# Definably amenable groups in continuous logic

Juan Felipe Carmona  Universidad Nacional de Colomba, Cra 45 No 30, Bogotá,
Colombia<EMAIL_ADDRESS>and  Alf Onshuus  Universidad de los Andes,
Cra 1 No 18A-10, Bogotá, Colombia<EMAIL_ADDRESS>
# Definably amenable groups in continuous logic

Juan Felipe Carmona  Universidad Nacional de Colomba, Cra 45 No 30, Bogotá,
Colombia<EMAIL_ADDRESS>and  Alf Onshuus  Universidad de los Andes,
Cra 1 No 18A-10, Bogotá, Colombia<EMAIL_ADDRESS>
###### Abstract.

We generalize the notions of definable amenability and extreme definable
amenability to continuous structures and show that the stable and ultracompact
groups are definable amenable. In addition, we characterize both notions in
terms of fixed-point properties. We prove that, for dependent theories,
definable amenability is equivalent to the existence of a good $S_{1}$ ideal.
Finally, we show the randomizations of first-order definable amenable groups
are extreme definably amenable.

##  1\. Introduction

The notion of amenability has been an important tool in model theory. Starting
with the work of [HPP08] for abelian groups definable in an o-minimal
expansion of a field, it was used to understand the quotient of such groups
$G$ by their smallest type-definable subgroup of bounded index $G^{00}$.
Afterwards, it has been central in understanding topological properties of
definable groups in several contexts. In this paper we propose and study a
notion of amenability in the context of continuous logic.

Amenability was introduced first in connection with the Banach-Tarski paradox
and paradoxical decompositions. This notion was generalized to locally compact
topological groups (the discrete case coinciding with the original definition)
and it is defined by any of the following equivalent conditions (see [Pat00]).

######  Fact 1.1.

Let $G$ be a topological locally compact group. Then the following conditions
are equivalent.

  1. (1)

There is a finitely additive probability left invariant measure $\mu$ on
$\mathcal{B}(G)$, the set of Borel subsets of $G$ which is absolutely
continuous with respect to the Haar measure.

  2. (2)

There is a left invariant mean $m$ on $C_{B}(G)$, the set of real-valued
bounded continuous functions on $G$.

  3. (3)

Any separately continuous and affine action of $G$ on a convex compact set $X$
has a fixed point.

  4. (4)

Any separately continuous action of $G$ on a compact set $X$ admits an
invariant probability measure $\mu$ on $X$.

  5. (5)

There is a left invariant mean $\mathfrak{m}$ on $RUCB(G)$, the set of real-
valued bounded uniformly continuous functions on $G$.

Here we say that a mean $\mathfrak{m}$ is invariant if for any function $f$ in
the domain of $\mathfrak{m}$ we have $\mathfrak{m}(f)=\mathfrak{m}(_{g}f)$
where ${}_{g}f(x)=f(g^{-1}x)$, and a finitely additive measure $\mu$ is left
invariant if $\mu(A)=\mu(gA)$ for any $A$ in the domain of $\mu$.

Conditions (3) and (4) are always equivalent, condition (3) is known as the
Fixed Point Property, and are key when studying amenable locally compact
topological groups. Condition (3) gives rise to the following definition:

######  Definition 1.2.

A topological group $G$ is _extremely amenable_ if every separately continuous
action of $G$ over a compact set $X$ has a fixed point.

In the context of non locally-compact groups, these conditions are not
equivalent any more: The group $U(\ell^{2}(\mathbb{R}))$, the unitary group of
the space $\ell^{2}(\mathbb{R})$ with the strong operator topology is
extremely amenable ([GM83]) but it does not satisfy condition (2). It does
however, satisfy condition (5) (also (3) and (4) since they are direct
consequences of extreme amenability). These and other examples indicate that
conditions (3), (4), and (5) provide the correct generalization of amenability
for non-locally compact topological groups. Thus, in the literature, one has
the following definitions for general topological groups:

######  Definition 1.3.

Let $G$ be a topological group. Then $G$ is _amenable_ if there is a left
invariant mean $\mathfrak{m}$ on $RUCB(G)$, the set of real-valued bounded
uniformly continuous functions on $G$. $G$ is _$\mathcal{B}$ -amenable_ if
there is a left invariant mean $\mathfrak{m}$ on $C_{B}(G)$, the set of real-
valued bounded continuous functions on $G$.

The universe $M$ of a structure in continuous logic is a metric space, and the
predicates are bounded uniformly continuous functions from the space to the
real numbers. The interesting examples of models of continuous logic are
usually non-locally compact and many of the definitions are based on bounded
uniformly continuous functions from the universe of the structures to the real
numbers. It therefore makes sense to suggest the following definitions:

######  Definition 1.4.

Let $G$ be a structure in continuous logic with a definable group structure
(see Section 2 for definition of definable sets in continuous logic). We say
that $G$ is _definably amenable_ if there is a mean $\mathfrak{m}$ on the set
$\mathcal{P}_{\bar{M}}^{x}|_{G}$ of unary predicates contained in $G$ and
definable with parameters $M$, such that $\mathfrak{m}(P)=\mathfrak{m}(g\cdot
P)$.

This generalizes the definition of extreme definable amenability from [BM21]
where a group is _extremely definably amenable_ if it satisfies the above
definition, with the added requirement that $\ker\mathfrak{m}$ is a complete
type, or, in simpler terms, that it has a $G$-invariant global type.

The structure of the paper is as follows. In Section 2 we recall the basic
definitions of continuous logic and show that the notions of definable
amenability and extreme definable amenability are logically sound (in the
sense that they are preserved under elementary equivalence) and that any
amenable group will be definably amenable when interpreted as a continuous
logic structure (same for extremely amenable groups). In section 3 we recall
some important properties about the space of means in continuous logic and
give characterizations of (extreme) definable amenability in terms of fixed-
point properties. This generalizes the authors work from [CDOZ21] to the
continuous setting.

Section 4 starts the study of the relation between amenable groups and the
structure of $G/G^{00}$. This is inspired by first order equivalences proved
in [HPP08] and [CS15], we characterize definably amenable dependent groups in
terms of the existence of an f-generic type, modulo the existence of a wide-
type, (whose existence in the classical case is trivial, but we ignore if it
always exists in the continuous setting); the extremely definable case,
however, is very well understood.

In Section 6 we prove that any randomization of a first order amenable group
is extremely definably amenable as a continuous logic structure, generalizing
the results of Berenstein and Muñoz in the dependent setting. Thus, providing
a body of examples of extremely definably amenable groups outside dependent
theories. Finally, in Section 5 we prove that any bounded stable group is
definably amenable.

We hope this work starts interesting directions for future work. On the one
hand, we know that all the pseudocompact groups are definably amenable groups
but we do not have any examples (if any) of such groups that are extremely
definably amenable. This question is interesting even in the First-order
setting for the case of pseudofinite groups. Another interesting question is
the existence of wide-types given an ideal S. The existence of such types will
prove the existence of $f$-generic types in definably amenable dependent
groups, which would give a characterization of definable amenability in the
continuous dependent logic context, very much in line with what is known in
the first order case.

##  2\. Preliminaries, notation and first results

First, we summarize the notation we will use throughout the paper and recall
some key theorems from continuous logic. For a more detailed exposition the
reader may check [YBHU08].

Note: in this paper we are not assuming that the truth values are in the
interval $[0,1]$; instead, every formula may have its own (bounded) range in
the reals. This does not make any difference in the theory since we can always
normalize the formulas and their semantic remains intact, but it allows us to
treat the set of formulae and predicates as a vector space, which facilitates
the translation between logic and functional analysis.

A language $L$ is a triplet $(\mathcal{S},F,R)$ which contains the following
data:

  1. (1)

$\mathcal{S}$ is the set of sorts; for each sort $S\in\mathcal{S}$, there is a
symbol $d_{S}$ meant to be interpreted as a metric bounded by a positive
number $M_{S}$.

  2. (2)

$\mathcal{F}$ is the set of function symbols; for each function symbol $f\in
F$, we formally specify $\mathop{dom}(f)$ as a sequence $(S_{1},...,S_{n})$
from $\mathcal{S}$ and $rng(f)=S$ for some $S\in\mathcal{S}$. We will want $f$
to be interpreted as a uniformly continuous function. To this effect we
additionally specify, as part of the language, functions
$\delta^{f}_{i}:\mathbb{R}^{+}\to\mathbb{R}^{+}$, for $i\leq n$. These
functions are called uniform continuity moduli.

  3. (3)

$\mathcal{R}$ is the set of relation symbols; for each relation symbol
$R\in\mathcal{R}$ we formally specify $\mathop{dom}(R)$ as a sequence
$(S_{1},...,S_{n})$ from $S$ and $rng(R)=K_{R}$ for some bounded interval
$K_{R}inR$. As with function symbols, we additionally specify, as part of the
language, functions $\delta^{R}_{i}\mathbb{R}^{+}\to\mathbb{R}^{+}$, for
$i\leq n$, called uniform continuity moduli.

For each sort $S\in\mathcal{S}$, we have infinitely many variables $x^{S}_{i}$
Terms are defined inductively in the same way as in first order:

  1. (1)

A variable $x^{S}_{i}$ is a term with domain and range $S$

  2. (2)

If $f\in\mathcal{F}$, $dom(f)=(S_{1},...,S_{n})$ and $\tau_{1},...,\tau_{n}$
are terms with $rng(\tau_{i})=S_{i}$ then $f(\tau_{1},...,\tau_{n}$) is a term
with range the same as $f$ and domain determined by the $\tau_{i}$’s.

Formulas are also defined inductively, but the definition of connectives and
quantifiers require more attention:

  1. (1)

If $R$ is a relation symbol (possibly a metric symbol) with domain
$(S_{1},...,S_{n})$ and $\tau_{1},...,\tau_{n}$ are terms with ranges
$S_{1},...,S_{n}$ respectively then $R(\tau_{1},...,\tau_{n})$ is a formula.
Both the domain and uniform continuity moduli of $R(\tau_{1},...,\tau_{n})$
can be determined naturally from $R$ and $\tau_{1},...,\tau_{n}$. These are
the atomic formulas.

  2. (2)

(Connectives) If $f:\mathbb{R}^{n}\to\mathbb{R}$ is a uniformly continuous
function and $\varphi_{1},...,\varphi_{n}$ are formulas then
$f(\varphi_{1},...,\varphi_{n})$ is a formula. The domain and uniform
continuity moduli are determined from $f$ and $\varphi_{1},...,\varphi_{n}$.

  3. (3)

(Quantifiers) If $\varphi$ is a formula and $x$ is a variable of sort $S$,
then both $\inf_{x\in S}\varphi$ and $\sup_{x\in S}\varphi$ are formulas.

Given a language $L=(\mathcal{S},\mathcal{F},\mathcal{R})$, a metric structure
will be a collection of metric spaces, one for each sort $S$, with a metric
$d^{M}_{S}$ of diameter at most $M_{S}$. Together with a collection of
functions $F^{M}_{i}$ and relations $R^{M}_{i}$ interpreted in the natural
way.

The set of formulas with free variables in $\bar{x}$ in the language
$\mathcal{L}$ is denoted by $\mathfrak{F}_{\mathcal{L}}^{\bar{x}}$. The set of
sentences $\mathfrak{F}^{()}_{\mathcal{L}}$ is denoted by
$Sent_{\mathcal{L}}$.

When interpreted in a metric structure $M$, formulas become bounded uniformly
continuous functions $\varphi^{M}(\bar{x}):M^{n}\to\mathbb{R}$. If the formula
$\varphi$ is a sentence, then $\varphi^{M}$ is just a real number.

A _theory_ $T$ is a set of sentences. The _theory_ of a metric structure $M$
is the set of sentences $\varphi$ such that $\varphi^{M}=0$. We say that
$M\models T$ if $T\subset\mathop{Th}(M)$.

Given a theory $T$, we may endow $\mathfrak{F}_{\mathcal{L}}^{\bar{x}}$ with
the pseudometric

$|\varphi(\bar{x})|:=\sup\\{|\varphi(\bar{a})|:\bar{a}\in M^{n},M\models
T\\}.$

The quotient of $\mathfrak{F}_{\mathcal{L}}^{\bar{x}}$ is a normed vector
space whose completion is denoted as $\mathcal{P}^{\bar{x}}_{T}$, the set of
_definable predicates_. In other words, a function $P(\bar{x})$ is a definable
predicate if there is a sequence of formulas $\varphi(\bar{x})$ that converges
uniformly to $P(\bar{x})$ in every model $M\models T$.

We will denote as $\mathcal{P}_{T(A)}^{\bar{x}}$ the set of predicates with
parameters in the set $A$. If the theory $T$ is clear from the context, we
simply write $\mathcal{P}_{A}^{\bar{x}}$.

######  Definition 2.1.

A closed set $D$ of $M^{n}$ is a definable set if the relation
$dist(\bar{x},D)$ is a definable predicate (which we will denote simply by
$D(\bar{x})$).

######  Definition 2.2.

Let $D_{1}\subset M^{n}$ and $D_{2}\subset M^{k}$ be definable sets. A
function $f:D_{1}\to D_{2}$ is _definable_ if there is a definable predicate
$P:M^{n+k}\to[0,1]$ such that, for $(x,y)\in D_{1}\times D_{2}$, we have that
$P(x,y)=d(f(x),y)$.

######  Remark 2.3 (See [YBHU08]).

Let $D\subset M$ be a closed set. The following are equivalent:

  * •

$D$ is a definable set.

  * •

There exists a definable predicate $P(x)$ such that $D$ is the zero-set of
$P(x)$ and for every $\epsilon>0$ there exists $\delta>0$ such that, if
$P(x)<\delta$ then $d(x,D)<\epsilon$.

  * •

For every formula $\psi(x,y)$, the functions

$\sup_{x\in D}(\psi(x,y)\mbox{ and }\inf_{x\in D}\psi(x,y)$

are definable predicates.

  * •

There exists a definable predicate $P(x)$ such that $D$ is the zero-set of
$P(x)$ and for every ultraproduct $M_{\mathcal{U}}=\prod_{\mathcal{U}}M_{i}$
of models of $Th(M)$, we have that

$P(M_{\mathcal{U}})=\prod_{\mathcal{U}}P(M_{i}).$

A _partial type_ $\pi$ (over a set $A$) is any consistent set of definable
predicates. If the set is maximal, we say that the type is _complete_. In this
case, $p$ is the kernel of a functional
$\hat{p}:\mathcal{P}^{\bar{x}}\to\mathbb{R}$ such that, for some model $M$,
there exists a tuple $\bar{a}$ in $M^{|\bar{x}|}$ such that $\hat{p}(P)=P(a)$.

The following theorem probably is too obvious to have been stated anywhere
else, so we write down the proof here:

######  Theorem 2.4.

A functional $\hat{p}:\mathcal{P}^{\bar{x}}\to\mathbb{R}$ is Banach-algebra
homomorphism if and only if its kernel $p$ is a complete type.

###### Proof.

($\Rightarrow$). Clear.

($\Leftarrow$) We need to show that the kernel of $\hat{p}$ is finitely
consistent. First of all, notice that if a predicate $\varphi\geq 0$ satisfies
$\varphi(x)>0$ in every model $M$, then $\varphi(x)\geq\epsilon$ for some
$\epsilon>0$. Whence $\hat{p}(\varphi)\geq\epsilon$. Therefore, if
$\varphi\geq 0$ and $\hat{p}(\varphi)=0$, then there exists $a$ such that
$\varphi(a)=0$. Now, take $\varphi_{1},...,\varphi_{n}\in ker(\hat{p})$. Since
$\hat{p}$ is an homomorphism, then

$(\hat{p})(\sum_{i=1}^{n}\varphi_{i}^{2})=(\sum_{i=1}^{n}\hat{p}(\varphi))^{2}-2\sum_{i\neq
j}\hat{p}(\varphi_{i})\hat{p}(\varphi_{j})=0.$

Therefore, $\sum_{i=1}^{n}\varphi_{i}^{2}$ has a realization $a$. In
particular, $\varphi_{i}(a)=0$ for all $i$.

∎

######  Theorem 2.5 (Compactness [YBHU08]).

The space of types $S^{\bar{x}}$ is compact with the weak∗ topology induced
from the dual of $\mathcal{P}^{\bar{x}}$.

Analogous notions of amenability and extremely amenability for definable
groups and definable theories in first-order logic have been studied widely in
[HPP08] and [HKP19]. In this paper, we propose such notions for the metric
case.

Let us recall that a _definable group_ $(G,\cdot)$ in a structure $M$ is a
group such that the set $G\subset M^{n}$ is definable and the multiplication
is a definable function $\cdot:G\times G\to G$ (so there is a definable
predicate $Q(x,y,z)$ such that $Q(g_{1},g_{2},h)=d(g_{1}\cdot g_{2},h)$ for
every $g_{1},g_{2},h\in G$).

If $P(\bar{x})$ is any definable predicate contained in $G(\bar{x})$, then we
may define a new predicate $g\cdot P(x):=\inf_{y}(P(y)+Q(g^{-1},x,y))$. Notice
that if $x\in G$, then $g\cdot P(x)=P(g^{-1}\cdot x)$.

If $G$ is a definable group and $p$ is a complete type in $\mathcal{L}(M)$
such that $G(x)\in p$, then, for each $h\in G$, we define $q=h\cdot p$ to be
the type of $h\cdot g$, where $g$ is any realization of $p$.

######  Definition 2.6.

A functional $\mathfrak{m}$ on $\mathcal{P}_{M}^{\bar{x}}$ is called a _mean_
if $\mathfrak{m}(P)\geq 0$ when $P\geq 0$ and $\mathfrak{m}(\textbf{1})=1$,
where 1 is the constant function of value $1$.

Notice that, for any type $p$, the function $\hat{p}$ is a mean.

######  Definition 2.7.

A mean over $\mathcal{P}_{\bar{M}}^{\bar{x}}$ is _M-definable_ if it is
M-invariant and for every $M$-formula $\varphi(\bar{x},\bar{y})$, and
$r,s\in\mathbb{R}$, the set

$\\{q\in S_{y}(M):r<\mathfrak{m}(\varphi(\bar{x};\bar{b}))<s\mbox{ for any
}b\in\bar{M};b\models q\\}$

is an open subset of $S_{y}(M)$.

######  Remark 2.8.

A complete type $p$ is definable over $M$ (as a mean) if for each $M$-formula
$\varphi(\bar{x},\bar{y})$ there exists a predicate $P(\bar{y})$ definable
over $M$ such that for all suitable tuples b in M we have that $P(b)=r$ if and
only if $\hat{p}(\varphi(\bar{x},\bar{b}))=r$.

Quoting known results of continuous logic, we can now proof some easy facts of
our notions of amenability and extreme amenability. This provide evidence that
they are the correct ones in this context.

######  Remark 2.9.

If a metric group $G$ is (extremely) amenable, then it is (extremely)
definably amenable

###### Proof.

If $G$ is amenable, then there is a mean on $RUCB(C)$. Since the formulas are
all bounded right uniformly continuous, we have that $G$ is definably
amenable.

If $G$ is extremely amenable, then its action on the space of types $S(G)$ has
a fixed point, this is an invariant type, hence $G$ is definably extremely
amenable. ∎

######  Theorem 2.10.

Let $\\{M_{i}\\}_{i\in I}$ be a family of metric structures elementary
equivalent and let $G$ be a definable group. If $G(M_{i})$ is (extremely)
definably amenable for every $i$, then $G(\prod_{\mathcal{U}}M_{i})$ is
(extremely) definably amenable.

###### Proof.

Let $\mathfrak{m}_{i}$ be an invariant mean (type) on $G(M_{i})$. Then, it is
easy to check that the function $\hat{\mathfrak{m}}$, defined as

$\hat{\mathfrak{m}}(\varphi([x],[a_{i}]))=\lim_{\mathcal{U}}(\mathfrak{m}(\varphi(x,a_{i})))$

is an invariant mean (type) on $G(\prod_{\mathcal{U}}M_{i})$. ∎

######  Corollary 2.11.

If $M\prec N$ and $G$ is a definable group, then $G(M)$ is (extremely)
definably amenable if and only if $G(N)$ is.

###### Proof.

If $G(M)$ is (extremely) definably amenable, then by Keisler-Shelah theorem,
there exists $\prod_{\mathcal{U}}M$ an ultrapower of $M$ such that
$N\prec\prod_{\mathcal{U}}M$. By the previous theorem,
$G(\prod_{\mathcal{U}}M)$ is (extremely) definably amenable. ∎

##  3\. The space of means and fixed point theorems

In order to prove the existence of invariant means, it is useful to look at
the space of means in general and take advantage of its topological
properties.

As before, $\mathcal{P}^{x}$ denotes the set of unary definable predicates,
which is a normed space with the norm given by

$|\varphi|=\sup_{x\in M}|\varphi(x)|.$

Let us denote by $\Sigma(M)$ the space of means over $M$ and, as usual,
$S_{M}(x)$ denotes the space of complete types over $M$. By the previous
remark, we know that $S_{M}(x)\subset\Sigma(M)$.

By Alaouglu’s theorem, the unit ball of the dual set of
$\mathcal{P}_{\mathcal{L}}^{x}$ is compact with the weak∗-topology. Moreover,
the $\Sigma(M)$ is closed in the unit ball: it is the preimage of $1$ of the
(continuous) function $v^{\prime}\to v^{\prime}(\mathbf{1})$, whence it is
compact as well. Finally, notice that both the $S_{M}(x)$ and $\Sigma(M)$ are
closed as well, therefore compact.

Clearly, the set $\Sigma(M)$ is convex, hence the closure of the convex hull
of $S_{M}(x)$ is contained in $\Sigma(M)$. We will show that the converse is
also true. This is a direct corollary of the following theorem:

######  Theorem 3.1 (Phelps [Phe63]).

Let $A$ and $B$ be algebras of real valued functions on the sets $X$ and $Y$
respectively and suppose that $1\in A$. Let $K_{0}^{\prime}(A,B)$ be the
convex set of all linear operators $T$ from $A$ to $B$ which satisfy $T\geq 0$
and $T(1)\leq 1$. Then $T$ is an extreme point of $K_{0}^{\prime}(A,B)$ if and
only if $T$ is multiplicative.

######  Theorem 3.2.

The closure of the convex hull of the $S_{M}(x)$ is $\Sigma(M)$.

###### Proof.

Clearly, $\Sigma(M)$ is convex and compact. Now, the extreme points of
$\Sigma(M)$ are precisely $S_{M}(x)$ by the previous theorem and Remark 2.4;
since $(\mathcal{P}_{\mathcal{L}}^{x})^{\prime}$ is locally convex, then by
Choquet’s theorem [CM63], the closure of the convex hull of $S_{M}(x)$ is
$\Sigma(M)$. ∎

######  Definition 3.3.

  * •

Let $D$ be a definable set in a structure $M$ and $K$ be a compact Hausdorff
space. We say that $f:D\to K$ is _definable_ if, for every $C\subset U\subset
K$, with $C$ closed and $U$ open, there exists a definable predicate $P(x)$
and $\epsilon>0$ such that

$f^{-1}(C)\subset P(x)=0\subset P(x)<\epsilon\subset f^{-1}(U).$

  * •

A _definable action_ of $G$ on $X$ (or _definable $G$-flow_) is an action
$G\times K\to K$ such that

    * –

For every $g\in G$, $x\to gx$ is an homeomorphism.

    * –

The map from $G$ to the Roelcke compactification of $Homeo(X)$, where
$Homeo(X)$ has the compact open topology, is definable.

If all the types based on $G$ are definable, then the action of $G$ onto its
space of types is definable. In general, this hypothesis is too strong because
a theory $T$ is stable if and only if all its types over all models are
definable. Since we are not assuming stability here, we propose a weaker
definition of definable action:

######  Definition 3.4.

A _weak definable action_ of $G$ on a compact space $K$ (or _weak definable
$G$-flow_) is an action $G\times K\to K$ such that

  * •

For every $g\in G$, $x\to gx$ is an homeomorphism.

  * •

For some $k_{0}\in K$, $g\to k_{0}x$ is definable.

In this case, the triple $(K,G,k_{0})$ is called a _weak definable ambit_.

It is easy to check that the natural action of $G$ on its space of types makes
$(S(G),G,tp(e))$ a weak definable ambit. Moreover, the action of $G$ on its
space of means makes $(\Sigma(G),G,\hat{tp(e)})$ a weak definable ambit as
well. If all the types (respectively means) are definable, then
$(S(G),G,tp(e))$ ($(\Sigma(G),G,\hat{tp(e)})$ respectively) is a definable
ambit.

The following result has been proved for the first-order case (when all the
types are definable [[GPP14]]), but the proof can be easily transferable to
this context:

######  Fact 3.5.

The weak definable ambit $(S(G),G,tp(e))$ is universal in the category of weak
definable ambits. In particular, if all the types are definable, then
$(S(G),G,tp(e))$ is universal in the category of definable ambits.

As a corollary, we get:

######  Theorem 3.6.

Let $G$ be a definable group. Then

  * •

$G$ is extremely definably amenable if and only if every weak definable action
over a compact set has a fixed point.

Moreover, if all the types in $S(G)$ are definable, then:

  * •

$G$ is extremely definably amenable if and only if every definable action over
a compact set has a fixed point.

Using Theorem 3.2 we get the following corollaries:

######  Theorem 3.7.

The weak definable ambit $(\Sigma(G),G,\hat{tp(e)})$ is universal in the
category of weak definable ambits $(K,G,k_{0})$, where $K$ is a compact convex
set of a locally convex vector space.

######  Theorem 3.8.

Let $G$ be a definable group.

  * •

$G$ is definably amenable if and only if every weak definable action over a
compact convex subset on a locally convex space has a fixed point.

Moreover, if all the means in $\Sigma(G)$ are definable, then:

  * •

$G$ is definably amenable if and only if every definable action over a compact
convex subset on a locally convex space has a fixed point.

##  4\. Amenability, S1 ideals, f-generic types and $G^{00}$

This section is an attempt to recover some of the results in [HPP08] and
[CS15] relating (in the dependent first-order context) amenability with the
structure of $G/G^{00}$ in terms of stabilizers of $f$-generic types.

We will start with some definitions.

######  Definition 4.1.

A type $\pi(x;B_{0})$ _divides_ over $C$ if there exists an indiscernible
sequence $\\{B_{i}\\}_{i<\omega}$ such that the $\bigcup\pi(x;B_{i})$ is not
satisfiable.

We say that $\pi(x,b_{0})$ _forks_ over $C$ if there exists
$D\supset\\{b_{0}\\}\cup C$ such that every complete type $p$ over $D$, with
$p\supset\pi(x,b_{0})$, divides over $C$.

Let $T$ denote a complete continuous theory and $\bar{M}$ a
$\kappa$-saturated, $\kappa$-strongly homogeneous model of $T$, for $\kappa$
large enough.

######  Definition 4.2.

A formula $\varphi(x,y)$ has the independence property (IP) if there are
$r<s$, sequences $\\{a_{i}\\}_{i<\omega}$ and $\\{b_{I}\\}_{I\subset\omega}$
in $\bar{M}$ such that $\varphi(a_{i},b_{I})<r$ for $i\in I$ and
$\varphi(a_{i},b_{I})>s$ for $i\notin I$. A theory $T$ is _dependent_ if no
formula has the independence property.

Let $G$ be a definable group, by $G^{00}_{A}$ we denote the intersection of
all type-definable subgroups over $A$ of bounded index. If $G^{00}_{A}$ does
not depend on $A$ we say that $G^{00}$ exists and it is equal to $G^{00}_{A}$.
In the same way, we define $G^{\infty}_{A}$ as the smallest $A$-invariant
subgroup of bounded index. If the group does not depend on $A$, then we say
that $G^{\infty}$ exists (and it is equal to $G^{\infty}_{A}$).

The existence of $G^{00}$ in continuous dependent theories can be proved
analogously to the classic case (see [BM21], Theorem 2.19).

If the group is extremely definably amenable, we know that $G^{00}=G$,
however, for the definably amenable case, things get much more complicated.

We start with the following theorem. The proof is analogous to the classic one
[Sim15].

######  Theorem 4.3.

Let $T$ be a dependent theory. A global type $p\in S(M)$ is $A$-invariant if
and only if $p$ does not fork over $A$.

###### Proof.

Since $p$ is a global type, not forking over $A$ is equivalent to not dividing
over $A$. Hence every $A$-invariant type does not fork over $A$ (this
direction does not require dependent). Now, assume that $p$ is not
$A$-invariant. Hence

$r=\hat{p}(\varphi(x,a_{0}))\neq\hat{p}(\varphi(x,a_{1})=s$

for some $A$-indiscernible sequence $(a_{i})_{i<\omega}$. By $dependence$,
there does not exist any element $b$ such that $\varphi(b,a_{2n})=r$ for and
$\varphi(b,a_{2n+1})=s$, whence, the formula
$(\varphi(x,a_{0})-r)(\varphi(x,a_{1})-s)$ in $p$ divides. ∎

These properties are not known outside the dependent context. It is for this
reason that we will initially work under the assumption of dependence. We will
also need the notion of $f$-generic types, defined as in [HPP08]:

######  Definition 4.4.

A global type $p$ is _$f$ -generic_ if there is a small model $M_{0}$ such
that neither $p$ nor its translates (under the group action) fork over
$M_{0}$.

The following fact is well known in the first-order setting, but the proof can
be adapted almost verbatim to the continuous setting:

######  Fact 4.5 (Corollary 8.20 in [Sim15]).

If $G$ admits an f-generic type over some model $M$, then it admits an
f-generic type over any model $M_{0}$.

###  4.1. Stabilizers and amenability in dependent theories

In this section, we will assume that $G$ is definable in a structure $M$ and
$Th(M)$ is dependent.

######  Theorem 4.6.

If $G$ is a definable group with an $f$-generic type $p$, then:

  1. (1)

$stab(p)=G^{00}=G^{000}$.

  2. (2)

$G$ is definably amenable.

###### Proof.

Let $p$ be the $f$-generic type over $M_{0}$. First we will show that
$stab(p)$ is a type-definable group. More precisely, it is defined by

$stab(p)=\\{g_{1}^{-1}g_{2}|\mathop{tp}(g_{1}/M_{0})=\mathop{tp}(g_{2}/M_{0})\\}:$

Since $gp$ is $Aut_{M_{0}}(M)$ invariant, then, for
$\mathop{tp}(g_{1}/M_{0})=\mathop{tp}(g_{2}/M_{0})$, we have that
$g_{1}p=f(g_{1}p)=f(g_{1})p=g_{2}p$, for any $f\in Aut_{M_{0}}(M)$. Whence
$g_{1}^{-1}g_{2}p=p$.

On the other hand, if $h\in stab(p)$, then $ha=b$ for some realizations of $p$
outside $M$. This implies that $ha^{\prime}=b^{\prime}$ for some realizations
of $p|M_{0}$ inside $M$. By naming $g_{1}=b^{-1}$ and $g_{2}=a^{-1}$ we get
the desired result.

Now, $stab(p)$ is a type-definable group and its index is bounded by

$|S_{M_{0}}(G)|<2^{|M_{0}|}<\bar{\kappa}.$

This implies that $stab(p)$ contains $G^{00}$. On the other hand, since

$\\{g_{1}^{-1}g_{2}|tp(g_{1}/M_{0})=tp(g_{2}/M_{0})\\}$

generates a bounded group, then $G^{\infty}$. Whence

$stab(p)\subset G^{\infty}\subset G^{00}\subset stab(p).$

In order to prove that $G$ is definably amenable, let us assume first that $T$
is a countable theory over a countable language and let
$\varphi:M\to\mathbb{R}$ be any formula defined over a model $M_{0}$ of
countable density character. Assume that $p$ is f-generic over $M_{0}$ as
well.

Let us define $f_{\varphi}:G/G^{00}\to\mathbb{R}$ as follows:

$f_{\varphi}([g])=\hat{p}(g\varphi).$

Notice that $f_{\varphi}$ is well-defined since $p$ is $G^{00}$ invariant.

Claim 1. $f_{\varphi}$ is measurable:

###### Proof.

We want to check that $\\{[g]\in G/G^{00}:r<\hat{p}(g\varphi(x))<s\\}$ is
Borel in $G/G^{00}$. It suffices to show that $\\{g\in
G:\hat{p}(g\varphi(x))\in(r,s)\\}$ is a countable union of closed sets of $G$
(because the projection of $S(G)$ onto $G/G^{00}$ is closed).

If $(a_{i})_{i<\omega}\models p^{\omega}_{M_{0}}$, then
$\lim_{i\to\infty}\psi(a_{i})=\hat{p}(\psi(x))$. Thus
$\hat{p}(\psi(x))\in(r,s)$ if and only if

(*) There exists a rational number $q\in(r,s)$ such that, for

$\epsilon_{q}=\frac{1}{2}\min\\{|q-r|,|q-s|\\}$

there exists $N<\omega$ such that $|\psi(a_{N})-q|<\epsilon_{q}$ and
$|\psi(a_{i})-\psi(a_{j})|<\epsilon_{q}$ for $i,j>N$.

Fix an element $g\in G$. By dependence, for every $\epsilon$ there is a
maximal $N$ such that $(a_{i})_{i<N}\models p^{N}_{M_{0}}$ and
$|g\varphi(a_{i})-g\varphi(a_{i+1})|\geq\epsilon$ for $i<N$.

Let

$\Phi^{q}_{N}(g)=\Big{\\{}\inf\limits_{x_{1},...,x_{N}}|\varphi_{i}(x_{1},...,x_{n})|\cdot\left[\prod_{i<N}\left(\epsilon_{q}\mathbin{\vphantom{+}\text{\ooalign{\kern-1.50696pt\cr$\smash{\cdot}$\cr\kern
1.50696pt\cr$-$\cr}}}|g\varphi(x_{i})-g\varphi(x_{i+1})|\right)\right]\cdot$

$|g\varphi(x_{N})-q|\mathbin{\vphantom{+}\text{\ooalign{\kern-1.50696pt\cr$\smash{\cdot}$\cr\kern
1.50696pt\cr$-$\cr}}}\epsilon_{q}:\varphi_{i}\in p\Big{\\}}.$

(This type says that there are $N$ realizations of the type $p^{N}_{M_{0}}$
satisfying (*)).

Hence,

$f_{\varphi}(g)=\hat{p}(g\varphi)\in(r,s)\Longleftrightarrow
g\in\bigcup_{q\in\mathbb{Q}\cap(r,s)}(\bigcup_{N<\omega}(\Phi^{q}_{N}(g)\cap\neg(\Phi^{q}_{N+1}(g))).$

Finally, since $L$ is countable and $M_{0}$ has a countable dense set, we may
assume that $(\Phi^{q}_{N+1}(g))$ is countable. Thus, $\\{g\in
G:\hat{p}(g\varphi(x))\in(r,s)\\}$ is a countable union of closed sets and
$\\{[g]\in G/G^{00}:r<\hat{p}(g\varphi(x))<s\\}$ is Borel in $G/G^{00}$.
Therefore $f_{\varphi}$ is measurable.

∎

Claim 2. $f_{g_{0}\varphi}=g_{0}\cdot f_{\varphi}$.

###### Proof.

$f_{g_{0}\varphi}([g])=p(gg_{0}\varphi)=f_{\varphi}([g]g_{0})=g_{0}\cdot
f_{\varphi}([g])$. ∎

Claim 3. $f_{\varphi+\psi}=f_{\varphi}+f_{\psi}$.

###### Proof.

$f_{\varphi+\psi}[g]=p(g(\varphi+\psi))=p(g\varphi+g\psi)$

$=p(g(\varphi)+p(g\psi))=(f_{\varphi}+f_{\psi})[g]$. ∎

Let us define $\mathfrak{m}$ as

$\mathfrak{m}(\varphi)=\int f_{\varphi}.$

Claim 4: $\mathfrak{m}$ is a $G$-invariant functional
$\mathfrak{m}:Form\to\mathbb{R}$ and $\mathfrak{m}(1)=1$.

###### Proof.

  * •

$\mathfrak{m}$ is $G$-invariant: $\mathfrak{m}(g\varphi)=\int
f_{g\varphi}=\int g\cdot f_{\varphi}=\int f_{\varphi}$.

  * •

$\mathfrak{m}$ is a functional:
$\mathfrak{m}(\varphi+\psi)=\int(f_{\varphi+\psi})=\int
f_{\varphi}+f_{\psi}=\int f_{\varphi}+\int f_{\psi}$.

  * •

$\mathfrak{m}(1)=1$: $f_{1}[g]=p(g\cdot 1)=p(1)=1$, hence
$\mathfrak{m}(1)=\int f_{1}=\int 1=1$.

∎

Therefore, we conclude that if $G$ has an $f$-generic type, then it is
definably amenable. ∎

###  4.2. Towards proving the existence of an $f$-generic types in definably
amenable groups.

In this section we prove a partial converse to Theorem 4.6. So we assume
throughout this subsection that $G$ is a definably amenable group with mean
$\mathfrak{m}$ and all the predicates live in $G$. For the results we do
manage to prove, we don’t need any further hypothesis on the theory. We will
first show the existence of an $S1$ ideal of _small positive predicates_. We
then show that if a type contains only formulas outside the ideal, then it is
$f$-generic. So the only missing piece is to show that in any definably
amenable group, there are types which avoid all small positive predicates.

######  Definition 4.7 (Classical First-Order).

An ideal $I$ of formulas is $S_{1}$ if for any formula $\varphi(x,y)$ and
indiscernible $(a_{i})_{i\in\mathbb{N}}$, if
$\varphi(x,a_{i})\cap\varphi(x,a_{j})\in I$ for $i\neq j$, then some
$\varphi(x,a_{i})\in I$.

In the classical first-order setting, it can be shown that if $G$ is definably
amenable, then the set of small formulas is $S1$. In this context, by a small
formula we mean a formula of measure 0.

We will prove an analogue of this fact in the continous setting, however, the
notion of small needs to be adjusted: intuitively, we want to define a
predicate $P(x)$ to be small if its set of zeros is small. Let us keep this
assumption for now and assume that $P(x)\geq 0$, since, for almost all $x$,
$P(x)>0$, we may find some $\lambda$ big enough such that $\lambda P(x)>1$
almost everywhere. Hence,
$\mathfrak{m}(1\mathbin{\vphantom{+}\text{\ooalign{\kern-1.50696pt\cr$\smash{\cdot}$\cr\kern
1.50696pt\cr$-$\cr}}}\lambda P(x)\to 0$ as $\lambda\to\infty$. Since in
practice we do not have any measure on the subsets of the model, we adopt this
as a definition.

From now on, we will be working almost exclusively with positive predicates
with parameters.

######  Definition 4.8.

Let $P(x)$ be a positive predicate. We define $\overline{P}(x)$ as
$1\mathbin{\vphantom{+}\text{\ooalign{\kern-1.50696pt\cr$\smash{\cdot}$\cr\kern
1.50696pt\cr$-$\cr}}}(1\mathbin{\vphantom{+}\text{\ooalign{\kern-1.50696pt\cr$\smash{\cdot}$\cr\kern
1.50696pt\cr$-$\cr}}}P(x))$. This is, $\overline{P}(x)=P(x)$ if $P(x)<1$,
otherwise $\overline{P}(x)=1$.

######  Remark 4.9.

If $P$ and $Q$ are positive predicates, then
$\overline{P}\,\overline{Q}\leq\overline{PQ}$ and

$\frac{1}{2}(\overline{P}+\overline{Q})\leq\overline{P+Q}\leq\overline{P}+\overline{Q}.$

###### Proof.

Since $\overline{P}\leq P$ and $\overline{Q}\leq Q$, then
$\overline{P}\,\overline{Q}\leq PQ$, so
$\overline{P}\,\overline{Q}\leq\overline{PQ}$. The other property is also easy
to check. ∎

######  Definition 4.10.

Let us denote $\lim\limits_{\lambda\to\infty}\mathfrak{m}(\overline{\lambda
P})$ as $\mathfrak{m}_{\infty}(P)$. We say that $P$ is _small_ if it is a
positive predicate and $\mathfrak{m}_{\infty}(P)=1$.

######  Lemma 4.11.

If $P$ and $Q$ are small, then $P+Q$ and $PQ$ are small.

###### Proof.

By linearity of $\mathfrak{m}$ and Remark 4.9, we have that $P+Q$ is small.
For $PQ$, let $\epsilon>0$, take $\lambda_{1}$ such that
$1-\mathfrak{m}(\overline{\lambda_{1}P})<\epsilon$ and
$1-\mathfrak{m}(\overline{\lambda_{2}Q})<\epsilon$. Notice that
$\overline{\lambda_{1}P}\,\overline{\lambda_{2}Q}\leq\overline{\lambda_{1}\lambda_{2}PQ}$,
therefore:

$\displaystyle 1-\mathfrak{m}(\overline{\lambda_{1}\lambda_{2}PQ})$
$\displaystyle=1+\mathfrak{m}(-\overline{\lambda_{1}\lambda_{2}PQ})$
$\displaystyle=1+\mathfrak{m}(\overline{\lambda_{1}P}-\overline{\lambda_{1}P}-\overline{\lambda_{1}\lambda_{2}PQ})$
$\displaystyle\leq
1+\mathfrak{m}(\overline{\lambda_{1}P}-\overline{\lambda_{1}P}-\overline{\lambda_{1}P}\,\overline{\lambda_{2}Q})$
$\displaystyle=1-\mathfrak{m}(\overline{\lambda_{1}P})+\mathfrak{m}(\overline{\lambda_{1}P}(1-\overline{\lambda_{2}Q}))$
$\displaystyle\leq
1-\mathfrak{m}(\overline{\lambda_{1}P})+\mathfrak{m}(1-\overline{\lambda_{2}Q})$
$\displaystyle\leq\epsilon+\epsilon$

∎

######  Lemma 4.12.

For every positive predicates $P$ and $Q$ we have

$\overline{P\,Q}\leq
1+\overline{P}+\overline{Q}-\overline{P+Q}-\overline{1/2(P+Q)}.$

###### Proof.

If $P\geq 1$ then the inequality would be

$\overline{Q}\leq 1+1+\overline{Q}-1-\overline{1/2(P+Q)}$

which is trivially true. The same holds if $Q\geq 1$, then we may assume that
$P$ and $Q$ are both less than 1.

Notice that $P=\overline{P}$, $Q=\overline{Q}$ and
$1/2(P+Q)=\overline{1/2(P+Q)}$. Now, since $P$ and $Q$ are both in $[0,1]$,
then $PQ\leq\sqrt{PQ}$, therefore

$PQ\leq 1/2(P+Q).$

Now, since $\overline{P+Q}\leq 1$, then

$PQ\leq(1-\overline{P+Q})+1/2(P+Q)=1+P+Q-\overline{P+Q}-1/2(P+Q)$

which is what we wanted to prove. ∎

######  Definition 4.13.

Let $I$ be a set of positive predicates. We say that $I$ is an _ideal_ , if:

  1. (1)

Whenever $P\in I$ and $Q\geq P$, then $Q\in I$.

  2. (2)

If $P$ and $Q$ are in $I$ then $PQ$ is in $I$.

  3. (3)

$\lambda\in I$ for $\lambda>0$.

The ideal is _invariant over $A$_ (_$A$ -invariant_), if for every
$P(\bar{x},\bar{a})\in I$ and $\bar{a}\equiv_{A}\bar{a}^{\prime}$, we have
that $P(\bar{x},\bar{a}^{\prime})\in I$.

We say that an $A$-invariant ideal $I$ is _$S_{1}$ over $A$_ if, for every
$A$-indiscernible sequence of predicates $(P_{i})_{i<\omega}$, if
$P_{i}+P_{j}\in I$ for every ($i\neq j$), then $P_{i}\in I$ for every $I$.

######  Theorem 4.14.

If $G$ is definably amenable and the measure is $M$-invariant, then the set
$I$ of small positive predicates is an $S_{1}$-ideal over $M$.

###### Proof.

It is clear that $I$ is an ideal and $M$-invariant. Let us prove that it is
$S_{1}$:

Let $(P_{i})_{i<\omega}$ be a $M$-indiscernible sequence of predicates, such
that $P_{i}+P_{j}$ is in $I$ for $i\neq j$. Let us show that $P_{i}$ is in
$I$. Since $P_{i}$ is in $I$ if and only if $\overline{P_{i}}$ is, we are
going to assume that $P_{i}\leq 1$ for every $i$.

Suppose that $P_{i}\notin I$, then there exists some $\epsilon>0$ such that
$\mathfrak{m}_{\infty}(P_{i})=1-\epsilon$ for every $i$. We will prove that:

  1. (1)

$(P_{i}P_{j})_{i,j<\omega}$ is an indiscernible sequence (with the
lexicographic order).

  2. (2)

The predicates $P_{i}P_{j}+P_{k}P_{l}$ are in $I$.

  3. (3)

$\mathfrak{m}_{\infty}(P_{i}P_{j})\leq 1-2\epsilon$.

###### Proof.

  1. (1)

Clear.

  2. (2)

We need to show that $P_{i}P_{j}+P_{k}P_{l}$ is small. Since $(P_{i}+P_{k})$,
$(P_{i}+P_{l})$, $(P_{j}+P_{k})$ and $(P_{j}+P_{l})$ are small, then
$(P_{i}+P_{k})(P_{i}+P_{l})(P_{j}+P_{k})(P_{j}+P_{l})$ is. Notice that this
multiplication has 16 terms, 8 of those have as a factor $P_{i}P_{j}$ and the
other 8 has $P_{k}P_{l}$ as a factor, therefore
$(P_{i}+P_{k})(P_{i}+P_{l})(P_{j}+P_{k})(P_{j}+P_{l})\leq
8(P_{i}P_{j}+P_{k}P_{l})$; hence, $8(P_{i}P_{j}+P_{k}P_{l})$ is in $I$. In
particular, $(P_{i}P_{j}+P_{k}P_{l})\in I$.

  3. (3)

Using the previous lemma, we have that

$\overline{\lambda\,P\,Q}\leq
1+\overline{\sqrt{\lambda}P}+\overline{\sqrt{\lambda}Q}-\overline{\sqrt{\lambda}P+\sqrt{\lambda}Q}-\overline{1/2(\sqrt{\lambda}P+\sqrt{\lambda}Q)}$

therefore:

$\mathfrak{m}(\overline{\lambda\,P\,Q})\leq\mathfrak{m}(\overline{\lambda
1})+\mathfrak{m}(\overline{\sqrt{\lambda}P})+\mathfrak{m}(\overline{\sqrt{\lambda}Q})-\mathfrak{m}(\overline{\sqrt{\lambda}(P+Q)})-\mathfrak{m}(\overline{\sqrt{\lambda}/2(P+Q)})$

By making $\lambda\to\infty$, we get:

$\mathfrak{m}_{\infty}(\,P\,Q)\leq
1+(1-\epsilon)+(1-\epsilon)-1-1=1-2\epsilon$

∎

If we proceed inductively, we can show that
$\mathfrak{m}_{\infty}(P_{1}...P_{n})\leq 1-2n\epsilon$ for every $n$. This is
a contradiction.

∎

######  Definition 4.15.

A type $p$ is $I$-_wide_ if $p\cap I=\emptyset$.

Following the notation here [Hru15], we say that an ideal is _good_ if, for
every $\varphi\notin I$, there exists a complete type $p\ni\varphi$ such that
$p$ is $I$-wide. (In [Hru15] the ideal is a collection of type-definable sets,
but the definition is the same).

In first-order, it is easy to check that any ideal of formulas is _good_.
However, even for the case of this precise ideal, the existence of wide types
is open:

######  Question 4.16.

Assume that $G$ is definably amenable and $\mathfrak{m}$ is $M$-invariant. Is
the set of small formulas a _good_ ideal?

We can now relate all to $f$-generic types.

######  Lemma 4.17.

If $I$ is an ideal and $P(x,a)>0$ in $G$, then $P\in I$.

###### Proof.

By saturation, $P(x,a)\geq\lambda$ for some $\lambda>0$, thus $P$ is in $I$. ∎

######  Theorem 4.18.

If $I$ is a $S_{1}$-ideal over $M$ and $p$ is $I$-wide, then it is
$f$-generic.

###### Proof.

Clearly, if $p$ is $I$-wide, all its conjugates are, so it suffices to show
that $p$ does not fork (divide) over $M$: otherwise, there exists a formula
$\varphi(x,a_{0})\in p$ and an indiscernible sequence $(a_{i})_{i<\omega}$
such that $\\{\varphi(x,a_{i})\\}_{i<\omega}$ is $k$-inconsistent for some
$k$. This implies that $\sum_{i=0}^{k-1}|\varphi(x,a_{i})|>0$, thus, by the
previous lemma, it is small. Therefore, by the $S_{1}$ property, each
$|\varphi(x,a_{i})|$ is in $S_{1}$, this is a contradiction. ∎

######  Corollary 4.19.

If $G$ is definably amenable with an $M$-invariant mean and $I$ is the ideal
of small formulas. Then any $I$-wide global type is $f$-generic.

It follows of course that a positive answer to Question 4.16 would imply that
a definable group is definably amenable if and only if it has an $f$-generic
type, with all the consequences this would have through Theorem 4.6. We have
not been able to prove this yet.

##  5\. Stable Groups

In this section, we show that stable groups are definably amenable. The proof
is rather straightforward and it relies on several results proved by Ben-
Yaacov in [Yaa10], but as far we know this result has not been stated
explicitly before. Let us recall the definitions and results we need:

######  Definition 5.1.

A formula $\varphi(x,y)$ is _stable_ if for every $(a_{i})_{i<\omega}$ and
$(b_{j})_{j<\omega}$, we have

$\lim_{i\to\infty}\lim_{j\to\infty}\varphi(a_{i},b_{j})=\lim_{j\to\infty}\lim_{i\to\infty}\varphi(a_{i},b_{j}).$

A theory $T$ is _stable_ if all its formulas are stable in every model
$M\models T$.

A group $G$ is _stable_ if $G$ is definable in some stable theory $T$.

If $G$ is a definable group, then $G$ acts continuously on $S(G)$:
$gp(x)=p(g^{-1}x)$. This will help us define generic types in continuous
logic:

######  Definition 5.2 (Ben Yaacov).

Let $S$ be a $G$-space. A set $X\subset S$ is _generic_ if finitely many
$G$-translates of $X$ cover $S$. A type $p$ is _generic_ if every open set of
$S(G)$ containing $p$ is generic.

######  Example 5.3.

The group $S_{1}$ is compact (hence stable). The formulas $|x|<\epsilon$ are
clearly generics, but the set $|x|=0$ is not generic.

A group is _definably connected_ if $G=G^{00}$. It is known that
$stab(p)\subset G^{00}_{M}$ for any type $p$ (see, for example, [Wag00], Lemma
4.1.23). Thus, if $G$ is EDA, then it is definably connected.

######  Theorem 5.4 (Ben-Yaacov).

Let $G$ be a stable group, then:

  * •

The set of generic types $Gen$ of $G$ is non-empty.

  * •

The group $G$ has a connected component $G^{00}$, this is, there exists the
smallest type-definable group of bounded index.

  * •

The group $G$ acts in $Gen$ and this action is homeomorphic to the action on
$G$ on $G/G^{00}$.

The proof of the following fact is analogue to the first-order case (see, for
example, Lemma 8.10 in [Sim15]).

######  Lemma 5.5.

The group $G/G^{00}$ is a compact group.

######  Theorem 5.6.

Stable groups are definably amenable.

###### Proof.

Since $G/G^{00}$ is a compact group, it has an invariant mean $m^{*}$ on all
its bounded uniformly continuous functions. Let $\psi:G/G^{00}\to Gen$ the
isomorphism (as $G$-sets), let us define a function $m$ on all the formulas
$m(\varphi)$ as $m^{*}(\psi\circ\varphi^{*})$, where
$\varphi^{*}:Gen\to\mathbb{R}$: $\varphi^{*}(p)=p(\varphi)$. It is easy to
check that $m^{*}$ is an invariant mean on all the formulas. ∎

######  Example 5.7.

Let $(\Omega,\mu)$ be a probability space. We can define a metric group
$(B,*,d)$, where $B$ is the set of measurable sets of $\Omega$, the operation
group is given by $X*Y=X\triangle Y$ and the distance is given by
$d(X,Y)=\mu(X\triangle Y)$. This is an $\omega$-stable group, hence definably
amenable.

We call this group the _probability group_ of $(\Omega,\mu)$.

We will see in the following section that this group is, in fact, extremely
definably amenable.

##  6\. Randomizations

In this section, we prove that the randomization or first-order definably
amenable groups are extremely definably amenable. This generalizes the results
from Berenstein and Muñoz [BM21] that was restricted to dependent groups.

Fix $(\Omega,B,\mu)$ a probability space and $L$ a first-order language.

We define a new language $L^{R}$ as
$L^{R}=\\{K,B,\top,\bot,\sqcap,\sqcup,\neg,\llbracket\varphi_{i}(x)\rrbracket\\},$
where $K$ and $B$ are sorts, $\top,\bot,\sqcap,\sqcup,\neq$ are symbols for
the boolean operations and $\llbracket\varphi_{i}(x)\rrbracket$ are functions

$\llbracket\varphi_{i}(x)\rrbracket:K^{n}\to B,$

where $n$ is the number of free variables of $\varphi_{i}$.

Given a first order $L$-structure $M$, let $(\Omega,\mathcal{B},\mu)$ be an
atomless finitely additive probability space and $\mathcal{K}$ a set of
functions

$\\{f_{i}:\Omega\to M|i\in I\\}.$

Both $K$ and $B$ are pre-metric spaces with

$d_{K}(X,Y)=\mu(\\{m\in M|X(m)\neq Y(m)\\})$

and

$d_{B}(A,B)=\mu(A\triangle B).$

The pair $(\mathcal{B},K)$, is called a _randomization_ of $M$ if, as a
premetric structure in $L^{R}$, satisfies the following:

  1. (1)

For each $L$-formula $\varphi(x)$ and $f$ in $K^{n}$, we have

$\llbracket\varphi_{i}(X)\rrbracket=\\{m\in M:M\models\varphi(X(m))\\}$

  2. (2)

For every $B\in\mathcal{B}$ and $\epsilon>0$ there are $f,g\in\mathcal{K}$
such that

$\mu(\llbracket f=g\rrbracket\triangle B)<\epsilon$

.

  3. (3)

For every $L$-formula $\varphi(x;y)$, $g$ in $\mathcal{K}^{n}$ and
$\epsilon>0$ there is $f\in\mathcal{K}$ such that

$\mu\llbracket\varphi(f,g)\rrbracket\triangle\llbracket\exists\varphi(x,g)\rrbracket<\epsilon$

The interpretation of $\top,\bot,\sqcap,\sqcup,\neq$ are the usual boolean
operations on $B$.

The completion of $K$ and $B$ are continuous structures.

One associates an $L^{R}$ continuous theory $T^{R}$ to every first order $L$
theory $T$, see [BYJK09]. This theory $T^{R}$ is the common theory of all the
randomizations of models of $T$.

######  Fact 6.1 ([BYJK09],).

$T$ is complete/stable/dependent if and only if $T^{R}$ is.

######  Definition 6.2.

Let $(\Omega,\mathcal{B},\mu)$ be an atomless finitely additive probability
space and let $M$ be a $L$-structure.

A $\mathcal{B}$-simple random element of $M$ is a $\mathcal{B}$-measurable
function in $M$ with finite range.

A $\mathcal{B}$-countable random element of $M$ is a $\mathcal{B}$-measurable
function in $M^{\Omega}$ with countable range.

It is proven in [Kei99] that the set $K_{S}$ of $B$-simple random elements of
$M$ is a full-randomization of $M$ and that $K_{C}$, the set of countable
random elements is a full-randomization of $M$ that is pre complete. Moreover
$K_{S}$ is dense in $K_{C}$ ([BYJK09]).

Now we prove the main theorem of this section:

######  Theorem 6.3.

Let $G$ be a definably amenable group with measure $\nu$. Then $G^{R}$ is
extremely definably amenable.

###### Proof.

Since EDA is a property of the theory of $G^{R}$, we may assume that
$M^{R}=K_{C}$ where $K_{C}$ is induced by the countable random elements.
Moreover, without loss of generality, we may assume that $\Omega=[0,1]$. Let
$p$ be the following type:

$p=\\{\mu\llbracket\varphi(x,\bar{a})\rrbracket=\sum_{m_{i}\in
ran(a)}\nu(\varphi(x,m_{i}))\mu(\bar{a}^{-1}(m_{i}))|\bar{a}\in K_{S}\\}$

By quantifier elimination and density of $K_{S}$, this determines a unique
complete type.

Notice that the definition of $p$ is quite natural: if $\bar{a}$ has constant
value $m$, then
$\hat{p}(\llbracket\varphi(x,\bar{a})\rrbracket)=\nu(\varphi(x,m))$. The
extension to non-constant variables $\bar{a}$ is basically the integral over
that function (which is equal to a finite sum since $\bar{a}$ is simple.

Let us check that $p$ is consistent: take $p_{0}$ a finite subset of $p$,

$p_{0}=\\{\mu\llbracket\varphi_{j}(x,\bar{a}_{j})\rrbracket=\sum_{m_{i,j}\in
ran(a_{j})}\nu(\varphi_{j}(x,m_{i,j})\mu(\bar{a}_{j}^{-1}(m_{i,j})):j\leq
n\\}$

Let us define $A_{i,j}=\bar{a}_{j}^{-1}(m_{i,j})$. Notice that
$\\{A_{i,j}\\}_{m_{i,j}\in ran(a_{j})}$ is a partition of $\Omega$. Hence, the
non-empty intersections of the form $\bigcap_{j\leq n}A_{i_{j}}$ form a
partition of $\Omega$. We build $f$, a realization of $p_{0}$, as follows: For
every

$I=\\{(i_{1},1),...,(i_{n},n)\\}.$

We name any of such intersections as

$A_{I}=A_{i_{1},1}\cap...\cap A_{i_{n},n}.$

Take $B_{1},...,B_{l}$ the atoms of the Boolean algebra generated by
$\varphi_{j}(x,m_{i_{j},j})$ and choose any partition of $A_{I}$ of the form
$\\{A_{I,r}\\}_{r\leq l}$ such that $\mu(A_{I,r})=\nu(B_{r})\mu(A_{I})$, (it
exists since $\sum\nu(B_{r})=1$).

Now, for each $A_{I,r}$, we take a point $b_{I,r}$ satisfying $B_{r}$ and
define $f:\Omega:\to G$ such that $f[A_{I,r}]=b_{I,r}$. Since $\\{A_{I,r}\\}$
is a measurable partition of $\Omega$, then $f\in K_{S}$.

Let us check that $f\models p_{0}$:

$\displaystyle\mu\llbracket\varphi_{j}(f,\bar{a}_{j})\rrbracket$
$\displaystyle=\mu\\{\omega:\models\varphi_{j}(f(\omega),\bar{a}_{j}(\omega))\\}$
$\displaystyle=\sum_{m_{i,j}\in ran(\bar{a}_{j})}\mu\\{\omega\in
A_{i,j}:\models\varphi_{j}(f(\omega),m_{i,j})\\}$
$\displaystyle=\sum_{m_{i,j}\in
ran(\bar{a}_{j})}\sum_{I\ni(i,j)}\mu\\{\omega\in
A_{I}:\models\varphi_{j}(f(\omega),m_{i,j})\\}$
$\displaystyle=\sum_{m_{i,j}\in
ran(\bar{a}_{j})}\sum_{I\ni(i,j)}\sum_{r}\mu\\{\omega\in
A_{I,r}:\models\varphi_{j}(f(\omega),m_{i,j})\\}$
$\displaystyle=\sum_{m_{i,j}\in
ran(\bar{a}_{j})}\sum_{I\ni(i,j)}\sum_{r}\mu\\{\omega\in
A_{I,r}:\models\varphi_{j}(b_{I,r},m_{i,j})\\}$
$\displaystyle=\sum_{m_{i,j}\in
ran(\bar{a}_{j})}\sum_{I\ni(i,j)}\sum_{B_{r}\subset\varphi_{j}(x,m_{i,j})}\mu(A_{I,r})$
$\displaystyle=\sum_{m_{i,j}\in
ran(\bar{a}_{j})}\sum_{I\ni(i,j)}\mu(A_{I})\sum_{B_{r}\subset\varphi_{j}(x,m_{i,j})}\nu(B_{r})$
$\displaystyle=\sum_{m_{i,j}\in
ran(\bar{a}_{j})}\mu(A_{i,j})\nu(\varphi_{j}(x,m_{i,j}))$
$\displaystyle=\sum_{m_{i,j}\in
ran(\bar{a}_{j})}\nu(\varphi_{j}(x,m_{i,j})\mu(\bar{a}_{j}^{-1}(m_{i,j}))$

Now, let us check that $p$ is $G^{R}$-invariant. Again, since $K_{S}$ is
dense, it is enough to prove that $p$ is invariant under the simple random
elements in $G^{R}$:

Let $g_{R}$ a simple random element in $G^{R}$ and let $h_{R}=g_{R}^{-1}$,
then, in $p$, we have that

$\displaystyle\mu\llbracket g_{R}\varphi(x,\bar{a})\rrbracket$
$\displaystyle=\mu\llbracket\varphi(h_{R}x,\bar{a})\rrbracket$
$\displaystyle=\sum_{h_{i}\in ran(h_{R}),m_{j}\in
ran(\bar{a})}\nu(\varphi(h_{i}x,m_{j})(\mu(h_{R}^{-1}(h_{i}))+\mu(\bar{a}^{-1}(m_{j}))$
$\displaystyle=\sum_{m_{j}\in
ran(\bar{a})}\nu(\varphi(h_{i}x,m_{j}))\sum_{h_{i}\in
ran(h_{R})}(\mu(h_{R}^{-1}(h_{i}))+\mu(\bar{a}^{-1}(m_{j}))$

Finally, by amenability of $G$ we have that

$\nu(\varphi(h_{i}x,m_{j}))=\nu(\varphi(x,m_{j}))$

and, since $\mu(\Omega)=1$

$\sum_{h_{i}\in ran(h_{R})}(\mu(h_{R}^{-1}(h_{i})))=1$

So we get that:

$\mu\llbracket g_{R}\varphi(x,\bar{a})\rrbracket=\sum_{m_{i}\in
ran(\bar{a})}\nu(\varphi(x,m_{i})\mu(\bar{a}^{-1}(m_{i}))$

Which is what we wanted to prove. ∎

By definition, every extremely amenable metric group is amenable, but also
extremely definably amenable. Hence, we have the following diagram:

${A}$${EA}$${A\cap EDA}$${A\cup
EDA}$${DA}$${EDA}$$\subseteq$$\subseteq$$\subseteq$$\subseteq$$\subseteq$$\subseteq$

All the inclusions are strict as it may be seen from First-order examples:

  * •

$(\mathbb{R},+,<)\in A\cap EDA\setminus EA$: it is abelian and the type at
$+\infty$ is invariant, but since it is locally compact, it cannot be
extremely amenable.

  * •

The non-abelian free group is stable [Sel03] and connected [Pil08], hence it
is extremely definably amenable but not amenable.

  * •

Any non-trivial finite group is amenable but not extremely definable amenable.

  * •

$\mathbb{F}_{2}\times\mathbb{F}_{2}$ is definably amenable, since it is
stable, but it is not extremely definable amenable (because its connected
component is not trivial), nor amenable.

Finally, let us recall some interesting examples from continuous logic:

  * •

Since $\mathbb{R}$ is definably amenable, its randomization
$L_{1}(\Omega,\mathbb{R})$ is extremely definably amenable. However [Pes02],
it is not extremely amenable with the $L_{1}$ metric. Since
$L_{1}(\Omega,\mathbb{R})$ is abelian, we have that
$L_{1}(\Omega,\mathbb{R})\in A\cap EDA\setminus EA$.

  * •

Let $M_{n}(\mathbb{C})$ be the full matrix C*-algebra over the complex
numbers. Since its unitary group is compact, it is amenable, (hence definably
amenable). However, if $\mathcal{U}$ is a non-principal ultrafilter over
$\mathbb{N}$, the ultraproduct $\prod_{\mathcal{U}}M_{n}(\mathbb{C})$ is not
nuclear (nor elementary equivalent to a nuclear algebra, see Proposition
7.2.5. [FHL+16]). This implies that the unitary group of
$\prod_{\mathcal{U}}M_{n}(\mathbb{C})$ is not amenable with the weak topology
(in particular, it cannot be amenable with the metric topology). However,
definable amenability is preserved under ultraproducts (Theorem 2.11), so we
have an example of a definably amenable group that is not purely amenable.

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