Another instance where $\rho$ is BP-integrable is if
the Banach space $V$ is reflexive.

_(ii)_ Given a topological group $G$, a Baire probability measure
$\mu\in\mathcal{P}_{\sigma}(G)$, and a uniformly bounded $\mu$-integrable
Banach $G$-module $(V,\rho)$, Lemma 1.1 implies that the operator
$\rho_{\mu}\in\mathcal{B}(V)$ is contained in the point-weak closed convex
hull of $\\{\rho_{g}\mid g\in G\\}\subseteq\mathcal{B}(V)$. An application of
the Hahn-Banach separation theorem furthermore implies that for every $v\in V$
the element $\rho_{\mu}(v)$ is contained in the norm closed convex hull
$\mathcal{C}$ of $G.v:=\\{\rho_{g}(v)\mid g\in G\\}\subseteq V$. Hence, for
every $g\in G$,

$\rho_{\mu}(\rho_{g}(v))\in\overline{\text{conv}}^{\|\cdot\|}\\{\rho_{hg}(v)\mid
h\in G\\}=\mathcal{C},$

so that, by the linearity and norm continuity of the Markov operator,
$\mathcal{C}$ must be invariant under $\rho_{\mu}$. We inductively obtain that
for finitely many Baire probability measures
$\mu_{1},...,\mu_{i}\in\mathcal{P}_{\sigma}(G)$ the element
$(\rho_{\mu_{1}}\circ...\circ\rho_{\mu_{i}})(v)$ is contained in
$\mathcal{C}$. In particular,
$\|(\rho_{\mu_{1}}\circ...\circ\rho_{\mu_{i}})(v)\|\leq M\|v\|$ for every
$v\in V$, where $M:=\sup_{g\in G}\|\rho_{g}\|$.

We furthermore introduce bounded linear operators
$\rho_{\mu}^{\ast}\in\mathcal{B}(V^{\ast})$ and
$\rho_{\mu}^{\ast\ast}\in\mathcal{B}(V^{\ast\ast})$ via
$\rho_{\mu}^{\ast}(\phi):=\phi\circ\rho_{\check{\mu}}$ and
$\rho_{\mu}^{\ast\ast}(f):=f\circ\rho_{\check{\mu}}^{\ast}$ for $\phi\in
V^{\ast}$, $f\in V^{\ast\ast}$, where $\check{\mu}\in\mathcal{P}_{\sigma}(G)$
is the _symmetric opposite_ of $\mu$ given by $\check{\mu}(E):=\mu(E^{-1})$
for $E\in\mathcal{B}a(G)$. Note that

$\left(\rho_{\mu}^{\ast\ast}(\widehat{v})\right)(\phi)=\widehat{v}\circ\rho_{\check{\mu}}^{\ast}(\phi)=\widehat{v}(\phi\circ\rho_{\mu})=\phi\circ\rho_{\mu}(v)=\widehat{\rho_{\mu}(v)}(\phi)$
(2.2)

for all $v\in V$, $\phi\in V^{\ast}$, i.e.
$\rho_{\mu}^{\ast\ast}\circ\iota_{V}=\iota_{V}\circ\rho_{\mu}$.

######  Proposition 2.3.

Let $G$ be a topological group, $\mu\in\mathcal{P}_{\sigma}(G)$ a Baire
probability measure, and $(V,\rho)$ a uniformly bounded $\mu$-integrable
Banach $G$-module. Then there exists a bounded linear operator
$\mathbb{E}_{\mu}$ in the point-weak∗ closure of the convex hull
_$\text{conv}\\{(\rho_{\mu}^{\ast\ast})^{i}\mid
i\in\mathbb{N}\\}\subseteq\mathcal{B}(V^{\ast\ast})$_ satisfying
$\rho_{\mu}^{\ast\ast}\mathbb{E}_{\mu}=\mathbb{E}_{\mu}=\mathbb{E}_{\mu}\rho_{\mu}^{\ast\ast}$,
_$\text{im}(\mathbb{E}_{\mu})=\\{f\in
V^{\ast\ast}\mid\rho_{\mu}^{\ast\ast}(f)=f\\}$_ and
$\mathbb{E}_{\mu}^{2}=\mathbb{E}_{\mu}$.

###### Proof.

From Remark 2.2 (ii) it follows that every element $T$ in the point-weak∗
closure $\mathcal{C}$ of the convex hull of
$\\{(\rho_{\mu}^{\ast\ast})^{i}\mid i\in\mathbb{N}\\}$ in
$\mathcal{B}(V^{\ast\ast})$ satisfies $\|T\|\leq M$ where $M:=\sup_{g\in
G}\|\rho_{g}\|$. Furthermore, the map

$\mathcal{C}\rightarrow\prod_{f\in V^{\ast\ast}:\|f\|\leq 1}\prod_{\phi\in
V^{\ast}:\|\phi\|\leq
1}\mathbb{C},\,T\mapsto\left(\left(T(f)\right)(\phi)\right)_{f,\phi}$

is a homeomorphism onto its image, where the right-hand side is equipped with
the product topology. One easily checks that the image of this map is closed
and contained in the compact subset $\prod_{f\in V^{\ast\ast}:\|f\|\leq
1}\prod_{\phi\in V^{\ast}:\|\phi\|\leq 1}[-M,M]$. We obtain that
$\mathcal{C}\subseteq\mathcal{B}(V^{\ast\ast})$ is point-weak∗ compact. The
multiplication map $T\mapsto\rho_{\mu}^{\ast\ast}T$ maps $\mathcal{C}$ to
itself with $\rho_{\mu}^{\ast\ast}T=T\rho_{\mu}^{\ast\ast}$ for every
$T\in\mathcal{C}$ and is continuous with respect to the subspace topology. The
Tychonoff fixed point theorem hence implies the existence of an element
$\mathbb{E}_{\mu}\in\mathcal{C}$ with
$\rho_{\mu}^{\ast\ast}\mathbb{E}_{\mu}=\mathbb{E}_{\mu}=\mathbb{E}_{\mu}\rho_{\mu}^{\ast\ast}$.
It in particular follows that $\text{im}(\mathbb{E}_{\mu})\subseteq\\{f\in
V^{\ast\ast}\mid\rho_{\mu}^{\ast\ast}(f)=f\\}$. Conversely, if $f\in
V^{\ast\ast}$ with $\rho_{\mu}^{\ast\ast}(f)=f$, then $T(f)=f$ for every
$T\in\text{conv}\\{(\rho_{\mu}^{\ast\ast})^{i}\mid i\in\mathbb{N}\\}$ via
induction and hence also (by a continuity argument)
$f=\mathbb{E}_{\mu}(f)\in\text{im}(\mathbb{E}_{\mu})$. By a similar argument
we obtain that $\mathbb{E}_{\mu}^{2}=\mathbb{E}_{\mu}$. ∎

For a Baire probability measure $\mu$ on $G$ and a uniformly bounded
$\mu$-integrable Banach $G$-module $(V,\rho)$ we define the corresponding
_Laplacians_

$\displaystyle\Delta_{\mu}$ $\displaystyle:=$
$\displaystyle(\text{id}_{V}-\rho_{\mu})\in\mathcal{B}(V),$
$\displaystyle\Delta_{\mu}^{\ast}$ $\displaystyle:=$
$\displaystyle(\text{id}_{V^{\ast}}-\rho_{\mu}^{\ast})\in\mathcal{B}(V^{\ast}),$
$\displaystyle\Delta_{\mu}^{\ast\ast}$ $\displaystyle:=$
$\displaystyle(\text{id}_{V^{\ast\ast}}-\rho_{\mu}^{\ast\ast})\in\mathcal{B}(V^{\ast\ast}).$

######  Lemma 2.4.

Let $G$ be a topological group, $\mu\in\mathcal{P}_{\sigma}(G)$ a Baire
probability measure, let $(V,\rho)$ be a uniformly bounded $\mu$-integrable
Banach $G$-module, and $\mathbb{E}_{\mu}$ an operator as in Proposition 2.3.
We then have that
_$\overline{\text{im}(\Delta_{\mu}^{\ast\ast})}^{\|\cdot\|}\subseteq\ker(\mathbb{E}_{\mu})$_
and
_$\text{ker}(\mathbb{E}_{\mu})\cap\text{im}(\iota_{V})=\overline{\text{im}(\Delta_{\mu}^{\ast\ast})}^{\|\cdot\|}\cap\text{im}(\iota_{V})$_.

###### Proof.

The first inclusion follows from
$\rho_{\mu}^{\ast\ast}\mathbb{E}_{\mu}=\mathbb{E}_{\mu}=\mathbb{E}_{\mu}\rho_{\mu}^{\ast\ast}$,
as for every $f\in V^{\ast\ast}$

$\mathbb{E}_{\mu}(\Delta_{\mu}^{\ast\ast}(f))=\mathbb{E}_{\mu}(f)-\mathbb{E}_{\mu}(\rho_{\mu}^{\ast\ast}(f))=\mathbb{E}_{\mu}(f)-\mathbb{E}_{\mu}(f)=0.$

By the boundedness of $\mathbb{E}_{\mu}$ it can then be deduced that
$\mathbb{E}_{\mu}(f)=0$ for all
$f\in\overline{\text{im}(\Delta_{\mu}^{\ast\ast})}^{\|\cdot\|}$. This also
implies that
$\overline{\text{im}(\Delta_{\mu}^{\ast\ast})}^{\|\cdot\|}\cap\text{im}(\iota_{V})\subseteq\ker(\mathbb{E}_{\mu})\cap\text{im}(\iota_{V})$.

For the inclusion
$\ker(\mathbb{E}_{\mu})\cap\text{im}(\iota_{V})\subseteq\overline{\text{im}(\Delta_{\mu}^{\ast\ast})}^{\|\cdot\|}\cap\text{im}(\iota_{V})$
let $\phi$ be an arbitrary non-trivial linear functional on $V^{\ast\ast}$
that vanishes on $\overline{\text{im}(\Delta_{\mu}^{\ast\ast})}^{\|\cdot\|}$.
It follows that $\phi(f)=\phi(\rho_{\mu}^{\ast\ast}(f))$ for every $f\in
V^{\ast\ast}$. We have $\phi\circ\iota_{V}\in V^{\ast}$ and then also
$\widehat{v}(\phi\circ\iota_{V})=\left(T(\widehat{v})\right)(\phi\circ\iota_{V})$
for every $v\in V$, $T\in\text{conv}\\{(\rho_{\mu}^{\ast\ast})^{i}\mid
i\in\mathbb{N}\\}$. By a continuity argument we obtain that
$\widehat{v}(\phi\circ\iota_{V})=\left(T(\widehat{v})\right)(\phi\circ\iota_{V})$
for every $T$ in the point-weak∗ closure of the convex hull. In particular, if
$\widehat{v}\in\ker(\mathbb{E}_{\mu})\cap\text{im}(\iota_{V})$,

$\phi(\widehat{v})=\widehat{v}(\phi\circ\iota_{V})=\left(\mathbb{E}_{\mu}(\widehat{v})\right)(\phi\circ\iota_{V})=0.$

By the Hahn-Banach theorem (and since $\phi\in V^{\ast\ast\ast}$ with
$\phi|_{\overline{\text{im}(\Delta_{\mu}^{\ast\ast})}^{\|\cdot\|}}\equiv 0$
was arbitrary), this means that
$\widehat{v}\in\overline{\text{im}(\Delta_{\mu}^{\ast\ast})}^{\|\cdot\|}$ and
therefore
$\ker(\mathbb{E}_{\mu})\cap\text{im}(\iota_{V})\subseteq\overline{\text{im}(\Delta_{\mu}^{\ast\ast})}^{\|\cdot\|}\cap\text{im}(\iota_{V})$.
∎

The lemma implies an ergodic theorem for elements in $\text{im}(\iota_{V})$
whose image under $\mathbb{E}_{\mu}$ is again contained in
$\text{im}(\iota_{V})$.

######  Theorem 2.5.

Let $G$ be a topological group, $\mu\in\mathcal{P}_{\sigma}(G)$ a Baire
probability measure, let $(V,\rho)$ be a uniformly bounded $\mu$-integrable
Banach $G$-module, and $\mathbb{E}_{\mu}$ an operator as in Proposition 2.3.
For every $v\in V$ with
_$\mathbb{E}_{\mu}(\widehat{v})\in\text{im}(\iota_{V})$_ we then have that

$\mathbb{E}_{\mu}(\widehat{v})=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v}),$

where the sequence converges in norm.

###### Proof.

Note that for every $f\in V^{\ast\ast}$

$\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}\left(\Delta_{\mu}^{\ast\ast}(f)\right)=\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(f)-\frac{1}{n}\sum_{i=1}^{n}(\rho_{\mu}^{\ast\ast})^{i}(f)=\frac{1}{n}(\text{id}_{V^{\ast\ast}}-(\rho_{\mu}^{\ast\ast})^{n})(f),$

which norm converges to zero, so by continuity
$\frac{1}{n}\sum_{k=0}^{n-1}\rho_{\mu^{k}}^{\ast\ast}(f)\rightarrow 0$ for
every $f\in\overline{\text{im}(\Delta_{\mu}^{\ast\ast})}^{\|\cdot\|}$. For
$v\in V$ with $\mathbb{E}_{\mu}(\widehat{v})\in\text{im}(\iota_{V})$ we have
that
$\widehat{v}-\mathbb{E}_{\mu}(\widehat{v})\in\ker(\mathbb{E}_{\mu})\cap\text{im}(\iota_{V})$.
In combination with Lemma 2.4 this gives

$\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v})-\mathbb{E}_{\mu}(\widehat{v})=\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v}-\mathbb{E}_{\mu}(\widehat{v}))\rightarrow
0.$

This finishes the proof. ∎

For a Banach $G$-module $(V,\rho)$ we write $V^{G}$ for the corresponding
_subspace of fixed points_ under $\rho$. Furthermore, denote the distance of
an element $v\in V$ to a closed subspace $W\subseteq V$ by $\text{dist}(v,W)$,
i.e. $\text{dist}(v,W):=\inf\\{\|v-w\|\mid w\in W\\}$.

The proof of the following theorem uses the argument in [21, Theorem 5.1]
(which again relies on the proof of Furstenberg’s conjecture by Kaimanovich
and Vershik in [23, Theorem 4.3]). We will make use of it in the later
sections. We do not know whether the statement generalizes to non-separable
Banach spaces.

######  Theorem 2.6.

Let $G$ be a topological group and $(V,\rho)$ a uniformly bounded weakly
operator continuous Banach $G$-module with $V$ being separable (so that
$(V,\rho)$ is in particular BP-integrable). Furthermore, assume that for any
$\varepsilon>0$, $v\in V$ there exists a Baire probability measure
$\mu\in\mathcal{P}_{\sigma}(G)$ such that
_$\text{dist}\left(\rho_{\mu}(v),V^{G}\right) <\varepsilon$_. Then there also
exists $\mu\in\mathcal{P}_{\sigma}(G)$ with
$\mathbb{E}_{\mu}(\widehat{v})\in(V^{\ast\ast})^{G}$ for every $v\in V$, where
$\mathbb{E}_{\mu}$ is an operator as in Proposition 2.3.

###### Proof.

Let $(v_{i})_{i\in\mathbb{N}}\subseteq V$ be a sequence for which
$\\{v_{1},v_{2},...\\}$ is dense in the unit ball of $V$. Furthermore, fix an
arbitrary sequence $(n_{i})_{i\in\mathbb{N}}\subseteq\mathbb{N}_{\geq 1}$ with
$(\sum_{j=1}^{i}2^{-j})^{n_{i}}<2^{-i}$ for all $i\in\mathbb{N}_{\geq 1}$. By
our assumption we can inductively choose Baire probability measures
$(\mu_{i})_{i\in\mathbb{N}}\in\mathcal{P}_{\sigma}(G)$ such that for all
$i\in\mathbb{N}_{\geq 1}$, $1\leq j,j_{1},...,j_{k}<i$ and $k<n_{i}$ the
inequality

$\text{dist}\left(\rho_{\mu_{i}}\rho_{\mu_{j_{k}}}...\rho_{\mu_{j_{1}}}(v_{j}),V^{G}\right)<2^{-i}$

holds. We claim that
$\mu:=\sum_{i=1}^{\infty}2^{-i}\mu_{i}\in\mathcal{P}_{\sigma}(G)$ satisfies
$\text{dist}\left(\rho_{\mu}^{i}(v),V^{G}\right)\rightarrow 0$ for every $v\in
V$. Indeed, for $v\in V$ with $\|v\|\leq 1$ we find $i\in\mathbb{N}$ with
$\|v-v_{i}\|<2^{-i}$. Then, by Remark 2.2 (ii),

$\displaystyle\text{dist}\left(\rho_{\mu}^{n_{i}}(v),V^{G}\right)$
$\displaystyle\leq$
$\displaystyle\|\rho_{\mu}^{n_{i}}(v)-\rho_{\mu}^{n_{i}}(v_{i})\|+\text{dist}\left(\rho_{\mu}^{n_{i}}(v_{i}),V^{G}\right)$
$\displaystyle\leq$ $\displaystyle
2^{-i}M+\text{dist}\left(\rho_{\mu}^{n_{i}}(v_{i}),V^{G}\right),$

where $M:=\sup_{g\in G}\|\rho_{g}\|$. The expression
$\text{dist}\left(\rho_{\mu}^{n_{i}}(v_{i}),V^{G}\right)$ can be estimated via

$\displaystyle\text{dist}\left(\rho_{\mu}^{n_{i}}(v_{i}),V^{G}\right)$
$\displaystyle=$
$\displaystyle\text{dist}\left(\sum_{j_{1},...,j_{n_{i}}=1}^{\infty}\frac{1}{2^{j_{1}+...+j_{n_{i}}}}\rho_{\mu_{j_{1}}}...\rho_{\mu_{j_{n_{i}}}}(v_{i}),V^{G}\right)$
$\displaystyle\leq$ $\displaystyle\sum_{\max(j_{1},...,j_{n_{i}})\leq
i}\frac{1}{2^{j_{1}+...+j_{n_{i}}}}\left\|\rho_{\mu_{j_{1}}}...\rho_{\mu_{j_{n_{i}}}}(v_{i})\right\|$
$\displaystyle\qquad+\sum_{\max(j_{1},...,j_{n_{i}})>i}\frac{1}{2^{j_{1}+...+j_{n_{i}}}}\text{dist}\left(\rho_{\mu_{j_{1}}}...\rho_{\mu_{j_{n_{i}}}}(v_{i}),V^{G}\right)$
$\displaystyle\leq M$
$\displaystyle\left(\sum_{j=1}^{i}2^{-j}\right)^{n_{i}}\left\|v_{i}\right\|+\sum_{\max(j_{1},...,j_{n_{i}})>i}\frac{1}{2^{j_{1}+...+j_{n_{i}}}}\text{dist}\left(\rho_{\mu_{j_{1}}}...\rho_{\mu_{j_{n_{i}}}}(v_{i}),V^{G}\right).$

For every tuple $(j_{1},...,j_{n_{i}})$ with $\max(j_{1},...,j_{n_{i}})>i$
there exists a largest integer $1\leq k\leq n_{i}$ for which $j_{k}>i$. Hence,
by the choice of the sequence
$(\mu_{j})_{j\in\mathbb{N}}\in\mathcal{P}_{\sigma}(G)$,

$\text{dist}\left(\rho_{\mu_{j_{1}}}...\rho_{\mu_{j_{n_{i}}}}(v_{i}),V^{G}\right)\leq
M\text{dist}\left(\rho_{\mu_{j_{k}}}...\rho_{\mu_{j_{n_{i}}}}(v_{i}),V^{G}\right)<2^{-j_{k}}M<2^{-i}M.$

Here the first inequality follows from the fact that the representation $\rho$
induces a (well-defined) linear representation
$\widetilde{\rho}:G\rightarrow\text{GL}(V/V^{G})$ on the quotient space with
uniform bound $M$. With $(\sum_{j=1}^{i}2^{-j})^{n_{i}}<2^{-i}$ and
$\|v_{i}\|\leq 1$ we obtain
$\text{dist}\left(\rho_{\mu}^{n_{i}}(v_{i}),V^{G}\right)<2^{-i}(1+M)$, thus

$\text{dist}\left(\rho_{\mu}^{n_{i}}(v),V^{G}\right)<2^{-i}(1+2M).$

For $n>n_{i}$ this implies

$\text{dist}\left(\rho_{\mu}^{n}(v),V^{G}\right)\leq
M\text{dist}\left(\rho_{\mu}^{n_{i}}(v),V^{G}\right)\leq 2^{-i}M(1+2M)$

and therefore $\text{dist}\left(\rho_{\mu}^{n}(v),V^{G}\right)\rightarrow 0$.
We can hence choose a sequence $(w_{n})_{n\in\mathbb{N}}\subseteq V^{G}$ with
$\|\rho_{\mu}^{n}(v)-w_{n}\|\rightarrow 0$. By Remark 2.2 (ii) we have
$\|\rho_{\mu}^{n}(v)\|\leq M\|v\|$ for every $n\in\mathbb{N}$. This gives that
the sequence $(\widehat{w_{n}})_{n\in\mathbb{N}}\subseteq V^{\ast\ast}$ is
bounded so that we can go over to a subnet
$(\widehat{w_{n_{\lambda}}})_{\lambda\in\Lambda}$ with weak∗-limit $f\in
V^{\ast\ast}$. It is easy to check that $f\in(V^{\ast\ast})^{G}$ and hence

$\mathbb{E}_{\mu}(\widehat{v})-f=\lim_{\lambda}(\mathbb{E}_{\mu}\circ\iota_{V})\left(\rho_{\mu}^{n_{\lambda}}(v)-w_{n_{\lambda}}\right)=0.$

We obtain that $\mathbb{E}_{\mu}(\widehat{v})=f\in(V^{\ast\ast})^{G}$, as
claimed. ∎

######  Remark 2.7.

_(i)_ If $V^{G}$ is finite-dimensional, one can employ the norm compactness of
the unit ball to deduce that, under the same assumptions as in Theorem 2.6,
$\mathbb{E}_{\mu}(\widehat{v})\in\iota_{V}(V^{G})$ for every $v\in V$.

_(ii)_ Let $G$ be a topological group and $(V,\rho)$ a uniformly bounded BP-
integrable Banach $G$-module. Assume that $V^{\ast}$ is separable and that for
any $\varepsilon>0$, $\phi\in V^{\ast}$ there exists a Baire probability
measure $\mu\in\mathcal{P}_{\sigma}(G)$ such that
$\text{dist}\left(\rho_{\mu}^{\ast}(\phi),(V^{\ast})^{G}\right)<\varepsilon$.
A simple modification of the proof of Theorem 2.6 then implies that there
exists $\mu\in\mathcal{P}_{\sigma}(G)$ with
$\mathbb{E}_{\mu}(\widehat{v})\in(V^{\ast\ast})^{G}$ for every $v\in V$, where
$\mathbb{E}_{\mu}$ is an operator as in Proposition 2.3.

######  Example 2.8.

A unital C∗-algebra $A$ with center $\mathcal{Z}(A):=\\{a\in A\mid ab=ba\text{
for all }b\in A\\}$ is said to have the _Dixmier property_ if for every $a\in
A$ the norm closure of the convex hull of $\\{uau^{\ast}\mid
u\in\mathcal{U}(A)\\}$ intersects $\mathcal{Z}(A)$ non-trivially. Here
$\mathcal{U}(A)$ denotes the unitary group of $A$. If we equip
$\mathcal{U}(A)$ with the restriction of the weak topology on $A$, we obtain a
weakly operator continuous uniformly bounded linear representation
$\rho:\mathcal{U}(A)\rightarrow\text{Aut}(A)\subseteq\text{GL}(A)$ via
$\rho_{u}(a):=uau^{\ast}$. Note that $\mathcal{Z}(A)=A^{\mathcal{U}(A)}$ so
that, in the case where $A$ is separable and has the Dixmier property, Theorem
2.6 implies the existence of a Baire probability measure
$\mu\in\mathcal{P}_{\sigma}(\mathcal{U}(A))$ with
$\mathbb{E}_{\mu}(\widehat{a})\in(A^{\ast\ast})^{\mathcal{U}(A)}$ for every
$a\in A$, with $\mathbb{E}_{\mu}$ as in Proposition 2.3. This class of
examples in particular covers all separable simple unital C∗-algebras with at
most one tracial state, see [20].

In a similar spirit, we obtain the following result.

Given a Baire probability measure $\mu$ on a topological group $G$ denote the
closed subgroup of $G$ generated by the set $\Sigma_{\mu}(G)$ of elements
$g\in G$ for which $\int_{G}fd\mu>0$ for every positive function $f\in
C_{b}(G,\mathbb{R})$ with $f(g)\neq 0$ by $G_{\mu}$. Furthermore, call a
function $f:G\rightarrow\mathbb{C}$ on $G$ _right-unifomly continuous_ if for
every $\varepsilon>0$ one can find a neighbourhood $\mathcal{U}$ of $e\in G$
with $|f(g)-f(h)|\leq\varepsilon$ for all $g,h\in G$ with
$hg^{-1}\in\mathcal{U}$. The set $\text{RUCB}(G)$ of all right-uniformly
continuous functions that are bounded with respect to the supremum norm
becomes a commutative unital C∗-algebra with respect to pointwise addition,
multiplication, and involution. For every $f\in\text{RUCB}(G)$ and
$\mu\in\mathcal{P}_{\sigma}(G)$ the function
$\Phi_{\mu}f:g\mapsto\int_{G}f(gh)d\mu(h)$ is again contained in
$\text{RUCB}(G)$. The function $f$ is called _$\mu$ -harmonic_, if
$f=\Phi_{\mu}f$. We say that the pair $(G,\mu)$ has the _Liouville property_
if for any $g\in G$ every $\mu$-harmonic function $f\in\text{RUCB}(G)$
satisfies $f(gh)=f(g)$ for $\mu$-almost all $h\in G$. The group $G$ is
furthermore said to be _Choquet–Deny_ if $(G,\mu)$ satisfies the Liouville
property for every $\mu\in\mathcal{P}_{\sigma}(G)$. Note that there are
several different definitions of these concepts in the literature. Here we
employ the ones that suit our purposes.

######  Proposition 2.9.

Let $G$ be a topological group and $\mu\in\mathcal{P}_{\sigma}(G)$ a Baire
probability measure for which the pair $(G,\check{\mu})$ has the Liouville
property. Let furthermore $(V,\rho)$ be a uniformly bounded, strongly operator
continuous, $\mu$-integrable Banach $G$-module. Then
$\mathbb{E}_{\mu}\circ\iota_{V}$ with $\mathbb{E}_{\mu}$ as in Proposition 2.3
is $G_{\mu}$-invariant.

###### Proof.

For fixed $\phi\in V^{\ast}$ we can define a bounded linear functional
$\psi_{\phi}\in V^{\ast}$ via
$\psi_{\phi}(v):=\mathbb{E}_{\mu}(\widehat{v})(\phi)$. For every $v\in V$ the
map $f_{v,\phi}:G\rightarrow\mathbb{C}$,
$f_{v,\phi}(g):=\rho_{g}^{\ast}(\psi_{\phi})(v)$ is contained in
$\text{RUCB}(G)$ with

$\int_{G}f_{v,\phi}(gh)d\breve{\mu}(h)=\int_{G}\rho_{gh}^{\ast}(\psi_{\phi})(v)d\breve{\mu}(h)=(\psi_{\phi}\circ\rho_{\mu})\left(\rho_{g^{-1}}(v)\right)=\rho_{g}^{\ast}(\psi_{\phi})(v)=f_{v,\phi}(g).$

It follows that $f_{v,\phi}$ is $\breve{\mu}$-harmonic and hence for all $g\in
G$, $f_{v,\phi}(gh)=f_{v,\phi}(g)$ for $\mu$-almost every $h\in G$. We in
particular obtain that the function $h\mapsto|f_{v,\phi}(h)-f_{v,\phi}(e)|$ is
positive, bounded and continuous with
$\int_{G}|f_{v,\phi}(h)-f_{v,\phi}(e)|d\mu(h)=0$. From the definition of
$\Sigma_{\mu}(G)$ it follows that $f_{v,\phi}(h)=f_{v,\phi}(e)$ for all
$h\in\Sigma_{\mu}(G)$ and hence, since $\phi\in V^{\ast}$, $v\in V$ where
chosen arbitrarily,
$\mathbb{E}_{\mu}\circ\iota_{V}\circ\rho_{h}=\mathbb{E}_{\mu}\circ\iota_{V}$.
We finally deduce that $\mathbb{E}_{\mu}\circ\iota_{V}$ is
$G_{\mu}$-invariant, as claimed. ∎

######  Remark 2.10.

The characterization in [39, Corollary 4.8] states that a Hausdorff second-
countable topological group $G$ is _amenable_ (i.e. the canonical action of
$G$ on the state space of $\text{RUCB}(G)$ has a fixed point) if and only if
it admits a fully supported, regular Borel probability measure $\mu$ for which
every $\mu$-harmonic function is constant. In combination with Proposition 2.9
this implies that every amenable topological group $G$ admits a Baire
probability measure $\mu\in\mathcal{P}_{\sigma}(G)$ such that for every
uniformly bounded, strongly operator continuous, BP-integrable Banach
$G$-module $(V,\rho)$ the map $\mathbb{E}_{\mu}\circ\iota_{V}$ is
$G$-invariant.

######  Corollary 2.11.

Let $G$ be a topological group, $\mu\in\mathcal{P}_{\sigma}(\mathcal{Z}(G))$ a
Baire probability measure, $(V,\rho)$ a uniformly bounded, strongly operator
continuous, $\mu$-integrable Banach $G$-module, and set $Z:=Z(G)_{\mu}$.
Assume that the product map $Z\times Z\rightarrow Z$ is measurable with
respect to the product $\sigma$-algebra
$\mathcal{B}a(Z)\otimes\mathcal{B}a(Z)$ and $\mathcal{B}a(Z)$. Then
$\mathbb{E}_{\mu}\circ\iota_{V}$ with $\mathbb{E}_{\mu}$ as in Proposition 2.3
is $Z$-invariant with
_$\text{im}(\mathbb{E}_{\mu}\circ\iota_{V})\subseteq(V^{\ast\ast})^{Z}$_.

###### Proof.

By Proposition 2.9 it suffices to show that the center of $G$ is Choquet–Deny.
Following the proof in [34], consider for
$\mu\in\mathcal{P}_{\sigma}(\mathcal{Z}(G))$ the bounded linear operator
$\Phi_{\mu}$ on $\text{RUCB}(\mathcal{Z}(G))$ defined above and define for
fixed $f\in\text{RUCB}(\mathcal{Z}(G))$ with $f=\Phi_{\mu}f$ a function
$h\in\text{RUCB}(\mathcal{Z}(G))$ via
$h(z):=\int_{\mathcal{Z}(G)}|f(z)-f(cz)|^{2}d\mu(c)$. In combination with
Fubini’s theorem and the Cauchy-Schwarz inequality we obtain that
$\Phi_{\mu}(h)(z)\geq h(z)$ for all $z\in\mathcal{Z}(G)$. By making use of the
identity $h=\Phi_{\mu}(|f|^{2})-|f|^{2}$, we then have

$h\leq\frac{1}{N}\sum_{i=0}^{N}\left(\Phi_{\mu}^{i+1}(|f|^{2})-\Phi_{\mu}^{i}(|f|^{2})\right)=\frac{1}{N}\left(\Phi_{\mu}^{N+1}(|f|^{2})-1\right)\rightarrow
0,$

so that $h$ vanishes. It follows that for every $z\in\mathcal{Z}(G)$ the
equality $f(z)=f(cz)$ holds $\mu$-almost everywhere. This finishes the proof.
∎

##  3\. The projection $\mathbb{E}_{\mu}$

Theorem 2.5 indicates that it is desirable to know more about the range of the
map $\mathbb{E}_{\mu}$ in Proposition 2.3. Studying it will be the aim of this
section.

###  3.1. Weakly uniquely stationary representations

For $\mu\in\mathcal{P}_{\sigma}(G)$ and a uniformly bounded $\mu$-integrable
linear representation $\rho:G\rightarrow\text{GL}(V)$ of a topological group
$G$ on a Banach space $V$ we write

$V^{\mu}:=\\{v\in V\mid\rho_{\mu}(v)=v\\}\subseteq V$

for the subspace of all _$\rho_{\mu}$ -invariant elements_ in $V$. Similarly,
$(V^{\ast})^{\mu}\subseteq V^{\ast}$ denotes the space of all
_$\rho_{\mu}^{\ast}$ -invariant functionals_ on $V$ and $(V^{\ast\ast})^{\mu}$
is the space of all _$\rho_{\mu}^{\ast\ast}$ -invariant elements_ in
$V^{\ast\ast}$. The equality (2.2) immediately implies that
$\iota_{V}(V^{\mu})\subseteq(V^{\ast\ast})^{\mu}$, whereas Proposition 2.3 in
particular gives that $\text{im}(\mathbb{E}_{\mu})=(V^{\ast\ast})^{\mu}$.

######  Definition 3.1.

Let $G$ be a topological group and $\mu\in\mathcal{P}_{\sigma}(G)$ a Baire
probability measure. We call a uniformly bounded linear representation
$\rho:G\rightarrow\text{GL}(V)$ of $G$ on a Banach space $V$ _weakly uniquely
$\mu$-stationary_ if it is $\mu$-integrable and if every non-trivial
functional $\phi\in(V^{\ast})^{\breve{\mu}}$ admits an element $v\in V^{\mu}$
with $\phi(v)\neq 0$. In this case we simply say that the Banach $G$-module
$(V,\rho)$ is _weakly uniquely $\mu$-stationary_.

The first three statements of the following proposition are essentially
contained in [24], [43], [40] (see also [26, Chapter 2.1]). We include a proof
for the convenience of the reader.

######  Proposition 3.2.

Let $G$ be a topological group. For a Baire probability measure
$\mu\in\mathcal{P}_{\sigma}(G)$ and a uniformly bounded $\mu$-integrable
Banach $G$-module $(V,\rho)$ the following statements are equivalent:

  1. (1)

$\rho$ is weakly uniquely $\mu$-stationary;

  2. (2)

$V$ admits a decomposition into norm closed $\rho_{\mu}$-invariant linear
subspaces _$V=\overline{\text{im}(\Delta_{\mu})}\oplus V^{\mu}$_ ;

  3. (3)

The sequence
$\left(\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v})\right)_{n\in\mathbb{N}}\subseteq
V^{\ast\ast}$ converges for every $v\in V$ in norm;

  4. (4)

_$\mathbb{E}_{\mu}(\widehat{v})\in\text{im}(\iota_{V})$_ for every $v\in V$,
where $\mathbb{E}_{\mu}$ is an operator as in Proposition 2.3.

###### Proof.

“$\text{(1)}\Rightarrow\text{(2)}$” Assume that the representation $\rho$ is
weakly uniquely $\mu$-stationary. Both
$\overline{\text{im}(\Delta_{\mu})}\subseteq V$ and $V^{\mu}\subseteq V$ are
obviously norm closed. As in the proof of Theorem 2.5 it can be deduced that
$\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v})\rightarrow
0$ in norm for every $v\in\overline{\text{im}(\Delta_{\mu})}$, whereas
$\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v})=\widehat{v}$
for all $n\in\mathbb{N}$, $v\in V^{\mu}$. Hence,
$\overline{\text{im}(\Delta_{\mu})}\cap V^{\mu}=\\{0\\}$. Define
$V_{0}:=\overline{\text{im}(\Delta_{\mu})}\oplus V^{\mu}\subseteq V$ and let
$\phi\in V^{\ast}$ be a functional on $V$ that vanishes on $V_{0}$. One in
particular has that $\phi\circ\rho_{\mu}(v)=\phi(v)$ for all $v\in V$, i.e.
$\phi\in(V^{\ast})^{\breve{\mu}}$. But $\rho$ is weakly uniquely
$\mu$-stationary, so by construction, the functional $\phi$ must be trivial.
The Hahn-Banach theorem finally implies that $V_{0}=V$.

“$\text{(2)}\Rightarrow\text{(3)}$” Assume that
$V=\overline{\text{im}(\Delta_{\mu})}\oplus V^{\mu}$ and let $v=v_{1}+v_{2}$
where $v_{1}\in\overline{\text{im}(\Delta_{\mu})}$ and $v_{2}\in V^{\mu}$. As
we have seen before,
$\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v_{1}})\rightarrow
0$ and
$\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v_{2}})\rightarrow\widehat{v_{2}}$
in norm. It follows that the sequence
$\left(\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v})\right)_{n\in\mathbb{N}}\subseteq
V^{\ast\ast}$ norm-converges for every $v\in V$.

“$\text{(3)}\Rightarrow\text{(4)}$” Assume that the sequence
$\left(\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v})\right)_{n\in\mathbb{N}}\subseteq
V^{\ast\ast}$ converges for every $v\in V$ in norm. The limit of this sequence
is $\rho_{\mu}^{\ast\ast}$-invariant so that

$\displaystyle\mathbb{E}_{\mu}(\widehat{v})$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\mathbb{E}_{\mu}\left((\rho_{\mu}^{\ast\ast})^{i}(\widehat{v})\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\mu}\left(\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v})\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v})\,$

is contained in $\text{im}(\iota_{V})$ as claimed.

“$\text{(4)}\Rightarrow\text{(1)}$” Assume that
$\mathbb{E}_{\mu}(\widehat{v})\in\text{im}(\iota_{V})$ for every $v\in V$. By
Theorem 2.5 we then have that for $v\in V$ and
$\phi\in(V^{\ast})^{\breve{\mu}}$

$\displaystyle\phi(v)$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\left(\phi\circ\rho_{\mu}^{i}\right)(v)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left(\phi\circ\iota_{V}^{-1}\right)\left(\frac{1}{n}\sum_{i=0}^{n-1}(\rho_{\mu}^{\ast\ast})^{i}(\widehat{v})\right)$
$\displaystyle=$
$\displaystyle\left(\phi\circ\iota_{V}^{-1}\right)\left(\mathbb{E}_{\mu}(\widehat{v})\right).$

But $(\iota_{V}^{-1}\circ\mathbb{E}_{\mu})(\widehat{v})\in V^{\mu}$. Hence
$\phi$ must be trivial if $\phi(v)=0$ for all $v\in V^{\mu}$. It follows that
$\rho$ is weakly uniquely $\mu$-stationary. ∎

Justified by Proposition 3.2, for a Baire probability measure
$\mu\in\mathcal{P}_{\sigma}(G)$ and a uniformly bounded weakly uniquely
$\mu$-stationary Banach $G$-module $(V,\rho)$, we will from now on denote
$\widetilde{\mathbb{E}}_{\mu}:=\iota_{V}^{-1}\circ\mathbb{E}_{\mu}\circ\iota_{V}$.

The following class of examples occurs in [22, Proposition 2.2].
Representations like this have been studied in [9]. More examples can be found
in Lemma 3.5, in Subsection 3.2, and in [22, Section 2].

######  Example 3.3.

Let $G$ be a countable discrete group equipped with a probability measure
$\mu$ satisfying $\\#G_{\mu}=\infty$, and let $(V,\rho)$ be a uniformly
bounded Banach $G$-module. Suppose that for all $v\in V$, $\phi\in V^{\ast}$
the function $f_{\phi,v}:g\mapsto\rho_{g}^{\ast}(\phi)(v)$ is contained in
$C_{0}(G)$. Then $(V,\rho)$ is weakly uniquely $\mu$-stationary. Indeed, by a
similar argument as in Proposition 2.9 one deduces that for
$\phi\in(V^{\ast})^{\breve{\mu}}$ and $v\in V$ the function $f_{\phi,v}\in
C_{0}(G)$ is $\breve{\mu}$-harmonic. The maximum principle (see e.g. [42,
Exercise 2.80]) then implies that $\phi\equiv 0$ and hence
$(V^{\ast})^{\breve{\mu}}=0$.

######  Proposition 3.4.

Let $G$ be a topological group, $\mu\in\mathcal{P}_{\sigma}(G)$ a Baire
probability measure and $(V,\rho)$ a uniformly bounded, weakly uniquely
$\mu$-stationary Banach $G$-module. Then all finite subsets $T\subseteq V$ and
$\varepsilon>0$ admit an element _$\delta\in\text{conv}\\{\rho_{g}\mid g\in
G\\}\subseteq\mathcal{B}(V)$_ with $\sup_{v\in
T}\|\widetilde{\mathbb{E}}_{\mu}(v)-\delta(v)\|<\varepsilon$.

###### Proof.

Let $\\{v_{1},...,v_{n}\\}\subseteq V$ be a finite subset and $\varepsilon>0$.
By Proposition 3.2 there exists an integer $N\in\mathbb{N}$ with $\max_{1\leq
i\leq
n}\|\widetilde{\mathbb{E}}_{\mu}(v_{i})-\frac{1}{N}\sum_{l=0}^{N-1}\rho_{\mu}^{l}(v_{i})\|<\frac{\varepsilon}{2}$.
Set $\mathbf{v}:=(v_{1},...,v_{n})\in V^{n}$ and consider the Banach space
$V^{n}:=\bigoplus_{1\leq i\leq n}V$ with the norm
$\|(w_{1},...,w_{n})\|_{n}:=\max_{1\leq i\leq n}\|w_{i}\|$ and the diagonal
representation $\overline{\rho}:G\rightarrow\text{GL}(V^{n})$. Then
$(V^{n},\overline{\rho})$ is a weakly operator continuous uniformly bounded
Banach $G$-module. By $(V^{n})^{\ast}\cong(V^{\ast})^{n}$ we furthermore have
that $(V^{n},\overline{\rho})$ is weakly uniquely $\mu$-stationary with
$\overline{\rho}_{\mu}(\mathbf{w})=(\rho_{\mu}(w_{1}),...,\rho_{\mu}(w_{n}))$
for all $\mathbf{w}:=(w_{1},...,w_{n})\in V^{n}$. Remark 2.2 (ii) implies the
existence of elements
$\overline{\delta}_{0},...,\overline{\delta}_{N-1}\in\text{conv}\\{\overline{\rho}_{g}\mid
g\in G\\}$ with $\max_{1\leq l\leq
N-1}\|\overline{\rho}_{\mu}^{l}(\mathbf{v})-\overline{\delta}_{l}(\mathbf{v})\|_{n}<\frac{\varepsilon}{2}$.
Denote the corresponding elements in $\text{conv}\\{\rho_{g}\mid g\in G\\}$ by
$\delta_{0},...,\delta_{N-1}$ and set
$\delta:=\frac{1}{N}\sum_{l=0}^{N-1}\delta_{l}$. Then,

$\displaystyle\max_{1\leq i\leq
n}\|\widetilde{\mathbb{E}}_{\mu}(v_{i})-\delta(v_{i})\|$ $\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{2}+\frac{1}{N}\sum_{l=0}^{N-1}\left(\max_{1\leq
i\leq n}\|\rho_{\mu}^{l}(v_{i})-\delta_{l}(v_{i})\|\right)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon}{2}+\frac{1}{N}\sum_{l=0}^{N-1}\|\overline{\rho}_{\mu}^{l}(\mathbf{v})-\overline{\delta}_{l}(\mathbf{v})\|_{n}$
$\displaystyle<$ $\displaystyle\varepsilon.$

This finishes the proof. ∎

Definition 3.1 is inspired by the notion of uniquely stationary actions of
groups on C∗-algebras, as introduced by Hartman and Kalantar in [21]. Recall
that for an action $\alpha:G\rightarrow\text{Aut}(A)$ of a countable discrete
group $G$ on a unital C∗-algebra $A$ by automorphism and
$\mu\in\text{Prob}(G)$ a state $\tau$ on $A$ is said to be _$\mu$
-stationary_, if $\tau=\sum_{g\in G}\mu(g)\alpha_{g}^{\ast}(\tau)$. By [21,
Proposition 4.2] every action $(G,\mu)\curvearrowright A$ admits a
$\mu$-stationary state. It is called _uniquely stationary_ if there exists a
unique $\mu$-stationary state on $A$.

Examples of uniquely stationary actions can be found in [21] and [1]. For
instance, by [21, Theorem 5.1] the reduced group C∗-algebra $C_{r}^{\ast}(G)$
of a countable discrete group $G$ is _simple_ (i.e. it contains no non-trivial
two-sided closed ideal) if and only if there exists a probability measure
$\mu$ on $G$ such that the action $(G,\mu)\curvearrowright C_{r}^{\ast}(G)$ by
inner automorphisms is uniquely stationary. Similarly, [1, Proposition 3.7]
states that for any non-amenable charmenable countable discrete group $G$ with
trivial amenable radical and any unitary representation
$\pi:G\rightarrow\mathcal{U}(\mathcal{H})$ of $G$ there exists a probability
measure $\mu$ on $G$ whose support generates the whole group and such that the
C∗-algebra generated by $\pi$ carries a unique $\mu$-stationary state.

The purpose of the following lemma is to clear up the connection between our
notion and the one of Hartman and Kalantar.

######  Lemma 3.5.

Let $A$ be a unital C∗-algebra and $\alpha:G\rightarrow\text{Aut}(A)$ an
action of a countable discrete group $G$ by automorphism. Let furthermore
$\mu$ be a probability measure on $G$. Then the following statements hold:

  1. (1)

If $(G,\breve{\mu})\curvearrowright A$ is uniquely stationary, then the
isometric linear representation $g\mapsto\alpha_{g}$ is weakly uniquely
$\mu$-stationary;

  2. (2)

If $A^{\mu}=\mathbb{C}1$, then $(G,\breve{\mu})\curvearrowright A$ is uniquely
stationary if and only if $g\mapsto\alpha_{g}$ is weakly uniquely
$\mu$-stationary.

###### Proof.

_About (1)_ : Assume that $(G,\breve{\mu})\curvearrowright A$ is uniquely
stationary and let $\tau\in\mathcal{S}(A)$ be the corresponding
$\breve{\mu}$-stationary state. Since $G$ is countable, the representation
$g\mapsto\alpha_{g}$ is BP-integrable. The uniqueness of the Jordan
decomposition furthermore implies that every functional
$\phi\in(A^{\ast})^{\breve{\mu}}\setminus\\{0\\}$ decomposes as
$\phi=\tau_{+}-\tau_{-}$, where both $\tau_{+}$ and $\tau_{-}$ are
$\breve{\mu}$-stationary positive functionals on $A$. But then, by our
assumption, $\phi$ must be a non-trivial multiple of $\tau$, so in particular
$\phi(1)\neq 0$. It follows that the representation is weakly uniquely
$\mu$-stationary.

_About (2):_ Assume that $A^{\mu}=\mathbb{C}1$. It remains to show that, if
$\alpha:g\mapsto\alpha_{g}$ is weakly uniquely $\mu$-stationary, the action
$(G,\breve{\mu})\curvearrowright A$ is uniquely stationary. So assume that
$g\mapsto\alpha_{g}$ is weakly uniquely $\mu$-stationary. By Proposition 3.2,
$\widetilde{\mathbb{E}}_{\mu}$ defines a well-defined $\breve{\mu}$-stationary
state on $A$. Assume that $\tau\in\mathcal{S}(A)$ is another such state, then
(by a similar calculation as in the proof of Proposition 3.2),

$\displaystyle\tau(a)$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\left(\tau\circ\alpha_{\mu}^{i}\right)(a)=\lim_{n\rightarrow\infty}\left(\tau\circ\iota_{A}^{-1}\right)\left(\frac{1}{n}\sum_{i=0}^{n-1}(\alpha_{\mu}^{\ast\ast})^{i}(\widehat{a})\right)$
$\displaystyle=$
$\displaystyle\left(\tau\circ\iota_{A}^{-1}\right)\left(\mathbb{E}_{\mu}(\widehat{a})\right)=\widetilde{\mathbb{E}}_{\mu}(a)$

for all $a\in A$. ∎

###  3.2. Weakly almost periodic representations

In [3] the study of reduced continuous cohomology of weakly almost periodic
representations was initiated.

######  Definition 3.6 ([3, Definition 1]).

A linear representation $\rho:G\rightarrow\text{GL}(V)$ of a topological group
$G$ on a Banach space $V$ is called _weakly almost periodic_ (or simply _wap_)
if for every $v\in V$ the $G$-orbit $G.v:=\\{\rho_{g}(v)\mid g\in
G\\}\subseteq V$ is relatively weakly compact in $V$.

By the discussion in [3], all linear representations on reflexive Banach
spaces are weakly almost periodic. Another source of examples is given by the
natural $L^{1}$-representations induced by actions via measure preserving
transformations on probability spaces, see [3, Example 10].

As the following theorem demonstrates, weakly almost periodic representations
fall within the setting of the previous sections.

######  Theorem 3.7.

Let $G$ be a topological group and $(V,\rho)$ a wap weakly operator continuous
Banach $G$-module. Then $(V,\rho)$ is BP-integrable and weakly uniquely
$\mu$-stationary for every $\mu\in\mathcal{P}_{\sigma}(G)$.

###### Proof.

It was argued in [3] that wap representations are uniformly bounded. Lemma 1.1
implies that for every $v\in V$ and $\mu\in\mathcal{P}_{\sigma}(G)$ the
operator $f_{v}\in V^{\ast\ast}$ given by
$f_{v}(\phi):=\int_{G}\phi\left(\rho_{g}(v)\right)d\mu(g)$ is contained in the
weak∗-closure of the convex hull of $\\{\rho_{g}^{\ast\ast}(\widehat{v})\mid
g\in G\\}\subseteq\mathcal{B}(V^{\ast\ast})$. We can hence choose a net
$(v_{\lambda})_{\lambda\in\Lambda}\subseteq\text{conv}(G.v)$ for which
$\widehat{v_{\lambda}}\rightarrow f_{v}$ in the weak∗-topology. With the
assumption that $(V,\rho)$ is weakly almost periodic, Krein’s theorem (see
e.g. [15, Theorem 3.133]) gives that the closed convex hull of $G.v$ is weakly
compact in $V$. By possibly going over to a subnet we can hence assume that
$v_{\lambda}$ weakly converges to an element $w\in V$. But then
$f_{v}(\phi)=\lim_{\lambda}\widehat{v_{\lambda}}(\phi)=\lim_{\lambda}\phi(v_{\lambda})=\phi(w)=\widehat{w}(\phi)$
for every $\phi\in V^{\ast}$ so that
$f_{v}=\widehat{w}\in\text{im}(\iota_{V})$. It follows that $(V,\rho)$ is BP-
integrable.

Let $\mathcal{C}$ be the norm closure of the convex hull of $G.v$. By Remark
2.2 (ii), $\rho_{\mu}^{i}(v)\in\mathcal{C}$ for every $i\in\mathbb{N}$. Since
the operator $\mathbb{E}_{\mu}$ from Proposition 2.3 is contained in the
point-weak∗ closure of $\text{conv}\\{(\rho_{\mu}^{\ast\ast})^{i}\mid
i\in\mathbb{N}\\}\subseteq\mathcal{B}(V^{\ast\ast})$, the image
$\mathbb{E}_{\mu}(\widehat{v})$ is an element in the weak∗-closure of
$\iota_{V}(\mathcal{C})\subseteq V^{\ast\ast}$. But this set coincides (by
applying Krein’s theorem once again) with the image of the weak closure of
$\text{conv}(G.v)$ under $\iota_{V}$ and is therefore contained in
$\text{im}(\iota_{V})$. It follows that
$\mathbb{E}_{\mu}(\widehat{v})\in\text{im}(\iota_{V})$ for every $v\in V$ and
hence, by Proposition 3.2, $(V,\rho)$ is weakly uniquely $\mu$-stationary. ∎

In combination with [3, Lemma 17], Theorem 2.6 and Theorem 3.7 immediately
imply the following corollary.

######  Corollary 3.8.

Let $G$ be a topological group and $(V,\rho)$ a wap Banach $G$-module with $V$
being separable. Then $(V,\rho)$ is BP-integrable and there exists a Baire
probability measure $\mu$ on $G$ with
$\mathbb{E}_{\mu}(\widehat{v})\in\iota_{V}(V^{G})$ for every $v\in V$, where
$\mathbb{E}_{\mu}$ is an operator as in Proposition 2.3.

##  4\. Application to cohomology of groups

In the following, we apply the results from the previous sections to the study
of the reduced continuous cohomology of uniformly bounded representations.

###  4.1. Complementation

In [31] Nowak initiated the study of the complementability of closed subspaces
that appear in the cohomological framework beyond the setting of unitary
representations on Hilbert spaces. For unitary representations results like
this occur e.g. in Hodge theory, in the deduction of the vanishing of certain
cohomology groups (see e.g. [18]), they were used in [32] to obtain a
functional analytic proof of Gromov’s polynomial growth theorem, and they are
applied to rigidity questions for groups (see e.g. [38]).

######  Definition 4.1.

A subspace $W$ of a topological vector space $V$ is _complemented_ , if there
exists a subspace $W^{\prime}\subseteq V$ such that $V$ decomposes as a
topological direct sum $V=W\oplus W^{\prime}$.

For linear representations on superreflexive Banach spaces and (more
generally) weakly almost periodic representations, the complementation of
$V^{G}$ in $V$ was studied in [2] and [3]. For $\mu$-integrable Banach
modules, Proposition 3.2 characterizes the complementability of the subspace
of elements that are invariant with respect to the convolution with a given
Baire probability measure.

The following statement is a refinement of the main theorem in [31] (see also
[37, Example 6.1]).

######  Proposition 4.2.

Let $G$ be a topological group, $(V,\rho)$ a uniformly bounded strongly
operator continuous Banach $G$-module, and $\mu\in\mathcal{P}_{\sigma}(G)$ a
Baire probability measure. Assume that $\mu$ admits a null set
$\mathcal{N}\subseteq G$ for which $G\setminus\mathcal{N}$ is contained in a
compact set. If furthermore $\|\rho_{\mu}\|<1$, then $B^{1}(G,\rho)\subseteq
Z^{1}(G,\rho)$ is a complemented closed subspace with

$Z^{1}(G,\rho)=B^{1}(G,\rho)\oplus\mathcal{H}_{\mu}^{1}(G,\rho),$

where $\mathcal{H}_{\mu}^{1}(G,\rho)$ is the subspace of all _$\mu$ -harmonic
$1$-cocycles_, i.e. the subspace of all $b\in Z^{1}(G,\rho)$ with
$\int_{G}b(h,e)d\mu(h)=0$.

###### Proof.

First note that Pettis’ measurability theorem implies the well-definedness of
the operator $\rho_{\mu}$. Furthermore, the function $h\mapsto b(h,e)$ is
Bochner integrable for every 1-cocycle $b\in Z^{1}(G,\rho)$. By
$\|\rho_{\mu}\|<1$ the operator $\Delta_{\mu}$ is (boundedly) invertible, so
that $f_{b}\in C^{1}(G,V)^{G}$ given by

$f_{b}(g):=\rho_{g}\circ\Delta_{\mu}^{-1}\left(\int_{G}b(h,e)d\mu(h)\right)$

is well-defined. Define a linear map $P:Z^{1}(G,\rho)\rightarrow
B^{1}(G,\rho)$ via $P(b):=\partial^{1}(f_{b})$. One checks that $P$ is
continuous with $P|_{B^{1}(G,\rho)}=\text{id}$. It follows that
$B^{1}(G,\rho)\subseteq Z^{1}(G,\rho)$ is closed with
$Z^{1}(G,\rho)=B^{1}(G,\rho)\oplus\ker(P)$. An element $b\in Z^{1}(G,\rho)$ is
contained in the kernel of $P$ if and only if
$\Delta_{\mu}^{-1}\left(\int_{G}b(h,e)d\mu(h)\right)\in V^{G}=\\{0\\}$, that
is $b\in\mathcal{H}_{\mu}^{1}(G,\rho)$. ∎

######  Remark 4.3.

_(i)_ Proposition 4.2 in particular applies to the case of isometric
representations without almost invariant vectors on uniformly convex Banach
spaces, see e.g. [37, Lemma 5.2].

_(ii)_ Note that the representation in Proposition 4.2 is weakly uniquely
$\mu$-stationary with $\mathbb{E}_{\mu}\equiv 0$.

_(iii)_ Every unitary representation of a locally compact group admits a
decomposition as in Proposition 4.2, see e.g. [19], [4], [10], [11].

In combination with the ideas in [3] our previous results can be used to
obtain complementation results for subspaces in higher degrees. Statements
like this have earlier been studied by Rosendal in [37] in a more restrictive
setting.

Motivated by an important ingredient in the proof of Shalom’s cohomological
rigidity results for unitary representations of locally compact groups in
[38], in [3], by applying the Ryll-Nardzewski fixed point Theorem, Bader,
Rosendal and Sauer proved an analog of the following theorem in the setting of
strongly operator continuous wap representations.

######  Theorem 4.4.

Let $G$ be a topological group containing subgroups $N,C\leq G$ with
$C\subseteq C_{G}(N)$, where $C_{G}(N)$ denotes the _centralizer_ of $N$ in
$G$. Let furthermore $\mu\in\mathcal{P}_{\sigma}(C)$ be a Baire probability
measure, $(V,\rho)$ a uniformly bounded, strongly operator continuous, weakly
uniquely $\mu$-stationary Banach $G$-module, and $b\in Z^{n}(G,\rho)$ a
$n$-cocycle. Then $\widetilde{\mathbb{E}}_{\mu}\circ(b|_{N^{n+1}})$ defines an
element in $Z^{n}(N,\rho|_{N})$ which is almost cohomologous to the
restriction $b|_{N^{n+1}}$.

###### Proof.

By $\mu\in\mathcal{P}_{\sigma}(C)$ the operator $\rho_{\mu}$ commutes with all
elements $\rho_{m}\in\text{GL}(V)$, $m\in N$. From Proposition 3.2 it follows
that for all $m,m_{1},...,m_{n+1}\in N$,

$\displaystyle(\widetilde{\mathbb{E}}_{\mu}\circ b)(mm_{1},...,mm_{n+1})$
$\displaystyle=$
$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=0}^{N-1}\rho_{\mu}^{i}\left(b(mm_{1},...,mm_{n+1})\right)$
$\displaystyle=$
$\displaystyle\lim_{N\rightarrow\infty}\rho_{m}\left(\frac{1}{N}\sum_{i=0}^{N-1}\rho_{\mu}^{i}\left(b(m_{1},...,m_{n+1})\right)\right)$
$\displaystyle=$
$\displaystyle\rho_{m}\left((\widetilde{\mathbb{E}}_{\mu}\circ
b)(m_{1},...,m_{n+1})\right),$

i.e. $\widetilde{\mathbb{E}}_{\mu}\circ(b|_{N^{n+1}})\in C(N^{n+1},V)^{N}$. It
is furthermore clear that
$\partial^{n+1}(\widetilde{\mathbb{E}}_{\mu}\circ(b|_{N^{n+1}}))=0$, so that
$\widetilde{\mathbb{E}}_{\mu}\circ(b|_{N^{n+1}})$ indeed defines an element in
$Z^{n}(N,\rho|_{N})$.

Since
$\text{im}(1-\widetilde{\mathbb{E}}_{\mu})=\overline{\text{im}(\Delta_{\mu})}$
is a $N$-invariant closed subspace of $V$, the element
$b^{\prime}:=(1-\widetilde{\mathbb{E}}_{\mu})\circ(b|_{N^{n+1}})$ is a
$n$-cocycle taking values in $\overline{\text{im}(\Delta_{\mu})}$. We may
assume that $N$ is infinite. Denote by $\mathcal{S}$ the set of non-empty
finite subsets of $N^{n+1}$, partially ordered by inclusion. For any
$T\in\mathcal{S}$, Proposition 3.4 implies the existence of an element
$\delta_{T}\in\text{conv}\\{\rho_{c}\mid c\in C\\}$ with $\max_{\mathbf{n}\in
T}\|\widetilde{\mathbb{E}}_{\mu}(b(\mathbf{n}))-\delta_{T}(b(\mathbf{n}))\|<(\\#T)^{-1}$.
For $c\in C$ define a homomorphism $h_{c}:C^{n+1}(G,V)^{G}\rightarrow
C^{n}(G,V)^{G}$ via

$\left(h_{c}(f)\right)(g_{1},...,g_{n}):=\sum_{i=1}^{n}(-1)^{i+1}f(g_{1},...,g_{i},g_{i}c,...,g_{n}c).$

By the proof of [3, Lemma 18],
$(1-\rho_{c})\circ(b|_{N^{n+1}})=\partial^{n}\left(h_{c}(b)|_{N^{n+1}}\right)$
and hence also $(1-\delta_{T})\circ(b|_{N^{n+1}})\in B^{n}(N,\rho|_{N})$. We
claim that $(1-\delta_{T})\circ(b|_{N^{n+1}})\rightarrow b^{\prime}$ in
$Z^{n}(N,\rho|_{N})$. Indeed, let $K\subseteq N^{n+1}$ be a norm compact set
and let $\varepsilon>0$. By the continuity of $b$ we may find finitely many
elements $\mathbf{m}_{1},...,\mathbf{m}_{k}\in K$ with
$\frac{1}{k}<\frac{\varepsilon}{3}$ and $\min_{1\leq i\leq
k}\|b(\mathbf{n})-b(\mathbf{m}_{i})\|<\frac{\varepsilon}{3M}$ for every
$\mathbf{n}\in K$, where $M:=\sup_{g\in G}\|\rho_{g}\|$. For
$T:=\\{\mathbf{m}_{1},...,\mathbf{m}_{k}\\}\in\mathcal{S}$ and $T\subseteq
T^{\prime}\in\mathcal{S}$,

$\displaystyle\sup_{\mathbf{n}\in
K}\|b^{\prime}(\mathbf{n})-(1-\delta_{T^{\prime}})\circ
b(\mathbf{n})\|=\sup_{\mathbf{n}\in
K}\|\widetilde{\mathbb{E}}_{\mu}(b(\mathbf{n}))-\delta_{T^{\prime}}(b(\mathbf{n}))\|$
$\displaystyle\leq$ $\displaystyle\frac{2\varepsilon}{3}+\max_{1\leq i\leq
k}\|\widetilde{\mathbb{E}}_{\mu}(b(\mathbf{m}_{i}))-\delta_{T^{\prime}}(b(\mathbf{m}_{i}))\|<\frac{2\varepsilon}{3}+\frac{1}{\\#T^{\prime}}<\varepsilon.$

This implies that

$b^{\prime}=\lim_{T\in\mathcal{S}}(1-\delta_{T})\circ
b\in\overline{B^{n}(N,\rho|_{N})},$

i.e. $\widetilde{\mathbb{E}}_{\mu}\circ(b|_{N^{n+1}})$ is almost cohomologous
to $b|_{N^{n+1}}$ in $Z^{n}(N,\rho|_{N})$. ∎

######  Remark 4.5.

By Theorem 3.7 the statement in Theorem 4.4 in particular applies to wap
representations and arbitrary measures. If $V$ is separable, Theorem 2.6
furthermore implies in combination with [3, Lemma 17] that $\mu$ can be chosen
in a way such that $b|_{N^{n+1}}$ is almost cohomologous to a cocycle taking
values in $V^{C}$.

######  Corollary 4.6.

Let $G$ be a topological group containing subgroups $N,C\leq G$ with
$N\subseteq C_{G}(N)$. Let furthermore $\mu\in\mathcal{P}_{\sigma}(C)$ be a
Baire probability measure and $(V,\rho)$ a uniformly bounded, strongly
operator continuous, weakly uniquely $\mu$-stationary Banach $G$-module. Then
$\overline{B^{1}(N,V_{0})}$ with _$V_{0}:=\overline{\text{im}(\Delta_{\mu})}$_
topologically embeds as a complemented subspace into the image of the
restriction map $Z^{1}(G,\rho)\rightarrow Z^{1}(N,\rho|_{N})$.

In particular, if $V^{\mu}\subseteq V^{N}$, then
$\overline{B^{1}(N,\rho|_{N})}$ is complemented in the image of the
restriction map $Z^{1}(G,\rho)\rightarrow Z^{1}(N,\rho|_{N})$.

###### Proof.

Denote the restriction map $Z^{1}(G,\rho)\rightarrow Z^{1}(N,\rho|_{N})$ by
Res and note that $V_{0}$ is a $N$-invariant closed subspace of $V$. The
linear map $C^{1}(N,V)^{N}\rightarrow C^{1}(G,V)^{G}$ given by
$f\mapsto\overline{f}$ with $\overline{f}(g):=\rho_{g}\circ f(e)$ for $g\in G$
is well-defined, continuous and bijective. It induces a topological embedding
of $\overline{B^{1}(N,\rho|_{N})}$ into $\text{Res}(Z^{1}(G,\rho))$ via
$\partial^{1}f\mapsto\text{Res}(\partial^{1}\overline{f})$ for $f\in
C^{1}(N,V)^{N}$. Similarly, $\overline{B^{1}(N,V_{0})}$ topologically embeds
as a subspace of $\text{Res}(Z^{1}(G,\rho))$ via
$\partial^{1}f\mapsto\text{Res}(\partial^{1}\overline{f})$ for $f\in
C^{1}(N,V_{0})^{N}\subseteq C^{1}(N,V)^{N}$.

By Theorem 4.4 and Lemma 2.4,
$(1-\widetilde{\mathbb{E}}_{\mu})\circ\text{Res}(b)\in\overline{B^{1}(N,V_{0})}$
for every $b\in Z^{1}(G,\rho)$. We thus obtain a map
$P:\text{Res}(Z^{1}(G,\rho))\rightarrow\overline{B^{1}(N,V_{0})}$ which is
well-defined and continuous. For every $f\in C^{1}(N,V_{0})^{N}$ and $m,n\in
N$ we have that

$P(\text{Res}(\partial^{1}\overline{f}))(m,n)=(1-\widetilde{\mathbb{E}}_{\mu})\left(f(n)-f(m)\right)=f(n)-f(m)=\text{Res}(\partial^{1}\overline{f})(m,n)$

so that $P$ restricts to the identity on $\overline{B^{1}(N,V_{0})}$. This
implies that $\overline{B^{1}(N,V_{0})}$ is complemented in
$\text{Res}(Z^{1}(G,\rho))$.

In the case where $V^{\mu}\subseteq V^{N}$, the space
$\overline{B^{1}(N,V_{0})}$ identifies with $\overline{B^{1}(N,\rho|_{N})}$ in
$\text{Res}(Z^{1}(G,\rho))$. Indeed, it is clear that
$\overline{B^{1}(N,V_{0})}\subseteq\overline{B^{1}(N,\rho|_{N})}$. For the
converse inclusion let $f\in C^{1}(N,V)^{N}$ be a function and note that by
Proposition 3.2 and the same argument as in the proof of Theorem 4.4,
$(1-\widetilde{\mathbb{E}}_{\mu})\circ f\in C^{1}(N,V_{0})^{N}$. For all
$m,n\in N$ we have that $\widetilde{\mathbb{E}}_{\mu}\left(f(n)-f(m)\right)=0$
so that
$\text{Res}(\partial^{1}\overline{f})=\text{Res}(\partial^{1}\overline{(1-\widetilde{\mathbb{E}}_{\mu})\circ
f})$. From this, we obtain the converse inclusion and therefore, by the
previous paragraph, that $\overline{B^{1}(N,\rho|_{N})}$ is complemented in
$\text{Res}(Z^{1}(G,\rho))$. ∎

If the subgroup is chosen to be the center of the group, the statement of
Corollary 4.6 extends to higher degrees.

######  Corollary 4.7.

Let $G$ be a topological group, $\mu\in\mathcal{P}_{\sigma}(\mathcal{Z}(G))$ a
Baire probability measure, and $(V,\rho)$ a uniformly bounded, strongly
operator continuous, weakly uniquely $\mu$-stationary Banach $G$-module. Then

$Z^{n}(G,\rho)=\overline{B^{n}(G,V_{0})}\oplus Z^{n}(G,V^{\mu}),$ (4.1)

where _$V_{0}:=\overline{\text{im}(\Delta_{\mu})}$_. In particular,
$\overline{H}_{c}^{n}(G,\rho)\cong\overline{H}_{c}^{n}(G,V^{\mu})$.

###### Proof.

Theorem 4.4 implies that $b\mapsto\widetilde{\mathbb{E}}_{\mu}\circ b$ induces
a well-defined linear map $P:Z^{n}(G,\rho)\rightarrow Z^{n}(G,V^{\mu})$ with
$(1-P)(b)\in\overline{B^{n}(G,V)}$ for every $b\in Z^{n}(G,\rho)$. The map is
obviously a continuous linear projection with $\text{im}(P)=Z^{n}(G,V^{\mu})$.
By Lemma 2.4 we furthermore have that
$\ker(P)=\text{im}(1-P)=\overline{B^{n}(G,V_{0})}$, from which the
decomposition (4.1) follows. ∎

An analog to Corollary 4.7 for strongly operator continuous wap
representations occurs in [3, Corollary 5]. The statement notably generalizes
a number of previous results, see [35], [28], [25], [29].

###  4.2. Cohomology of nilpotent groups

As before, given a Baire probability measure $\mu$ on a topological group $G$,
let $G_{\mu}$ be the closed subgroup of $G$ generated by the set
$\Sigma_{\mu}(G)$ of elements $g\in G$ for which $\int_{G}fd\mu>0$ for every
positive function $f\in C_{b}(G,\mathbb{R})$ with $f(g)\neq 0$. By combining
the statement in Proposition 2.11 with Corollary 4.7, one can formulate an
analog of [38, Corollary 3.7] and [3, Theorem 7] for weakly uniquely
stationary Banach modules. The proof is essentially the same as in [3, Theorem
7].

######  Proposition 4.8.

Let $G$ be a topological group, $\mu\in\mathcal{P}_{\sigma}(\mathcal{Z}(G))$ a
Baire probability measure, and $(V,\rho)$ a uniformly bounded, strongly
operator continuous, weakly uniquely $\mu$-stationary Banach $G$-module with
$V^{G}=\\{0\\}$. Set $Z:=\mathcal{Z}(G)_{\mu}$ and assume that the product map
$Z\times Z\rightarrow Z$ is measurable with respect to the product
$\sigma$-algebra $\mathcal{B}a(Z)\otimes\mathcal{B}a(Z)$ and
$\mathcal{B}a(Z)$. Then, $Z^{1}(G,V)\cong\overline{B^{1}(G,V_{0})}\oplus
Z^{1}(G/Z,V^{Z})$, where _$V_{0}:=\overline{\text{im}(\Delta_{\mu})}$_. In
particular,
$\overline{H}_{c}^{1}(G,\rho)\cong\overline{H}_{c}^{1}(G/Z,V^{Z})$.

###### Proof.

Since $(V,\rho)$ is weakly uniquely $\mu$-stationary, we have by Corollary
2.11 that $V^{\mu}=V^{Z}$. By Corollary 4.7 it thereby suffices to show that
$Z^{1}(G/Z,V^{Z})\cong Z^{1}(G,V^{Z})$. One easily checks that
$Z^{1}(G/Z,V^{Z})$ canonically embeds into $Z^{1}(G,V^{Z})$. For every $b\in
Z^{1}(G,V^{Z})$ and $z\in Z$, $g,h\in G$ we furthermore have

$b(g,zh)=b(g,e)+b(e,zh)=b(g,e)+b(e,h)-b(z,e)=b(g,e)+b(e,h)+b(e,z).$

For $g=h$ it in particular follows that $b(e,z)\in V^{G}=\\{0\\}$ and hence
$b(g,zh)=b(zg,h)=b(g,h)$. This means that $b$ is constant on cosets of $Z$, so
that $Z^{1}(G/Z,V^{Z})\cong Z^{1}(G,V^{Z})$. ∎

Under similar conditions, we deduce the following implication.

######  Proposition 4.9.

Let $G$ be a topological group, $\mu\in\mathcal{P}_{\sigma}(\mathcal{Z}(G))$ a
Baire probability measure, and $(V,\rho)$ a uniformly bounded, strongly
operator continuous, weakly uniquely $\mu$-stationary Banach $G$-module. Set
$Z:=\mathcal{Z}(G)_{\mu}$ and assume that the product map $Z\times
Z\rightarrow Z$ is measurable with respect to the product $\sigma$-algebra
$\mathcal{B}a(Z)\otimes\mathcal{B}a(Z)$ and $\mathcal{B}a(Z)$. Then, for any
1-cocycle $b\in Z^{1}(G,\rho)$ with $\lim_{n}n^{-1}\|b(e,z^{n})\|=0$ for all
$z\in Z$, the element $\widetilde{\mathbb{E}}_{\mu}\circ b\in Z^{1}(G,V^{Z})$
factors through $G/Z$.

###### Proof.

Since $(V,\rho)$ is weakly uniquely $\mu$-stationary we have by Corollary 2.11
that $V^{\mu}=V^{Z}$. Theorem 4.4 therefore implies that
$b^{\prime}:=\widetilde{\mathbb{E}}_{\mu}\circ b\in Z^{1}(G,V^{Z})$. By the
same calculation as in the proof of Proposition 4.8, $b(e,z^{n})=nb(e,z)$ for
every $z\in Z$, $n\in\mathbb{N}$, so that by our assumption
$\|b(e,z)\|=\lim_{n}n^{-1}\|b(e,z^{n})\|=0$. The claim then follows in the
same way as in the proof of Proposition 4.8. ∎

In the case of weakly almost periodic representations of finitely generated
nilpotent groups the following theorem strengthens [3, Theorem 8] and [16,
Theorem 1].

######  Theorem 4.10.

Let $G$ be a finitely generated nilpotent topological group and $(V,\rho)$ a
strongly operator continuous wap Banach $G$-module. Denote the Abelianization
of $G$ by $G^{\text{ab}}:=G/[G,G]$. Then

$Z^{1}(G,\rho)\cong Z^{1}(G^{\text{ab}},V^{[G,G]})\oplus W$ (4.2)

for some closed subspace $W\subseteq\overline{B^{1}(G,\rho)}$. The
identification induces a topological isomorphism
$\overline{H}_{c}^{1}(G,\rho)\cong\overline{H}_{c}^{1}(G^{\text{ab}},V^{[G,G]})$
and in the case where $V^{[G,G]}=V^{G}$ we have $W=\overline{B^{1}(G,\rho)}$.

###### Proof.

Let $G$ be a finitely generated $c$-step nilpotent group and let
$G=\gamma_{0}(G)\triangleright...\triangleright\gamma_{c}(G)=\\{e\\}$ be its
lower central series. We perform an induction over $c$.

For $c=1$ the group $G$ is Abelian so that $[G,G]=\\{e\\}$ and
$G^{\text{ab}}=G$. Therefore $Z^{1}(G,\rho)=Z^{1}(G^{\text{ab}},V^{[G,G]})$
and the claim immediately follows by choosing $W$ to be trivial.

For the induction step assume that for any finitely generated nilpotent
topological group of step $<c$ and every strongly operator continuous wap
Banach $G$-module the assertion of the theorem holds, and let $G$ be finitely
generated $c$-step nilpotent. The subgroup
$N:=\gamma_{c-1}(G)\leq\mathcal{Z}(G)$ is finitely generated, so that we may
choose a finite symmetric generating set $S$ of $N$. Consider the
corresponding Baire probability measure $\mu:=(\\#S)^{-1}\sum_{s\in
S}\delta_{s}$. By the main theorem in [36] the pair $(N,\mu)$ is Liouville so
that an application of Proposition 2.9, Theorem 3.7 and Corollary 4.7 give
$Z^{1}(G,V)\cong\overline{B^{1}(G,V_{0})}\oplus Z^{1}(G,V^{N})$, where
$V_{0}:=\overline{\text{im}(\Delta_{\mu})}$. Let $T$ be a finite generating
set of $G$ and denote the corresponding word length function on $G$ by
$\ell_{T}$. It is well-known (see e.g. [13, Lemma 14.15]) that every element
$g\in N$ is distorted in the sense that
$\lim_{n\rightarrow\infty}n^{-1}\ell_{T}(g^{n})=0$. In particular,

$\lim_{n\rightarrow\infty}n^{-1}\left\|b(e,g^{n})\right\|\leq
M\left(\max_{t\in
T}\|b(e,t)\|\right)\left(\lim_{n\rightarrow\infty}n^{-1}\left\|\ell_{T}(g^{n})\right\|\right)=0$

for all $b\in Z^{1}(G,\rho)$, $g\in N$, where $M:=\sup_{g\in G}\|\rho_{g}\|$.
The same reasoning as in the proof of Proposition 4.9 hence implies that for
any 1-cocycle $b\in Z^{1}(G,\rho)$ the element
$\widetilde{\mathbb{E}}_{\mu}\circ b\in Z^{1}(G,V^{N})$ factors through $G/N$.
It therefore follows that

$Z^{1}(G,\rho)\cong\overline{B^{1}(G,V_{0})}\oplus Z^{1}(G/N,V^{N}).$

The quotient $G/N$ is a finitely generated nilpotent topological group of step
at most $c-1$ and the induced quotient representation of $G/N$ is strongly
operator continuous and weakly almost periodic. Our induction assumption gives
a decomposition

$Z^{1}(G/N,V^{N})\cong Z^{1}\left((G/N)^{\text{ab}},V^{[G/N,G/N]}\right)\oplus
W$

with a suitable closed subspace
$W\subseteq\overline{B^{1}\left(G/N,V^{N}\right)}$. Note that the group
$(G/N)^{\text{ab}}$ is a quotient of $G^{\text{ab}}$ and that the kernel of
the quotient map acts trivially on $V^{[G,G]}$. This implies that
$Z^{1}\left((G/N)^{\text{ab}},V^{[G/N,G/N]}\right)\cong
Z^{1}(G^{\text{ab}},V^{[G,G]})$ canonically and hence

$Z^{1}(G,\rho)\cong\left(\overline{B^{1}(G,V_{0})}\oplus W\right)\oplus
Z^{1}(G^{\text{ab}},V^{[G,G]}).$

The space $\overline{B^{1}(G,V_{0})}\oplus W$ identifies with a subspace of
$\overline{B^{1}(G,\rho)}$, from which we conclude the first part of the
theorem. It furthermore immediately follows that
$\overline{H}_{c}^{1}(G,\rho)\cong\overline{H}_{c}^{1}(G^{\text{ab}},V^{[G,G]})$.

Now assume that $V^{[G,G]}=V^{G}$ so that $Z^{1}(G^{\text{ab}},V^{[G,G]})\cong
Z^{1}(G,V^{G})\subseteq Z^{1}(G,\rho)$. For $b\in Z^{1}(G,V^{G})$ and $g,h\in
G$ we have

$b(e,gh)=b(e,h)-b(g,e)=b(e,g)+b(e,h),$

which means that the map $g\mapsto b(e,g)$ is linear. An application of [8,
Corollary 3.3] gives that the map $g\mapsto b^{\prime}(e,g)$ is sublinear for
every $b^{\prime}\in\overline{B^{1}(G,\rho)}$. Thereby the intersection of
$Z^{1}(G,V^{G})$ and $\overline{B^{1}(G,\rho)}$ is trivial. With the
decomposition (4.2) we obtain the second part of the theorem. ∎

######  Remark 4.11.

By similar arguments and an application of Theorem 2.5 it can be shown that
for every finitely generated 2-step nilpotent discrete group $G$, every
uniformly bounded Banach $G$-module $(V,\rho)$, and every subgroup
$H\leq[G,G]$ the image of the restriction map
$\overline{H}_{c}^{1}(G,\rho)\rightarrow\overline{H}_{c}^{1}(H,\rho|_{H})$ is
zero. Compare this with [16, Theorem 8].

###  4.3. Cohomology of products of groups

The results in Subsection 4.1 indicate that the notion of weak unique
stationarity is particularly useful in combination with commuting structures.
Naturally, this motivates the study of the cohomology of products of groups.
In this setting complementation results have been particularly important for
the deduction of superrigidity statements for suitable unitary representations
on Hilbert spaces and Banach modules; see for instance [38], [2] and also
Subsection 4.4.

The following theorem is an application of Theorem 4.2.

######  Theorem 4.12.

Let $G:=G_{1}\times G_{2}$ be a product of topological groups and $(V,\rho)$ a
uniformly bounded strongly operator continuous Banach $G$-module. Assume that
$G_{1}$ and $G_{2}$ admit Baire probability measures
$\mu_{1}\in\mathcal{P}_{\sigma}(G_{1})$,
$\mu_{2}\in\mathcal{P}_{\sigma}(G_{2})$ and null sets
$\mathcal{N}_{1}\subseteq G_{1}$, $\mathcal{N}_{2}\subseteq G_{2}$ for which
both $G_{1}\setminus\mathcal{N}_{1}$ and $G_{2}\setminus\mathcal{N}_{2}$ are
contained in compact sets. Assume furthermore that
$\|\rho_{\mu_{1}}\rho_{\mu_{2}}\|<1$ and that $V^{\mu_{1}}=V^{G_{1}}$ and
$V^{\mu_{2}}=V^{G_{2}}$. Then there exists a topological isomorphism
$H_{c}^{1}(G_{1}\times G_{2},V)\cong H_{c}^{1}(G_{1},V^{G_{2}})\oplus
H_{c}^{1}(G_{2},V^{G_{1}})$.

###### Proof.

First recall that Pettis’ measurability theorem implies the well-definedness
of the operators $\rho_{\mu_{1}}$ and $\rho_{\mu_{2}}$, and that the functions
$G_{1}\ni g\mapsto b(g,e)$ and $G_{2}\ni h\mapsto b(h,e)$ are Bochner
integrable for every 1-cocycle $b\in Z^{1}(G,\rho)$. By the same argument as
in the proof of Proposition 4.2, we find that $Z^{1}(G,\rho)$ decomposes via
$B^{1}(G,\rho)\oplus\mathcal{H}^{1}(G,\rho)$, where $\mathcal{H}^{1}(G,\rho)$
consists of all elements $b\in Z^{1}(G)$ with
$\int_{G_{1}}(\int_{G_{2}}b(gh,e)d\mu_{2}(h))d\mu_{1}(g)=0$. Denote the
corresponding projection onto $\mathcal{H}^{1}(G,\rho)$ by $P$.

For $b\in\mathcal{H}^{1}(G,\rho)$ set $v:=\int_{G_{1}}b(g,e)d\mu_{1}(g)$ and
$w:=\int_{G_{2}}b(h,e)d\mu_{2}(h)$. We then have

$\rho_{\mu_{1}}(w)+v=\int_{G_{1}}\left(\int_{G_{2}}b(gh,e)d\mu_{2}(h)\right)d\mu_{1}(g)=0$

and similarly $\rho_{\mu_{2}}(v)+w=0$, so that
$\rho_{\mu_{1}}\rho_{\mu_{2}}(v)=\rho_{\mu_{1}}(-w)=v$. From this we obtain
with $\|\rho_{\mu_{1}}\rho_{\mu_{2}}\|<1$ that $v=0$. In the same way, $w=0$.
For $h\in G_{2}$,

$\displaystyle b(h,e)=\rho_{h}(v)+b(h,e)=\int_{G_{1}}b(gh,e)d\mu_{1}(g)$
$\displaystyle=$
$\displaystyle\int_{G_{1}}\rho_{g}\left(b(h,e)\right)d\mu_{1}(g)+v=\rho_{\mu_{1}}\left(b(h,e)\right),$

which means that $b(h,e)$ is invariant under $\rho_{\mu_{1}}$ and hence
$b|_{G_{2}\times G_{2}}\in Z^{1}(G_{2},V^{G_{1}})$. A similar calculation
leads to $b|_{G_{1}\times G_{1}}\in Z^{1}(G_{1},V^{G_{2}})$. We may therefore
introduce a well-defined linear map $\pi:H_{c}^{1}(G,\rho)\rightarrow
H_{c}^{1}(G_{1},V^{G_{2}})\oplus H_{c}^{1}(G_{2},V^{G_{1}})$ via
$[b]\mapsto[Pb|_{G_{1}\times G_{1}}]\oplus[Pb|_{G_{2}\times G_{2}}]$. It
remains to show that $\pi$ is an isomorphism of topological vector spaces.

  * •

_Injectivity_ : Let $b\in\mathcal{H}^{1}(G,\rho)$ with $Pb|_{G_{1}\times
G_{1}}\in B^{1}(G_{1},V^{G_{2}})$ and $Pb|_{G_{2}\times G_{2}}\in
B^{1}(G_{2},V^{G_{1}})$. Then there exist functions $f_{1}\in
C^{1}(G_{1},V^{G_{2}})^{G_{1}}$ and $f_{2}\in C^{1}(G_{2},V^{G_{1}})^{G_{2}}$
with $Pb|_{G_{1}\times G_{1}}=\partial^{1}f_{1}$ and $Pb|_{G_{2}\times
G_{2}}=\partial^{1}f_{2}$. For $g,g^{\prime}\in G_{1}$, $h,h^{\prime}\in
G_{2}$ we have

$\displaystyle b(gh,g^{\prime}h^{\prime})$ $\displaystyle=$ $\displaystyle
b(e_{G},g^{\prime}h^{\prime})-b(e_{G},gh)$ $\displaystyle=$
$\displaystyle\rho_{g^{\prime}}\left(b(e_{G},h^{\prime})\right)+b(e_{G},g^{\prime})-\rho_{g}\left(b(e_{G},h)\right)-b(e_{G},g)$
$\displaystyle=$ $\displaystyle
b(e_{G},h^{\prime})+b(e_{G},g^{\prime})-b(e_{G},h)-b(e_{G},g)$
$\displaystyle=$
$\displaystyle\partial^{1}f_{2}(e,h^{\prime})+\partial^{1}f_{1}(e,g^{\prime})-\partial^{1}f_{2}(e,h)-\partial^{1}f_{1}(e,g)$
$\displaystyle=$ $\displaystyle
f_{1}(g^{\prime})+f_{2}(h^{\prime})-f_{1}(g)-f_{2}(h).$

It follows that $b\in B^{1}(G,\rho)$ and hence that $\pi$ is injective.

  * •

_Surjectivity_ : For $b_{1}\in Z^{1}(G_{1},V^{G_{2}})$ and $b_{2}\in
Z^{1}(G_{2},V^{G_{1}})$ define $b:G\times G\rightarrow V$ by
$b(gh,g^{\prime}h^{\prime}):=b_{1}(g,g^{\prime})+b_{2}(h,h^{\prime})$, where
$g,g^{\prime}\in G_{1}$, $h,h^{\prime}\in G_{2}$. For
$g,g_{1}^{\prime},g_{2}^{\prime}\in G_{1}$,
$h,h_{1}^{\prime},h_{2}^{\prime}\in G_{2}$ we have

$\displaystyle
b(ghg_{1}^{\prime}h_{1}^{\prime},ghg_{2}^{\prime}h_{2}^{\prime})$
$\displaystyle=$ $\displaystyle
b_{1}(gg_{1}^{\prime},gg_{2}^{\prime})+b_{2}(hh_{1}^{\prime},hh_{2}^{\prime})$
$\displaystyle=$
$\displaystyle\rho_{g}\left(b_{1}(g_{1}^{\prime},g_{2}^{\prime})\right)+\rho_{h}\left(b_{2}(h_{1}^{\prime},h_{2}^{\prime})\right)$
$\displaystyle=$
$\displaystyle\rho_{gh}\left(b_{1}(g_{1}^{\prime},g_{2}^{\prime})+b_{2}(h_{1}^{\prime},h_{2}^{\prime})\right)$
$\displaystyle=$
$\displaystyle\rho_{gh}\left(b(g_{1}^{\prime}h_{1}^{\prime},g_{2}^{\prime}h_{2}^{\prime})\right),$

and

$\displaystyle\partial^{2}b(gh,g^{\prime}h^{\prime},g^{\prime\prime}h^{\prime\prime})$
$\displaystyle=$ $\displaystyle
b(g^{\prime}h^{\prime},g^{\prime\prime}h^{\prime\prime})-b(gh,g^{\prime\prime}h^{\prime\prime})+b(gh,g^{\prime}h^{\prime})$
$\displaystyle=$
$\displaystyle\left(b_{1}(g^{\prime},g^{\prime\prime})-b_{1}(g,g^{\prime\prime})+b_{1}(g,g^{\prime})\right)$
$\displaystyle+\left(b_{2}(h^{\prime},h^{\prime\prime})-b_{2}(h,h^{\prime\prime})+b_{2}(h,h^{\prime})\right)$
$\displaystyle=$ $\displaystyle 0,$

so that $b\in Z^{1}(G,\rho)$ with $[Pb|_{G_{1}\times G_{1}}]=[b_{1}]$,
$[Pb|_{G_{2}\times G_{2}}]=[b_{2}]$. We obtain that $\pi$ is surjective.

  * •

_Homeomorphism_ : Consider the map $Z^{1}(G,\rho)\rightarrow
Z^{1}(G_{1},V^{G_{2}})\oplus Z^{1}(G_{2},V^{G_{1}})$ given by $b\mapsto
Pb|_{G_{1}\times G_{1}}\oplus Pb|_{G_{2}\times G_{2}}$. This map is clearly
continuous so that the induced map $Z^{1}(G,\rho)\rightarrow
H_{c}^{1}(G_{1},V^{G_{2}})\oplus H_{c}^{1}(G_{2},V^{G_{1}})$ is also
continuous. From the universal property of the quotient, it follows that $\pi$
must be continuous as well. The continuity of $\pi^{-1}$ follows in a similar
way by considering the map $\eta:Z^{1}(G_{1},V^{G_{2}})\oplus
Z^{1}(G_{2},V^{G_{1}})\rightarrow Z^{1}(G,\rho)$ given by $\eta(b_{1}\oplus
b_{2})(gh,g^{\prime}h^{\prime}):=b_{1}(g,g^{\prime})+b_{2}(h,h^{\prime})$ for
$b_{1}\in Z^{1}(G_{1},V^{G_{2}})$, $b_{2}\in Z^{1}(G_{2},V^{G_{1}})$,
$g,g^{\prime}\in G_{1}$, $h,h^{\prime}\in G_{2}$.

This finishes the proof. ∎

######  Remark 4.13.

Under the assumptions of Theorem 4.12, its assertion can alternatively be
formulated as follows: every 1-cocycle in $Z^{1}(G,\rho)$ is cohomologous to a
sum of the form $b_{1}+b_{2}$ with $b_{1},b_{2}\in Z^{1}(G,\rho)$, where
$b_{1}$ takes values in $V^{G_{2}}$ and where $b_{2}$ takes values in
$V^{G_{1}}$. Furthermore, $b_{1}$ and $b_{2}$ factor to elements in
$Z^{1}(G_{1},V^{G_{2}})$ and $Z^{1}(G_{2},V^{G_{1}})$.

######  Definition 4.14.

Let $G$ be a topological group. A uniformly bounded Banach $G$-module
$(V,\rho)$ is said to have _almost invariant unit vectors_ if for all compact
subsets $K\subseteq G$ and $\varepsilon>0$ there exists an element $v\in V$
with $\|v\|=1$ and $\|v-\rho_{g}(v)\|<\varepsilon$ for all $g\in K$.

Theorem 4.12 can be viewed as a partial generalization of [2, Theorem C], as
the following remark illustrates.

######  Remark 4.15.

By applying the results in [14], the assertion of Theorem 4.12 in particular
applies to isometric representations of compactly generated topological groups
without almost invariant unit vectors on uniformly convex Banach spaces.

######  Theorem 4.16.

Let $G:=G_{1}\times G_{2}$ be a product of topological groups,
$\mu_{1}\in\mathcal{P}_{\sigma}(G_{1})$,
$\mu_{2}\in\mathcal{P}_{\sigma}(G_{2})$ Baire probability measures and
$(V,\rho)$ a uniformly bounded strongly operator continuous Banach $G$-module.
Assume that $(V,\rho|_{G_{1}})$ is weakly uniquely $\mu_{1}$-stationary
without almost invariant unit vectors, and that $(V,\rho|_{G_{2}})$ is weakly
uniquely $\mu_{2}$-stationary. Then there exists an embedding of topological
vector spaces $H_{c}^{1}(G_{1}\times G_{2},V)\hookrightarrow
H_{c}^{1}(G_{1},V^{\mu_{2}})\oplus H_{c}^{1}(G_{2},V^{\mu_{1}})$ via
$[b]\mapsto[\widetilde{\mathbb{E}}_{\mu_{2}}\circ(b|_{G_{1}})]\oplus[\widetilde{\mathbb{E}}_{\mu_{1}}\circ(b|_{G_{2}})]$.

If furthermore $V^{\mu_{1}}=V^{G_{1}}$ and $V^{\mu_{2}}=V^{G_{2}}$, then the
embedding above is surjective, i.e. $H_{c}^{1}(G_{1}\times G_{2},V)\cong
H_{c}^{1}(G_{1},V^{G_{2}})$.

###### Proof.

Denote the map above by $\iota$. In the following, we demonstrate that $\iota$
is a well-defined linear homeomorphism onto its image.

  * •

_Well-defined_ : By $\mu_{2}\in\mathcal{P}_{\sigma}(G_{2})$ the operator
$\rho_{\mu_{2}}$ commutes with all elements $\rho_{g}$, $g\in G_{1}$. From
Proposition 3.2 it therefore follows that
$\widetilde{\mathbb{E}}_{\mu_{2}}\circ(f|_{G_{1}})\in
C^{1}(G_{1},V^{\mu_{2}})^{G_{1}}$ and similarly
$\widetilde{\mathbb{E}}_{\mu_{1}}\circ(f|_{G_{2}})\in
C^{1}(G_{2},V^{\mu_{1}})^{G_{2}}$ for every $f\in C^{1}(G,V)^{G}$. But then

$\displaystyle\widetilde{\mathbb{E}}_{\mu_{2}}\circ\left((\partial^{1}f)|_{G_{1}\times
G_{1}}\right)=\lim_{\lambda\in\Lambda}\partial^{1}\left(\widetilde{\mathbb{E}}_{\mu_{2}}\circ(f|_{G_{1}})\right)\in
B^{1}(G_{1},V^{\mu_{2}})$

and $\widetilde{\mathbb{E}}_{\mu_{1}}\circ\left((\partial^{1}f)|_{G_{2}\times
G_{2}}\right)\in B^{1}(G_{2},V^{\mu_{1}})$. This implies that $\iota$ is a
well-defined linear map.

  * •

_Injectivity_ : Let $b\in Z^{1}(G,\rho)$ be a 1-cocycle for which
$\widetilde{\mathbb{E}}_{\mu_{2}}\circ(b|_{G_{1}\times G_{1}})\in
B^{1}(G_{1},V^{\mu_{2}})$,
$\widetilde{\mathbb{E}}_{\mu_{1}}\circ(b|_{G_{2}\times G_{2}})\in
B^{1}(G_{2},V^{\mu_{1}})$ and let $f\in C^{1}(G,V^{\mu_{2}})^{G_{1}}$ with
$\widetilde{\mathbb{E}}_{\mu_{2}}\circ(b|_{G_{1}\times
G_{1}})=\partial^{1}(f|_{G_{1}})$. By replacing the 1-cocycle $b$ by
$b-\partial^{1}f$ we may assume that
$\widetilde{\mathbb{E}}_{\mu_{2}}\circ(b|_{G_{1}\times G_{1}})=0$. Now
consider the set $\mathcal{S}$ of non-empty finite subsets of $G_{1}\times
G_{1}\leq G\times G$, partially ordered by inclusion. By the proof of Theorem
4.4 we find a net
$(\delta_{T})_{T\in\mathcal{S}}\subseteq\text{conv}\\{\rho_{g}\mid g\in
G_{2}\\}$ with $(1-\delta_{T})\circ(b|_{G_{1}\times G_{1}})\in
B^{1}(G_{1},\rho|_{G_{1}})$ for every $T\in\mathcal{S}$ and
$(1-\delta_{T})\circ(b|_{G_{1}\times
G_{1}})\rightarrow(1-\widetilde{\mathbb{E}}_{\mu_{2}})\circ(b|_{G_{1}\times
G_{1}})$ in $Z^{1}(G_{1},\rho|_{G_{1}})$. For $T\in\mathcal{S}$ and $x\in
G_{2}$ let $0\leq\delta_{T}(x)\leq 1$ be the corresponding coefficient of
$\delta_{T}$, so that $\delta_{T}=\sum_{x\in G_{2}}\delta_{T}(x)\rho_{x}$.
Since $(V,\rho|_{G_{1}})$ has no almost invariant unit vectors, there exists a
non-trivial compact subset $K\subseteq G_{1}$ and $\varepsilon>0$ such that
$\sup_{g\in K}\|v-\rho_{g}(v)\|\geq\varepsilon\|v\|$ for all $v\in V$. Let
$K^{\prime}\subseteq G_{2}$ be a compact subset. For $g\in K$, $h\in
K^{\prime}$ we have

$\displaystyle b(e_{G},gh)$ $\displaystyle=$
$\displaystyle\rho_{h}\left((1-\widetilde{\mathbb{E}}_{\mu_{2}})\circ
b(e_{G},g)+\widetilde{\mathbb{E}}_{\mu_{2}}\circ b(e_{G},g)\right)+b(e_{G},h)$
$\displaystyle=$
$\displaystyle\rho_{h}\left((1-\widetilde{\mathbb{E}}_{\mu_{2}})\circ
b(e_{G},g)\right)+b(e_{G},h)$

so that

$\displaystyle\sup_{g\in K,h\in K^{\prime}}\left\|\sum_{x\in
G_{2}}\delta_{T}(x)\left(\rho_{h}\circ b(e_{G},x)-\rho_{h}\circ
b(e_{G},gx)\right)\right\|\rightarrow 0$

and hence by subtracting,

$\sup_{g\in K^{\prime},h\in K}\left\|(1-\rho_{g})\left(\sum_{x\in
G_{2}}\delta_{T}(x)\left(b(e_{G},hx)-b(e_{G},x)\right)\right)\right\|\rightarrow
0.$

The assumptions on $K$ then imply that the net of functions of the form
$G_{2}\ni h\mapsto\sum_{x\in
G_{2}}\delta_{T}(x)\left(b(e_{G},hx)-b(e_{G},x)\right)$ converges to 0
uniformly on compact subsets. Set $M:=\sup_{g\in G}\|\rho_{g}\|$. From the
estimate

$\displaystyle\|b(e_{G},gh)-\sum_{x\in G_{2}}\delta_{T}(x)(1-\rho_{gh})\circ
b(e_{G},x)\|$ $\displaystyle=$
$\displaystyle\|\rho_{h}\left((1-\widetilde{\mathbb{E}}_{\mu_{2}})\circ
b(e_{G},g)\right)+b(e_{G},h)-\sum_{x\in G_{2}}\delta_{T}(x)(1-\rho_{gh})\circ
b(e_{G},x)\|$ $\displaystyle=$
$\displaystyle\|\rho_{h}((1-\widetilde{\mathbb{E}}_{\mu_{2}})\circ
b(e_{G},g)-\sum_{x\in G_{2}}\delta_{T}(x)(1-\rho_{g})\circ b(e_{G},x))$
$\displaystyle\qquad+\sum_{x\in
G_{2}}\delta_{T}(x)\left(b(e_{G},hx)-b(e_{G},x)\right)\|$ $\displaystyle\leq$
$\displaystyle M\|(1-\widetilde{\mathbb{E}}_{\mu_{2}})\circ
b(e_{G},g)-(1-\delta_{T})\circ b(e_{G},g)\|$ $\displaystyle\qquad+\|\sum_{x\in
G_{2}}\delta_{T}(x)\left(b(e_{G},hx)-b(e_{G},x)\right)\|$

it then follows that $\partial^{1}\xi_{T}\rightarrow b$ in $Z^{1}(G,\rho)$,
where $\xi_{T}\in C^{1}(G,V)^{G}$ is given by $\xi_{T}(g):=\sum_{x\in
G_{2}}\delta_{T}(x)\rho_{g}\circ b(e_{G},x)$. Since $(V,\rho|_{G_{1}})$ admits
no almost invariant unit vectors, this implies that $b\in B^{1}(G,\rho)$. We
thereby obtain that the map $\iota$ is injective.

  * •

_Continuity_ : The continuity of the map $\iota$ and its inverse follow
similar to the proof of Theorem 4.12.

This implies the first part of the statement of the theorem. The surjectivity
of $\iota$ in the case where $V^{\mu_{1}}=V^{G_{1}}$ and
$V^{\mu_{2}}=V^{G_{2}}$ follows in the same way as in the proof of Theorem
4.12. ∎

######  Remark 4.17.

Similar to Remark 4.13, under the assumptions of the second part of Theorem
4.16, its assertion can alternatively be formulated as follows: every
1-cocycle in $Z^{1}(G,\rho)$ is cohomologous to a cocycle $b\in Z^{1}(G,\rho)$
which takes values in $V^{G_{2}}$ and which factors to an element in
$Z^{1}(G_{1},V^{G_{2}})$.

Theorem 4.16 has in combination with Theorem 3.8 the following immediate
consequence.

######  Corollary 4.18.

Let $G:=G_{1}\times G_{2}$ be a product of topological groups and $(V,\rho)$ a
strongly operator continuous wap Banach $G$-module. Assume that $V$ is
separable and that $(V,\rho|_{G_{1}})$ has no almost invariant unit vectors.
Then there exists a topological isomorphism ${H}_{c}^{1}(G_{1}\times
G_{2},V)\cong{H}_{c}^{1}(G_{1},V^{G_{2}})$.

###  4.4. Induction and superrigidity

In [38] and [2] decompositions as in Subsection 4.3 were used to obtain
superrigidity theorems for the continuous cohomology of suitable unitary
representations on Hilbert spaces and Banach modules. We finish this article
by following this path. For this, we first remind the reader of basic notions
surrounding the theory of lattices and induced representations.

####  4.4.1. Lattices

Let $G=G_{1}\times...\times G_{n}$ be a product of locally compact groups and
$\Gamma\leq G$ a discrete subgroup. Fix a Haar measure $\nu$ on $G$,
normalized so that the induced measure on $G/\Gamma$ has total mass one.
Recall that a subset $\mathcal{D}\subseteq G$ is called a _fundamental domain_
for $\Gamma$ if $G=\mathcal{D}\Gamma$ with
$\gamma\mathcal{D}\cap\gamma^{\prime}\mathcal{D}=\emptyset$ for all
$\gamma,\gamma^{\prime}\in\Gamma$ with $\gamma\neq\gamma^{\prime}$. The
discrete subgroup $\Gamma$ is called a _lattice_ if it admits a Borel
fundamental domain $\mathcal{D}$ with $\nu(D)<\infty$. A lattice is
_irreducible_ if the image of every projection $p_{i}:\Gamma\rightarrow G_{i}$
with $1\leq i\leq n$ has dense image in $G_{i}$.

Given a fundamental domain $\mathcal{D}$ of a lattice $\Gamma\leq
G_{1}\times...\times G_{n}$, we can consider the associated _cocycle_
$\chi_{\mathcal{D}}:G\times\mathcal{D}\rightarrow\Gamma$ defined by
$\chi_{\mathcal{D}}(g,x):=\gamma$, where $\gamma$ is the unique element in
$\Gamma$ with $gx\gamma\in\mathcal{D}$. One checks that
$\chi_{\mathcal{D}}(gh,x)=\chi_{\mathcal{D}}(h,x)\chi\left(g,hx\chi_{\mathcal{D}}(h,x)\right)$
for all $g,h\in G$, $x\in\mathcal{D}$.

######  Definition 4.19 ([2, Definition 8.2]).

For $p>0$ a lattice $\Gamma\leq G_{1}\times...\times G_{n}$ is called _$p$
-integrable_ if one of the following two statements holds:

  1. (1)

The lattice is _cocompact_ (i.e. the quotient $G/\Gamma$ is compact);

  2. (2)

The lattice it is finitely generated and for some finite generating set
$S\subseteq G$ there exists a Borel fundamental domain $\mathcal{D}\subseteq
G$ with
$\int_{\mathcal{D}}\ell_{S}^{p}\left(\chi_{\mathcal{D}}(g,x)\right)d\nu(x)<\infty$
for all $g\in G$. Here $\ell_{S}$ denotes the word length function associated
with $S$.

####  4.4.2. Induced representations

Let $G=G_{1}\times...\times G_{n}$ be a product of locally compact second
countable groups and $\Gamma\leq G$ a $p$-integrable lattice. As before, fix a
Haar measure $\nu$ on $G$, normalized so that the induced measure on
$G/\Gamma$ has total mass one, and denote the corresponding measure on
$G/\Gamma$ by $\widetilde{\nu}$. For a Banach space $V$ and $1<p<\infty$
consider the Banach space $L^{p}(G/\Gamma,V)$ of (equivalence classes of)
measurable functions $f$ on $G/\Gamma$ with coefficients in $V$ for which
$\|f\|$ is $p$-integrable. Given a strongly operator continuous isometric
linear representation $\rho:\Gamma\rightarrow\text{GL}(V)$ and a fundamental
domain $\mathcal{D}$ for $\Gamma$, we may introduce the induced representation
$\text{Ind}_{\Gamma}^{G}(\rho):G\rightarrow\text{GL}(L^{p}(G/\Gamma,V))$ by

$\left(\text{Ind}_{\Gamma}^{G}(\rho)_{g}(\xi)\right)(x\Gamma):=\rho_{\chi_{\mathcal{D}}(g^{-1},x)}\left(\xi\left(g^{-1}x\chi_{\mathcal{D}}(g^{-1},x)\Gamma\right)\right)$

for $x\in\mathcal{D}$, $\xi\in L^{p}(G/\Gamma,V)$. Then
$\text{Ind}_{\Gamma}^{G}(\rho)$ is again a strongly operator continuous linear
isometric representation. For notational convenience we will usually write
$\widetilde{\rho}:=\text{Ind}_{\Gamma}^{G}(\rho)$. For a 1-cocycle $b\in
Z^{1}(\Gamma,\rho)$ we furthermore define its _induced 1-cocycle_
$\widetilde{b}\in Z^{1}(G,\widetilde{\rho})$ via
$\widetilde{b}(g,h)(x\Gamma):=b\left(\chi_{\mathcal{D}}(g^{-1},x),\chi_{\mathcal{D}}(h^{-1},x)\right)$.
The assignment $b\mapsto\widetilde{b}$ induces a topological isomorphism
$H^{1}(\Gamma,\rho)\cong H^{1}(G,\widetilde{\rho})$. For more details see [38]
and [2].

####  4.4.3. Superrigidity

In the same way that Theorem 4.12 can be viewed as a partial generalization of
[2, Theorem C] (see Remark 4.15), the following theorem can be viewed as a
partial generalization of [2, Theorem D]. Its proof is essentially the same as
the one of [38, Theorem 4.1]. We sketch it for completeness. Under suitable
technical assumptions, the statement extends to arbitrary products of groups.

######  Theorem 4.20.

Let $G:=G_{1}\times G_{2}$ be a product of locally compact second countable
groups, $\Gamma\leq G$ an irreducible $p$-integrable lattice, and $(V,\rho)$ a
strongly operator continuous isometric Banach $\Gamma$-module. Assume that the
groups admit Baire probability measures
$\mu_{1}\in\mathcal{P}_{\sigma}(G_{1})$,
$\mu_{2}\in\mathcal{P}_{\sigma}(G_{2})$ and null sets
$\mathcal{N}_{1}\subseteq G_{1}$, $\mathcal{N}_{2}\subseteq G_{2}$ for which
$G_{1}\setminus\mathcal{N}_{1}$, $G_{2}\setminus\mathcal{N}_{2}$ are contained
in compact sets, for which
$\|\widetilde{\rho}_{\mu_{1}}\widetilde{\rho}_{\mu_{2}}\|<1$, and which
satisfy $\widetilde{V}^{\mu_{1}}=\widetilde{V}^{G_{1}}$,
$\widetilde{V}^{\mu_{2}}=\widetilde{V}^{G_{2}}$. Finally, let $b\in
Z^{1}(\Gamma,\rho)$ be a 1-cocycle. Then there exists for every
$i\in\\{1,2\\}$ a $\Gamma$-invariant closed subspace $V_{i}\subseteq V$ on
which the $\Gamma$-representation extends to a strongly operator continuous
isometric linear $G$-representation that factors through a representation of
$G_{i}$. Furthermore, there exist 1-cocycles $b_{1}\in Z^{1}(G,V_{1})$,
$b_{2}\in Z^{1}(G,V_{2})$ such that $b$ is cohomologous to
$b_{1}|_{\Gamma\times\Gamma}+b_{2}|_{\Gamma\times\Gamma}$ in $Z^{1}(\Gamma,V)$
and such that $b_{i}$ only depends on $G_{i}$ for $i\in\\{1,2\\}$.

###### Proof.

Let $\mathcal{D}$ be a Borel fundamental domain for $\Gamma$ with
$e\in\mathcal{D}$, which is either compact or satisfies the second condition
in Definition 4.19. Furthermore, fix a 1-cocycle $b\in Z^{1}(\Gamma,\rho)$,
consider the induced representation
$\widetilde{\rho}:=\text{Ind}_{\Gamma}^{G}(\rho):G\rightarrow\text{GL}(L^{p}(G/\Gamma,V))$
and let $\widetilde{b}\in Z^{1}(G,\widetilde{\rho})$ be the induced 1-cocycle.
Theorem 4.12 (or rather Remark 4.13) implies in combination with our
assumptions that $\widetilde{b}$ is cohomologous to a sum
$\widetilde{b}_{1}+\widetilde{b}_{2}$ with
$\widetilde{b}_{1},\widetilde{b}_{2}\in Z^{1}(G,\widetilde{\rho})$, where
$\widetilde{b}_{i}$ takes for every $i\in\\{1,2\\}$ values in
$G_{i}^{\prime}:=\prod_{j\neq i}G_{j}$ and $\widetilde{b}_{i}$ factors to an
element in $Z^{1}(G_{i},V^{G_{i}^{\prime}})$.

Now fix $i\in\\{1,2\\}$ and consider the 1-cocycle $\widetilde{b}_{i}\in
Z^{1}(G,\widetilde{\rho})$. As discussed in Subsection 4.4.2,
$H^{1}(\Gamma,\rho)\cong H^{1}(G,\widetilde{\rho})$ so that we find
$\beta_{i}\in Z^{1}(\Gamma,\rho)$ with
$\widetilde{b}_{i}-\widetilde{\beta}_{i}\in B^{1}(G,\widetilde{\rho})$. In
other words, there exists a vector $\xi\in L^{p}(G/\Gamma,V)$ with
$\widetilde{b}_{i}(g,h)=\widetilde{\beta}_{i}(g,h)+\widetilde{\rho}_{h}(\xi)-\widetilde{\rho}_{g}(\xi)$
for all $g,h\in G$. Since $\widetilde{b}_{i}$ factors to an element in
$Z^{1}(G_{i},V^{G_{i}^{\prime}})$, the 1-cocycle vanishes on $G_{i}^{\prime}$
so that
$\widetilde{\beta}_{i}(g,h)=\widetilde{\rho}_{g}(\xi)-\widetilde{\rho}_{h}(\xi)$
for all $g,h\in G_{i}^{\prime}$. Write $\gamma\cdot
v:=\rho_{\gamma}(v)+\beta_{i}(e,\gamma)$ for the affine $\Gamma$-action on $V$
associated with $\beta_{i}$ and consider the closed affine subspace
$V_{i}^{\prime}$ of $V$ consisting of all elements $v\in V$ for which
$\gamma_{j}\cdot v\rightarrow v$ for every sequence
$(\gamma_{j})_{j\in\mathbb{N}}\subseteq\Gamma$ with
$p_{i}(\gamma_{j})\rightarrow e$, where $p_{i}:G\rightarrow G_{i}$ denotes the
projection onto the $i$-th component of $G$. By the same argument as in [38,
Proposition 4.4] (which again relies on [27, Theorem 6.3] and [5, Lemma 1]),
we find that $V_{i}^{\prime}$ is non-empty. Note also that $V_{i}^{\prime}$ is
invariant under the affine $\Gamma$-action. We claim that the action
continuously extends to an action of $G$ on $V_{i}^{\prime}$ that factors
through $G_{i}$. Indeed, by the irreducibility of $\Gamma\leq G$ we may for
every $g\in G$ choose a sequence
$(\gamma_{j}(g))_{j\in\mathbb{N}}\subseteq\Gamma$ with
$p_{i}\left(\gamma_{j}(g)\right)\rightarrow p_{i}(g)$. Since
$(\gamma_{j}(g)\cdot v)_{j\in\mathbb{N}}\subseteq V_{i}^{\prime}$ is a Cauchy
sequence, for every $v\in V_{i}^{\prime}$ the limit $g\cdot
v:=\lim_{j\in\mathbb{N}}\gamma_{j}(g)\cdot v$ exists. By the above, it is
furthermore independent of the choice of
$(\gamma_{j}(g))_{j\in\mathbb{N}}\subseteq\Gamma$, it factors through $G_{i}$,
and it is continuous.

Now choose an element $v_{i}\in V^{\prime}$ and consider the closed subspace
$V_{i}:=V_{i}^{\prime}+v_{i}$ of $V$. For every $\gamma\in\Gamma$ we have that
$\rho_{\gamma}(v)=\gamma\cdot(v-v_{i})-\gamma\cdot v_{i}$. It thereby follows
in combination with the above that $\rho$ extends to a strongly operator
continuous isometric linear representation of $G$ on $V_{i}^{\prime}$. By a
similar argument, the 1-cocycle
$(\gamma,\gamma^{\prime})\mapsto\beta_{i}(\gamma,\gamma^{\prime})+\rho_{\gamma^{\prime}}(v_{i})-\rho_{\gamma}(v_{i})$
extends to an element $b_{i}$ in $Z^{1}(G,V_{i})$ that factors through
$G_{i}$.

By $H^{1}(\Gamma,\rho)\cong H^{1}(G,\widetilde{\rho})$, the sum
$\beta_{1}+\beta_{2}$ is cohomologous to $b$, so that $b$ must also be
cohomologous to $b_{1}|_{\Gamma\times\Gamma}+b_{2}|_{\Gamma\times\Gamma}$.
This implies the claim. ∎

Analogously, the argument in the proof of Theorem 4.20 implies with Remark
4.17 the following theorem.

######  Theorem 4.21.

Let $G:=G_{1}\times G_{2}$ be a product of locally compact second countable
groups, $\Gamma\leq G$ a $p$-integrable lattice,
$\mu_{1}\in\mathcal{P}_{\sigma}(G_{1})$,
$\mu_{2}\in\mathcal{P}_{\sigma}(G_{2})$ Baire probability measures, and
$(V,\rho)$ a uniformly bounded strongly operator continuous Banach
$\Gamma$-module. Assume that $(\widetilde{V},\widetilde{\rho}|_{G_{1}})$ is
weakly uniquely $\mu_{1}$-stationary without almost invariant unit vectors,
that $(\widetilde{V},\widetilde{\rho}|_{G_{2}})$ is weakly uniquely
$\mu_{2}$-stationary, and that
$\widetilde{V}^{\mu_{1}}=\widetilde{V}^{G_{1}},\widetilde{V}^{\mu_{2}}=\widetilde{V}^{G_{2}}$.
Finally, let $b\in Z^{1}(\Gamma,\rho)$ be a 1-cocycle. Then there exists a
$\Gamma$-invariant closed subspace $V_{0}\subseteq V$ on which the
$\Gamma$-representation extends to a strongly operator continuous isometric
linear $G$-representation that factors through a representation of $G_{1}$.
Furthermore, there exists a 1-cocycles $b_{0}\in Z^{1}(G,V_{0})$ such that $b$
is cohomologous to $b_{0}|_{\Gamma\times\Gamma}$ in $Z^{1}(\Gamma,V)$ and such
that $b_{0}$ only depends on $G_{1}$.

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