# Common Fixed Points of Semihypergroup Representations

Choiti Bandyopadhyay  Department of Mathematics, SRM University AP, Amaravati,
India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In a series of previous papers [2, 3, 5, 6], we initiated a systematic study
of semihypergroups and had a thorough discussion on certain analytic and
algebraic aspects associated to this class of objects. In particular, we
introduced the notion of semihypergroup actions on a general topological space
and discussed different continuity, equivalence and natural fixed point
properties of the same in [6]. Now in this article, we consider different
kinds of representations of a semihypergroup on compact convex subsets of a
locally convex space and explore equivalence relations between certain fixed-
point properties of such representations and amenability of the space of
almost periodic functions. Finally, we investigate how far these equivalence
relations can be strengthened when in particular, we consider representations
on the dual of a Banach space.

###### Key words and phrases:

semihypergroup, action, amenability, fixed points, almost periodic functions,
invariant mean, hypergroup, coset spaces, orbit spaces

###### 2020 Mathematics Subject Classification:

Primary 43A07, 43A62, 43A65, 47H10; Secondary 43A60, 43A85, 43A99, 46G12,
46E27

##  1\. Introduction

In the research of abstract harmonic analysis, left amenability of a given
function-space is determined by the existence of left translation invariant
means on the function-space under consideration. Since its inception, research
on this topic has been a pivotal area of investigation in this area due to its
vast applicability and deep interplay with other prominent areas of functional
analysis, operator algebras and analysis in general. On the other hand, in the
classical setting of topological semigroups and groups, fixed-point properties
of their respective representations on certain subsets of a Banach space or a
locally convex topological space, have been another pivotal area of research,
imperative for an in-depth understanding of the underlying space in
consideration.

It is well documented that invariance of this kind, i.e, left amenability is
intrinsically connected to the fixed point theory of different kinds of
actions of the underlying object on a certain topological space. In
particular, the amenability properties of a (semi)topological semigroup are
intrinsically related to the existence of fixed points of certain actions of
the semigroup on certain spaces, and in fact, one can completely characterize
amenability of different function-spaces of a semigroup using such fixed-point
properties (see [10, 14, 16, 18, 19, 20] for example, among others). In a
previous article [6], we introduced actions in the broader category of
semihypergroups and initiated an in-depth discussion on the interplay between
the existence of fixed points of different kinds of semihypergroup actions and
left amenability of a semihypergroup, in order to thereby realise where and
why this theory deviates from the classical theory of locally compact
semigroups and groups.

A semihypergroup, as one would expect, can be perceived simply as a
generalization of a locally compact semigroup, where the product of two points
is a certain compact set, rather than a single point. In a nutshell, a
semihypergroup is essentially a locally compact topological space where the
measure space is equipped with a certain convolution product, turning it into
an associative algebra (which unlike hypergroups, need not have an identity or
an involution).

The concept of semihypergroups, first introduced as ‘semiconvos’ by
Jewett[12], arises naturally in abstract harmonic analysis in terms of
left(right) coset spaces of locally compact groups, and orbit spaces of affine
group-actions which appear frequently in different areas of research in
mathematics, including Lie groups, coset theory, homogeneous spaces, dynamical
systems, and ordinary and partial differential equations, to name a few. These
objects arising from locally compact groups, although retain some interesting
structures from the parent category, often fail to be a topological group,
semigroup or even a hypergroup (see [2, 12] for detailed reasons). The fact
that semihypergroups admit a broader range of examples arising from different
fields of research compared to classical semigroup, group and hypergroup
theory, and yet sustains enough structure to allow an independent theory to
develop, makes it an intriguing and useful area of study with essentially a
broader range of applications.

However, unlike hypergroups, the category of (topological) semihypergroups
still lack an extensive systematic theory on it. In a series of previous
papers [2, 3, 5, 6] we initiated developing a systematic theory on
semihypergroups. Our approach towards it has been to define and explore some
basic algebraic and analytic aspects and areas on semihypergroups, which are
imperative for further development of any category of analytic objects in
general.

It has been a long-standing open problem to characterize certain coset spaces,
in general, semitopological semihypergroups, that have common fixed points
whenever they act on a nonempty weak* compact convex subset of a locally
convex space or the dual of a Banach space as weak* continuous and norm
nonexpansive mappings. In this article, we advance the theory by providing
some complete characterizations of the left amenability of the space of almost
periodic functions, in terms of existence of common fixed points of such
representations ofthe semihyperhypergroup. The rest of the article is
organized as the following.

In the next, i.e., second section of this article, we recall some preliminary
definitions and notations given by Jewett in [12], and some notions used in
[6] required for further development. We conclude the section with listing
some important examples of semihypergroups and hypergroups.

The third and final section contains proofs of the main results of the
article. We begin with discussing the concepts of certain kinds of
semihypergroup representations such as affine, equicontinuous, non-expansive
and orbit-bounded, to name a few. We first show in Theorem 3.3 that in this
much broader setting, common fixed points of separately continuous,
equicontinuous, affine actions on certain compact convex subsets completely
characterize the left amenability of the function space of almost periodic
functions. Later in Theorem 3.4 we prove that a complete characterization of
left amenability of the this function space can also be carried out in terms
of existence of common fixed points of separately continuous, non-expansive
semihypergroup actions on certain compact convex sets. Finally in Theorem 3.5
we investigate how the conditions of this equivalence can be strengthened if
instead of actions on compact subsets of locally convex spaces, we consider
actions on duals of Banach spaces. We see that a complete characterization of
left amenability is indeed possible in this case, provided the action in
consideration has at least one bounded orbit. We conclude the section with
discussing certain immediate open questions in this area of research.

##  2\. Preliminary

We first list a preliminary set of notations and definitions that we will use
throughout the text. All the topologies throughout this text are assumed to be
Hausdorff.

For any locally compact Hausdorff topological space $X$, we denote by $M(X)$
the space of all regular complex Borel measures on $X$, where
$M_{F}^{+}(X),M^{+}(X)$ and $P(X)$ respectively denote the subsets of $M(X)$
consisting of all non-negative measures with finite support, all finite non-
negative regular Borel measures and all probability measures on $X$. For any
measure $\mu$ on $X$, we denote by $supp(\mu)$ the support of the measure
$\mu$. Moreover, $B(X),C(X),C_{c}(X)$ and $C_{0}(X)$ denote the function
spaces of all bounded functions, bounded continuous functions, compactly
supported continuous functions and continuous functions vanishing at infinity
on $X$ respectively.

Unless mentioned otherwise, the space $M^{+}(X)$ is equipped with the cone
topology [12], i.e, the weak topology on $M^{+}(X)$ induced by
$C_{c}^{+}(X)\cup\\{\chi_{{}_{X}}\\}$. We denote the set of all compact
subsets of $X$ by $\mathfrak{C}(X)$, and consider the natural Michael topology
[17] on it, which makes it into a locally compact Hausdorff space. For any
element $x\in X$, we denote by $p_{x}$ the point-mass measure or the Dirac
measure at the point $x$.

For any three locally compact Hausdorff spaces $X,Y,Z$, a bilinear map
$\Psi:M(X)\times M(Y)\rightarrow M(Z)$ is called positive continuous if the
following properties hold true.

  1. (1)

$\Psi(\mu,\nu)\in M^{+}(Z)$ whenever $(\mu,\nu)\in M^{+}(X)\times M^{+}(Y)$.

  2. (2)

The map $\Psi|_{M^{+}(X)\times M^{+}(Y)}$ is continuous.

Now we state the formal definition for a (topological) semihypergroup. Note
that we follow Jewett’s notion [12] in terms of the definitions and notations,
in most cases.

######  Definition 2.1 (Semihypergroup).

A pair $(K,*)$ is called a (topological) semihypergroup if they satisfy the
following properties:

(A1):

$K$ is a locally compact Hausdorff space and $*$ defines a binary operation on
$M(K)$ such that $(M(K),*)$ becomes an associative algebra.

(A2):

The bilinear mapping $*:M(K)\times M(K)\rightarrow M(K)$ is positive
continuous.

(A3):

For any $x,y\in K$ the measure $(p_{x}*p_{y})$ is a probability measure with
compact support.

(A4):

The map $(x,y)\mapsto\mbox{ supp}(p_{x}*p_{y})$ from $K\times K$ into
$\mathfrak{C}(K)$ is continuous.

Note that for any $A,B\subset K$ the convolution of subsets is defined as the
following:

$A*B:=\cup_{x\in A,y\in B}\ supp(p_{x}*p_{y}).$

We define the concepts of left (resp. right) topological and semitopological
semihypergroups, in accordance with similar concepts in the classical
semigroup theory.

######  Definition 2.2.

A pair $(K,*)$ is called a left (resp. right) topological semihypergroup if it
satisfies all the conditions of Definition 2.1, with property ($A2$) replaced
by property ($A2^{\prime}$) (resp. property ($A2^{\prime\prime}$)), given as
the following:

(A2′):

The map $(\mu,\nu)\mapsto\mu*\nu$ is positive and for each $\omega\in
M^{+}(K)$ the map
$L_{\omega}:M^{+}(K)\rightarrow M^{+}(K)$ given by
$L_{\omega}(\mu)=\omega*\mu$ is continuous.

(A2′′):

The map $(\mu,\nu)\mapsto\mu*\nu$ is positive and for each $\omega\in
M^{+}(K)$ the map
$R_{\omega}:M^{+}(K)\rightarrow M^{+}(K)$ given by
$R_{\omega}(\mu)=\mu*\omega$ is continuous.

A pair $(K,*)$ is called a semitopological semihypergroup if it is both left
and right topological semihypergroup, i.e, if the convolution $*$ on $M(K)$ is
only separately continuous.

For any Borel measurable function $f$ on a (semitopological) semihypergroup
$K$ and each $x,y\in K$, we define the left translate $L_{x}f$ of $f$ by $x$
(resp. the right translate $R_{y}f$ of $f$ by $y$) as

$L_{x}f(y)=R_{y}f(x)=f(x*y):=\int_{K}f\ d(p_{x}*p_{y})\ .$

Unless mentioned otherwise, we will always assume the uniform (supremum) norm
$||\cdot||_{u}$ on $C(K)$ and $B(K)$. We denote by $\mathcal{B}_{1}$ the
closed unit ball of $C(K)^{*}$. Similarly, for any linear subspace
$\mathcal{F}$ of $C(K)$, we denote the closed unit ball of $\mathcal{F}^{*}$
as
$\mathcal{B}_{1}(\mathcal{F}^{*}):=\\{\omega\in\mathcal{F}^{*}:||\omega||\leq
1\\}$. Moreover, $\mathcal{F}$ is called left (resp. right) translation-
invariant if $L_{x}f\in\mathcal{F}$ (resp. $R_{x}f\in\mathcal{F}$) for each
$x\in K,f\in\mathcal{F}$. We simply say that $\mathcal{F}$ is translation-
invariant, if it is both left and right translation-invariant.

A function $f\in C(K)$ is called left (resp. right) uniformly continuous if
the map $x\mapsto L_{x}f$ (resp. $x\mapsto R_{x}f$) from $K$ to
$(C(K),||\cdot||_{u})$ is continuous. We say that $f$ is uniformly continuous
if it is both left and right uniformly continuous. The space consisting of all
such functions is denoted by $UC(K)$, which forms a norm-closed linear
subspace of $C(K)$.

The left (resp. right) orbit of a function $f\in C(K)$, denoted as
$\mathcal{O}_{l}(f)$ (resp. $\mathcal{O}_{r}(f)$), is defined as
$\mathcal{O}_{l}(f):=\\{L_{x}f:x\in K\\}$ (resp.
$\mathcal{O}_{r}(f):=\\{R_{x}f:x\in K\\}$). A function $f\in C(K)$ is called
left (resp. right) almost periodic if we have that $\mathcal{O}_{l}(f)$ (resp.
$\mathcal{O}_{r}(f))$ is relatively compact in $(C(K),||\cdot||_{u})$. We
showed in a previous work [2, Corollary 4.4] that a function $f$ on $K$ is
left almost periodic if and only if it is right almost periodic. Hence we
regard any left or right almost periodic function on $K$ simply as an almost
periodic function, and denote the space of all almost periodic functions on
$K$ as $AP(K)$. We further saw in [2] that $AP(K)$ is a norm-closed,
conjugate-closed (with respect to complex conjugation), translation-invariant
linear subspace of $C(K)$ containing constant functions, such that
$AP(K)\subseteq UC(K)$.

An element $e$ in a semihypergroup $K$ is called a two sided identity of $K$
if

$p_{x}*p_{e}=p_{e}*p_{x}=p_{x}$

for any $x\in K$. Note that a two sided identity of a semihypergroup is
necessarily unique [12].

######  Remark 2.3.

Given a Hausdorff topological space $K$, in order to define a continuous
bilinear mapping $*:M(K)\times M(K)\rightarrow M(K)$, it suffices to only
define the measures $(p_{x}*p_{y})$ for each $x,y\in K$. This is true since
one can then extend the convolution ‘$*$’ bilinearly to $M_{F}^{+}(K)$. As
$M_{F}^{+}(K)$ is dense in $M^{+}(K)$ in the cone topology [12], one can
further achieve a continuous extension of ‘$*$’ to $M^{+}(K)$ and hence to the
whole of $M(K)$ using bilinearity.

Now we list some well known examples [12, 23] of semihypergroups and
hypergroups. See [2, Section 3] for details on the constructions as well as
the reasons why most of the structures discussed there, although attain a
semihypergroup structure, fail to be hypergroups.

######  Example 2.4.

If $(S,\cdot)$ is a locally compact topological semigroup, then $(S,*)$ is a
semihypergroup where $p_{x}*p_{y}=p_{{}_{x.y}}$ for any $x,y\in S$. Similarly,
if $(G,\cdot)$ is a locally compact topological group with identity $e_{G}$,
then $(G,*)$ is a hypergroup with the same bilinear operation $*$, identity
element $e_{G}$ and the involution on $G$ defined as $x\mapsto x^{-1}$.

Note that $Z(S)=S$, $Z(G)=G$.

######  Example 2.5.

Take $T=\\{e,a,b\\}$ and equip it with the discrete topology. Define

$\displaystyle p_{e}*p_{a}$ $\displaystyle=$ $\displaystyle p_{a}*p_{e}\ =\
p_{a}$ $\displaystyle p_{e}*p_{b}$ $\displaystyle=$ $\displaystyle
p_{b}*p_{e}\ =\ p_{b}$ $\displaystyle p_{a}*p_{b}$ $\displaystyle=$
$\displaystyle p_{b}*p_{a}\ =\ z_{1}p_{a}+z_{2}p_{b}$ $\displaystyle
p_{a}*p_{a}$ $\displaystyle=$ $\displaystyle x_{1}p_{e}+x_{2}p_{a}+x_{3}p_{b}$
$\displaystyle p_{b}*p_{b}$ $\displaystyle=$ $\displaystyle
y_{1}p_{e}+y_{2}p_{a}+y_{3}p_{b}$

where $x_{i},y_{i},z_{i}\in\mathbb{R}$ such that
$x_{1}+x_{2}+x_{3}=y_{1}+y_{2}+y_{3}=z_{1}+z_{2}=1$ and
$y_{1}x_{3}=z_{1}x_{1}$. Then $(T,*)$ is a commutative hypergroup with
identity $e$ and the identity function on $T$ taken as involution. In fact,
any finite set can be given several (not necessarily equivalent)
semihypergroup and hypergroup structures.

######  Example 2.6.

Let $G$ be a locally compact group and $H$ be a compact subgroup of $G$ with
normalized Haar measure $\mu$. Consider the left coset space
$S:=G/H=\\{xH:x\in G\\}$ and the double coset space $K:=G//H=\\{HxH:x\in G\\}$
and equip them with the respective quotient topologies. Then $(S,*)$ is a
semihypergroup and $(K,*)$ is a hypergroup where the convolutions are given as
following for any $x,y\in G$ :

$p_{{}_{xH}}*p_{{}_{yH}}=\int_{H}p_{{}_{(xty)H}}\ d\mu(t),\ \ \ \
p_{{}_{HxH}}*p_{{}_{HyH}}=\int_{H}p_{{}_{H(xty)H}}\ d\mu(t).$

It can be checked [2] that the coset spaces $(S,*)$ fail to have a hypergroup
structure.

######  Example 2.7.

Let $G$ be a locally compact topological group and $H$ be any compact group
with normalized Haar measure $\sigma$. For any continuous affine action[12]
$\pi$ of $H$ on $G$, consider the orbit space $\mathcal{O}:=\\{x^{H}:x\in
G\\}$, where for each $x\in G$, $x^{H}=\\{\pi(h,x):h\in H\\}$ is the orbit of
$x$ under the action $\pi$.

Consider $\mathcal{O}$ with the quotient topology and the following
convolution:

$p_{{}_{x^{H}}}*p_{{}_{y^{H}}}:=\int_{H}\int_{H}p_{{}_{(\pi(s,x)\pi(t,y))^{H}}}\
d\sigma(s)d\sigma(t).$

Then $(\mathcal{O},*)$ becomes a semihypergroup. It can be shown [8, 12] that
$(\mathcal{O},*)$ becomes a hypergroup only if for each $h\in H$, the map
$x\mapsto\pi(h,x):G\rightarrow G$ is an automorphism.

##  3\. Main Results

Recall [5] that a semihypergroup action $\pi$ of $K$ on a locally convex
Hausdorff topological vector space $E$ (also denoted by $(E,Q)$ where $Q$ is
the family of seminorms inducing the topology on $E$) is a map $\pi:K\times
E\rightarrow E$ such that the following conditions are satisfied:

  1. (1)

For any $s,t\in K$, $x\in E$ we have

$\pi(s,\pi(t,x))=\int_{K}\pi(\zeta,x)\ d(\delta_{s}*\delta_{t})(\zeta).$

  2. (2)

Whenever $K$ has a two-sided identity $e$, for each $x\in E$ we have that

$\pi(e,x)=x\ .$

Such an action $\pi$ of $K$ on $E$ is called (separately) continuous, if the
map $\pi$ is (separately) continuous on $K\times E$. For each $s\in K$, we
denote by $\pi_{s}$ the co-efficient map

$x\mapsto\pi(s,x):E\rightarrow E\ .$

An action $\pi$ of $K$ is called affine if $\pi_{s}$ is an affine map for each
$s\in K$. We say that some $x_{0}\in E$ is a common fixed point of $\pi$ if
$\pi_{s}(x_{0})=x_{0}$ for each $s\in K$. In fact, using the fact that
commutative semihypergroups are amenable [2], one can prove the existence of
common fixed points for such semihypergroups.

######  Theorem 3.1 ([5]).

Let $K$ be commutative and $C$ be a compact convex subset of a locally convex
Hausdorff vector space $(E,Q)$. Then any separately continuous affine action
of $K$ on $C$ has a fixed point.

On the other hand, a stricter fixed point property serves as a sufficient
condition for any semihypergroup $K$ to admit a LIM on $AP(K)$.

######  Theorem 3.2 ([5]).

Assume that any jointly continuous affine action $\pi$ of $K$ on a compact
convex subset $C$ of a locally convex vector space $(E,Q)$ has a common fixed
point. Then there exists a LIM on $AP(K)$.

In what follows, we see that although the intrinsically different structures
of the underlying space and function-spaces in this broader setting warrants
further caution and evolved Banach algebraic techniques, one can indeed extend
a number of useful pivotal fixed-point characterizations of left amenability
to this broad setting of semihypergroups, from the classical setting of
semigroups [15, 16].

Recall that given a compact subset $C$ of a separated locally convex space
$E$, the subspace topology on $C$ is induced by a unique uniformity
$\mathcal{U}$ corresponding to $C$ [13, Chapter 6], namely the family of all
neighborhoods of the diagonal $\Delta=\\{(x,x):x\in C\\}$. We say that an
action $\pi$ of $K$ on $C$ is equicontinuous if for each $x\in C$ and
$U\in\mathcal{U}$, there exists some $V\in\mathcal{U}$ such that
$(\pi(s,x),\pi(s,y))\in U$ for each $s\in K$ whenever $(x,y)\in V$. An
important subclass of such actions consists of the non-expansive actions. An
action $\pi$ of $K$ on $(E,Q)$ is called non-expansive if we have that
$p(\pi(s,x)-\pi(s,y))\leq p(x-y)$ for each $p\in Q,s\in K$ and any $x,y\in E$.

Given a separately continuous action $\pi$ of $K$ on a compact Hausdorff space
$X$, for each $y\in X$ and $f\in C(X)$ we define a map
$T_{y}^{\pi}f:K\rightarrow\mathbb{C}$ as

$T_{y}^{\pi}f(s):=f(\pi(s,y))=f\circ\pi_{s}(y),$

for each $s\in K$. It is easy to see that $T_{y}^{\pi}f$ is a continuous
function. Moreover, it follows from the continuity of $f$ and $\pi_{s}$ that
the orbit of the function $T_{y}^{\pi}f$ is relatively compact in $C(K)$, i.e,
we will always have that $T_{y}^{\pi}f\in AP(K)$.

Now we proceed to deriving several pivotal characterizations of amenability of
the function space $AP(K)$ in terms of the existence of common fixed points of
such actions or representations.

######  Theorem 3.3.

$AP(K)$ is left amenable if and only if every separately continuous,
equicontinuous affine action of $K$ on a compact convex subset $C$ of a
separated locally convex space $E$ has a common fixed point.

###### Proof.

First, assume that any separately continuous, equicontinuous affine action
$\pi$ of $K$ on a compact convex subset $C$ of a separated locally convex
space $E$ has a common fixed point.

Recall [2] that the function $AP(K)$ is both left and right translation
invariant. For each $s\in K$, consider the dual $L_{s}^{*}$ of the Left
translation operator $L_{s}:AP(K)\rightarrow AP(K)$, where
$L_{s}^{*}:AP(K)^{*}\rightarrow AP(K)^{*}$ is canonically defined as

$L_{s}^{*}\phi(f)=\phi(L_{s}f)$

for each $\phi\in AP(K)^{*},f\in AP(K)$. Let $C\subseteq AP(K)^{*}$ denote the
set of all means on $AP(K)$ and $E:=AP(K)^{*}$. We equip $E$ with the weak∗
topology and define a map $\pi:K\times E\rightarrow E$ by

$\pi(s,\phi):=L_{s}^{*}\phi$

for each $s\in K,\phi\in E$. We know from [6, Theorem 4.4] that $\pi$ is an
affine, separately continuous action of $K$ on $E$, and $C$ is weak∗-compact
in $E$. Now to check if $\pi$ is equicontinuous, for each $f\in AP(K)$ let
$p_{f}$ be the seminorm on $AP(K)^{*}$ given by

$p_{f}(\phi):=sup\\{|\phi(L_{s}f)|:s\in K\\}$

for each $\phi\in AP(K)^{*}$. Set $Q:=\\{p_{f}:f\in AP(K)\\}$.

Let $\varepsilon>0$ be given. Given $f\in AP(K)$, choose $\phi,\psi\in
AP(K)^{*}$ such that

$sup_{s\in K}|(\phi-\psi)(L_{s}f)|=p_{f}(\phi-\psi)<\varepsilon.$

Then for each $x\in K$ and any $\phi,\psi\in AP(K)^{*},f\in AP(K)$ we have
that

$\displaystyle p_{f}(L_{x}^{*}\phi-L_{x}^{*}\psi)$ $\displaystyle=$
$\displaystyle sup_{s\in K}|(L_{x}^{*}\phi-L_{x}^{*}\psi)(L_{s}f)|$
$\displaystyle=$ $\displaystyle sup_{s\in K}|(\phi-\psi)(L_{x}(L_{s}f))|$
$\displaystyle=$ $\displaystyle sup_{s\in
K}\\{\Big{|}\int_{K}(\phi-\psi)(L_{u}f)\ d(p_{x}*p_{s})(u)\Big{|}$
$\displaystyle\leq$ $\displaystyle sup_{s\in K}\int_{K}|(\phi-\psi)(L_{u}f)|\
d(p_{x}*p_{s})(u)$ $\displaystyle\leq$ $\displaystyle sup_{s\in
K}\int_{K}p_{f}(\phi-\psi)\ d(p_{x}*p_{s})(u)$ $\displaystyle<$
$\displaystyle\varepsilon\ (p_{x}*p_{s})(K)\ =\ \varepsilon,$

as required. Now in particular, the given fixed point property provides us
with a common fixed point $m_{0}\in C$ of $\pi$, i.e, we get a mean $m_{0}$ on
$AP(K)$ such that $L_{s}^{*}m_{0}=m_{0}$ for each $s\in K$. Hence $m_{0}$
serves as a LIM on $AP(K)$.

Now conversely, let $AP(K)$ be left amenable and $m$ be a LIM on $AP(K)$.
Furthermore, let $\pi$ be a separately continuous, equicontinuous, affine
action of $K$ on a compact convex subset $C$ of a separated locally convex
space $(E,Q)$. We denote by $A_{f}(K)$ the closed subspace of $C(K)$
consisting of all real valued continuous affine functions on $K$. For each
$s\in K$, the evaluation map $E_{s}\in C(K)^{*}$ is canonically defined as
$E_{s}(f):=f(s)$ for each $f\in C(K)$.

Since $m$ is a mean and the evaluation maps serve as extreme points of the
unit ball, we know [6] that there exists a net $\\{\psi_{\alpha}\\}$ of finite
means such that

$\lim_{\alpha}\psi_{\alpha}(f)=m(f)$

for each $f\in AP(K)$. Here for each $\alpha$ there exists some
$n_{\alpha}\in\mathbb{N}$ such that

$\psi_{\alpha}=\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}E_{s_{i}^{\alpha}}$

where each $s_{i}^{\alpha}\in K$ and each $\lambda_{i}^{\alpha}>0$ such that
$\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}=1$. Now fix some $y\in C$ and
consider the net $\\{z_{\alpha}\\}\subseteq C$ where for each $\alpha$ we set

$z_{\alpha}:=\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}\pi(s_{i}^{\alpha},y).$

Since $C$ is compact, let $z$ be a cluster point of $\\{z_{\alpha}\\}$ in $C$.
On the other hand, for each bounded continuous function $f$ on $C$, consider
the map $T_{y}f:K\rightarrow\mathbb{C}$ given by

$T_{y}f(s):=f(\pi(s,y))=(f\circ\pi_{s})(y).$

It follows from the continuity of $f$ that $T_{y}f\in AP(K)$ (we provide a
general proof of this fact while discussing the proof of the next main
theorem, as well). Hence in particular, passing through a subnet if necessary,
without loss of generality, for each function $h\in A_{f}(C)$ we have that

$\displaystyle h(z)\ =\ h(\lim_{\alpha}\ z_{\alpha})$ $\displaystyle=$
$\displaystyle\lim_{\alpha}h\Big{(}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}\pi(s_{i}^{\alpha},y)\Big{)}$
$\displaystyle=$
$\displaystyle\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}h(\pi(s_{i}^{\alpha},y))$
$\displaystyle=$
$\displaystyle\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}T_{y}h(s_{i}^{\alpha})$
$\displaystyle=$
$\displaystyle\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}E_{s_{i}^{\alpha}}(T_{y}h)$
$\displaystyle=$ $\displaystyle\lim_{\alpha}\psi_{\alpha}(T_{y}h)$
$\displaystyle=$ $\displaystyle m(T_{y}h)\ =\ m\big{(}L_{s}(T_{y}f)\big{)}$

for each $s\in K$, where the second last equality holds true since $T_{y}h\in
AP(K)$ and the last equality follows from the left translation invariance of
the mean $m$. Hence for each $s\in K,f\in A_{f}(C)$ we attain the following
invariance:

$\displaystyle h(z)\ =\ m\big{(}L_{s}(T_{y}h)\big{)}$ $\displaystyle=$
$\displaystyle\lim_{\alpha}\psi_{\alpha}\big{(}L_{s}(T_{y}h)\big{)}$
$\displaystyle=$
$\displaystyle\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}E_{s_{i}^{\alpha}}\big{(}L_{s}(T_{y}h)\big{)}$
$\displaystyle=$
$\displaystyle\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}L_{s}(T_{y}h)(s_{i}^{\alpha})$
$\displaystyle=$
$\displaystyle\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}\int_{K}T_{y}h(\zeta)\
d(p_{s}*p_{s_{i}^{\alpha}})(\zeta)$ $\displaystyle=$
$\displaystyle\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}\int_{K}h(\pi_{\zeta}(y))\
d(p_{s}*p_{s_{i}^{\alpha}})(\zeta)$ $\displaystyle=$
$\displaystyle\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}h\Big{(}\int_{K}\pi_{\zeta}(y)\
d(p_{s}*p_{s_{i}^{\alpha}})(\zeta)\Big{)}$ $\displaystyle=$ $\displaystyle
h\Big{(}\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}\int_{K}\pi_{\zeta}(y)\
d(p_{s}*p_{s_{i}^{\alpha}})(\zeta)\Big{)}$ $\displaystyle=$ $\displaystyle
h\Big{(}\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}(\pi_{s}\circ\pi_{s_{i}^{\alpha}})(y)\Big{)}$
$\displaystyle=$ $\displaystyle
h\Big{(}\pi_{s}\big{(}\lim_{\alpha}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}\pi_{s_{i}^{\alpha}}(y)\big{)}\Big{)}$
$\displaystyle=$ $\displaystyle h\Big{(}\pi_{s}(\lim_{\alpha}\
z_{\alpha})\Big{)}\ =\ h(\pi(s,z)),$

where the seventh and eighth equality follows dince $h$ is affine and
continuous respectively and the third last equality follows since $\pi$ is
separately continuous and affine. But note that $A_{f}(C)$ separates the
points of $C$. Hence we must have that $\pi(s,z)=z$ for each $s\in K$, as
required ∎

######  Theorem 3.4.

Let $K$ be a semitopological semihypergroup. Then $AP(K)$ is left amenable if
and only if every separately continuous, non-expansive action of $K$ on a
compact convex subset $C$ of a separated locally convex space $E$ has a common
fixed point.

###### Proof.

First, let $\psi$ be a LIM on $AP(K)$, and $\pi$ be a separately continuous,
non-expansive action of $K$ on a compact convex subset $C$ of a separated
locally convex space $E$.

It follows from a standard application of Zorn’s lemma that there exists a
$X\subseteq C$ where $X$ is a non-empty minimal closed, convex,
$\pi$-invariant subset. Furthermore, using the same technique for the set $X$,
we can find a non-empty minimal closed $\pi$-invariant subset $F\subseteq
X\subseteq C$. We fix an element $y\in F$ and consider the map
$\phi:C(F)\rightarrow\mathbb{C}$ given by

$\phi(f):=\psi(T_{y}^{\pi}f)$

for each $f\in C(F)$. Note that the map is well defined since $T_{y}^{\pi}f\in
AP(K)$ as each $f$ is continuous. Since $\psi$ is a mean on $AP(K)$, it
follows by construction that $\phi$ is a mean on $C(F)$ as well.

Now for each $s\in K$, $f\in C(F)$ we define the map
$N_{s}f:F\rightarrow\mathbb{C}$ given by

$N_{s}f(x):=f(\pi(s,x))=f\circ\pi_{s}(x)=T_{x}^{\pi}f(s),$

for each $x\in F$. It is immediate that $N_{s}f$ is continuous since both
$f,\pi_{s}$ are continuous. Furthermore, note that
$T_{y}^{\pi}N_{s}f=L_{s}T_{y}^{\pi}f$ for any $s\in K$ since for any $s,t\in
K$ we have that

$\displaystyle T_{y}^{\pi}N_{s}f(t)$ $\displaystyle=$ $\displaystyle
N_{s}f(\pi(t,y))$ $\displaystyle=$ $\displaystyle
f\Big{(}\pi(s,\pi(t,y))\big{)}$ $\displaystyle=$
$\displaystyle\int_{K}f\big{(}\pi(\zeta,y)\big{)}\ d(p_{s}*p_{t})(\zeta)$
$\displaystyle=$ $\displaystyle\int_{K}T_{y}^{\pi}f(\zeta)\
d(p_{s}*p_{t})(\zeta)$ $\displaystyle=$ $\displaystyle T_{y}^{\pi}f(s*t)\ =\
L_{s}T_{y}^{\pi}f(t).$

Hence for each $s\in K$, $f\in C(F)$ we have the following equality:

$\displaystyle\phi(N_{s}f)$ $\displaystyle=$
$\displaystyle\psi(T_{y}^{\pi}N_{s}f)$ $\displaystyle=$
$\displaystyle\psi(L_{s}T_{y}^{\pi}f)$ $\displaystyle=$
$\displaystyle\psi(T_{y}^{\pi}f)\ =\ \phi(f),$

where the third equality follows since $\psi$ is left invariant on $AP(K)$.
Now, let $\lambda$ be the probability measure on $F$ such that

$\phi(f)=\int_{F}f\ d\lambda$

for each $f\in C(F)$. For each $s\in K$ and any Borel measurable set $B\subset
F$ consider the set

$\tilde{s}B:=\\{x\in F:\pi(s,x)\in B\\}.$

Note that $\tilde{s}B$ will also be a Borel set in $F$ since $\pi_{s}$ is
continuous, and hence induces a Borel measure $\tilde{s}\lambda$ on $F$ for
each $s\in K$ defined as

$\tilde{s}\lambda(B):=\lambda(\tilde{s}B)=\lambda\big{(}\pi_{s}^{-1}(B)\big{)}.$

In particular, for each $s\in K$ and any continuous function $f$ on $F$, since
$\phi(N_{s}f)=\phi(f)$, we have that

$\displaystyle\int_{F}f\ d\lambda$ $\displaystyle=$
$\displaystyle\int_{F}N_{s}f\ d\lambda$ $\displaystyle=$
$\displaystyle\int_{F}f\circ\pi_{s}(x)\ d\lambda(x)$ $\displaystyle=$
$\displaystyle\int_{F}f(z)\ d(\lambda(\pi_{s}^{-1}(z))\ =\ \int_{F}f\
d(\tilde{s}\lambda).$

Hence for each Borel set $B\subseteq F$ and any $s\in K$ we have that
$\tilde{s}\lambda(B)=\lambda(B)=\lambda(\pi_{s}^{-1}(B))$.

Now, let $\mathcal{F}$ be the family of all closed subsets $A\subseteq F$ such
that $\lambda(A)=1$. Consider the set

$F_{0}:=\bigcap_{A\in\mathcal{F}}A=supp(\lambda).$

In particular, for any $s\in K$ since $\tilde{s}A\in\mathcal{(}F)$ for each
$A\in\mathcal{F}$, we have that $F_{0}\subseteq\pi_{s}^{-1}(F_{0})$, i.e,
$\pi_{s}(F_{0})\subseteq F_{0}$. Hence by minimality of the closed
$\pi$-invariant subset $F\subseteq C$, we have that

$F=F_{0}=supp(\lambda).$

On the other hand, for each $s\in K$ we have that
$\lambda(\pi_{s}(F))=\lambda(\pi_{s}^{-1}(\pi_{s}(F)))=\lambda(F)=1$. Hence
$\pi_{s}(F)\in\mathcal{F}$, i.e, $F\supseteq\pi_{s}(F)\supseteq F_{0}=F$.

Thus we have that $\pi_{s}(F)=F$ for each $s\in K$. Hence if $F$ is singleton,
it serves as the required common fixed point of $\pi$. We claim that $F$ must
be singleton in this case, and prove this by contradiction.

If possible, let $F$ contains at least two distinct points. Then there exists
a continuous seminorm $p\in Q$ such that $r:=sup\\{p(x-y):x,y\in F\\}>0$. Then
there exists some $z_{0}\in\bar{co}F$ such that

$0<r_{0}:=sup\\{p(z_{0}-x):x\in F\\}<r.$

Now set $X_{0}:=X\cap\Big{(}\cap_{x\in F}B_{p}[x,r_{0}]\Big{)}$, where
$B_{p}[x,r_{0}]$ denotes the closed $p$-ball $\\{z\in X:p(z-x)\leq r_{0}\\}$.
Since $z_{0}\in X_{0}$ and $r_{0}<r$, we see that $X_{0}$ is a non-empty
closed, convex, proper subset of $X$.

In particular, $F\subseteq B_{p}[x,r_{0}]$ for each $x\in X_{0}$ since
$p(z-x)\leq r_{0}$ for each $z\in F$ by construction of $X_{0}$. Since $\pi$
is non-expansive, for any $s\in K$, $x\in X_{0},z\in F$ we have that

$p(\pi(s,x)-\pi(s,z))\leq p(x-z)\leq r_{0}.$

Therefore $F=\pi_{s}(F)\subseteq B_{p}[\pi(s,x),r_{0}]$, and hence
$p(z-\pi(s,x))\leq r_{0}$ for each $z\in F$, $s\in K,x\in X_{0}$.

Hence finally, for any $x\in X_{0}$ we have that $\pi_{s}(x)\in
B_{p}[z,r_{0}]$ for each $z\in F$, i.e, the closed, convex set $X_{0}$ is
$\pi$-invariant as well. This contradicts the minimality of $X$ since
$X_{0}\subseteq X$ is a proper subset. Therefore our claim that $F$ must be
singleton is true, providing us with a common fixed point of the action $\pi$.

The converse of the statement follows similar lines as the proof pf the
respective direction of the statement in Theorem 3.3.

∎

These fixed-point properties can further be improved by taking the locally
convex space to be a dual Banach space $X^{*}$ in particular, equipped with
the weak∗ topology.

######  Theorem 3.5.

$AP(K)$ is left amenable if and only if every separately continuous,
equicontinuous, affine action of $K$ on a dual Banach space $X^{*}$ which
admits an element $\omega_{0}\in X^{*}$ such that the orbit
$\\{\pi(s,\omega_{0}):s\in K\\}$ is bounded, has a common fixed point.

###### Proof.

First, assume that any separately continuous, equicontinuous, affine action
$\pi$ of $K$ on a dual Banach space $X^{*}$ for which we can find an element
$\omega_{0}\in X^{*}$ such that the orbit

$\mathcal{O}(\omega_{0})=\\{\pi(s,\omega_{0}):s\in K\\}$

is bounded, admits a common fixed point of the action $\pi$.

Given any $\mu\in M(K),f\in C(K)$, consider the averaging function

$L_{\mu}f:=\int_{K}L_{x}f\ d\mu(x).$

Then for any $s,t\in K$ we have that

$\displaystyle R_{t}(L_{\mu}f)(s)$ $\displaystyle=$ $\displaystyle
L_{\mu}f(s*t)$ $\displaystyle=$ $\displaystyle\int_{K}L_{\mu}f(u)\
d(p_{s}*p_{t})(u)$ $\displaystyle=$ $\displaystyle\int_{K}\int_{K}L_{x}f(u)\
d\mu(x)\ d(p_{s}*p_{t})(u)$ $\displaystyle=$
$\displaystyle\int_{K}\int_{K}f(x*u)\ d(p_{s}*p_{t})(u)\ d\mu(x)$
$\displaystyle=$ $\displaystyle\int_{K}L_{x}f(s*t)\ d\mu(x)\ =\
\int_{K}R_{t}(L_{x}f)(s)\ d\mu(x).$

Hence for any $t\in K$ we have that

$R_{t}(L_{\mu}f)=\int_{K}R_{t}(L_{x}f)\ d\mu(x)=\int_{K}L_{x}(R_{t}f)\
d\mu(x)=L_{\mu}(R_{t}f).$

Thus we see that if $f\in AP(K)$, then so is $L_{\mu}f$ for each $\mu\in M(K)$
since the map $f\mapsto L_{\mu}f:C(K)\rightarrow C(K)$ is continuous, as for
any $f,g\in C(K)$ we have that

$\displaystyle|L_{\mu}f(s)-L_{\mu}g(s)|$ $\displaystyle=$
$\displaystyle\Big{|}\int_{K}L_{x}(f-g)(s)\ d\mu(x)\Big{|}$
$\displaystyle\leq$ $\displaystyle\int_{K}|L_{x}(f-g)(s)|\ d|\mu|(x)$
$\displaystyle\leq$ $\displaystyle\int_{K}||L_{x}(f-g)||_{\infty}\ d|\mu|(x)$
$\displaystyle\leq$ $\displaystyle||f-g||_{\infty}|\mu|(K).$

Now we consider the map $\Phi:M(K)\times AP(K)\rightarrow AP(K)$ given by

$(\mu,f)\mapsto L_{\mu}f:=\int_{K}L_{x}f\ d\mu(x)$

for each $\mu\in M(K),f\in AP(K)$. We claim that $\Phi$ induces an
$M(K)$-module action on $AP(K)$. To prove this, we pick any two elements
$\mu,\nu\in M(K)$ and note that for any $s\in K$ we have that

$\displaystyle L_{\mu*\nu}f(s)$ $\displaystyle=$
$\displaystyle\int_{K}L_{x}f(s)\ d(\mu*\nu)(x)$ $\displaystyle=$
$\displaystyle\int_{K}R_{s}f(x)\ d(\mu*\nu)(x)$ $\displaystyle=$
$\displaystyle\int_{K}\int_{K}R_{s}f(x*z)\ d\mu(x)\ d\nu(z)$ $\displaystyle=$
$\displaystyle\int_{K}\int_{K}L_{x}f(z*s)\ d\mu(x)\ d\nu(z)$ $\displaystyle=$
$\displaystyle\int_{K}L_{z}(L_{\mu}f)(s)\ d\nu(z)$ $\displaystyle=$
$\displaystyle L_{\nu}(L_{\mu}f)$

where the fourth equality follows since $f(x*y*z)=L_{x}f(y*z)=R_{z}f(x*y)$ for
any $x,y,z\in K$ and any bounded measurable function $f$ on $K$. Moreover, the
second last equality follows since

$\displaystyle L_{z}(L_{\mu}f)(s)\ =\ L_{\mu}f(z*s)$ $\displaystyle=$
$\displaystyle\int_{K}L_{\mu}f(t)\ d(p_{z}*p_{s})(t)$ $\displaystyle=$
$\displaystyle\int_{K}\int_{K}L_{x}f(t)\ d\mu(x)\ d(p_{z}*p_{s})(t)$
$\displaystyle=$ $\displaystyle\int_{K}L_{x}f(t)\ d(p_{z}*p_{s})(t)\ d\mu(x)\
=\ \int_{K}L_{x}f(z*s)\ d\mu(x).$

Hence $E:=AP(K)$ is indeed a right $M(K)$-module with the module action
$\Phi:\mu\mapsto L_{\mu}$. Now for any scalar $\alpha\in\mathbb{C}$, let
$\alpha\mathbf{1}$ denote the constant function taking $\alpha$ on $K$. Then

$\displaystyle L_{\mu}(\alpha\mathbf{1})(s)$ $\displaystyle=$
$\displaystyle\int_{K}L_{x}(\alpha\mathbf{1})(s)\ d\mu(x)$ $\displaystyle=$
$\displaystyle\int_{K}(\alpha\mathbf{1})(x*s)\ d\mu(x)$ $\displaystyle=$
$\displaystyle\int_{K}\int_{K}(\alpha\mathbf{1})(z)\ d(p_{x}*p_{s})\ d\mu(x)$
$\displaystyle=$ $\displaystyle\alpha\int_{K}(p_{x}*p_{s})(K)\ d\mu(x)\ =\
\alpha\mu(K).$

Thus $\mathbb{C}\mathbf{1}$ is a submodule of $E$, and hence the quotient
$X:=E/\mathbb{C}\mathbf{1}$ is a Banach right $M(K)$-module. Now we consider
the Banach left $M(K)$-module $X^{*}$. Note that we can identify $X^{*}$ with
the left submodule $\mathbb{C}\mathbf{1}^{\perp}$ of $E^{*}$ given by

$\mathbb{C}\mathbf{1}^{\perp}:=\\{u\in E^{*}:u(\mathbf{1})=0\\}.$

where the standard left module action is given by $\mu\mapsto L_{\mu}^{*}$.

Next, we equip $X^{*}$ with the weak∗ topology and consider the map
$\Psi:M(K)\times(X^{*},weak^{*})\rightarrow(X^{*},weak^{*})$ given by

$\Psi(\mu,\phi):=L_{\mu}^{*}\phi.$

In particular, for each $\mu\in M(K),\phi\in X^{*},f\in X$ we have that
$L_{\mu}^{*}\phi(f)=\phi(L_{\mu}f)$. Now for any $\mu,\nu\in M(K)$ and
$\phi,\psi\in X^{*}$ the following holds true for each $f\in X$:

$\displaystyle\Big{|}\Big{(}\Psi(\mu,\phi)-\Psi(\nu,\psi)\Big{)}(f)\Big{|}$
$\displaystyle=$ $\displaystyle\Big{|}\phi(L_{\mu}f)-\psi(L_{\nu}f)\Big{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}\phi(L_{\mu}f)-\phi(L_{\nu}f)\Big{|}+\Big{|}\phi(L_{\nu}f)-\psi(L_{\nu}f)\Big{|}$
$\displaystyle\leq$ $\displaystyle||\phi||\
||L_{\mu}f-L_{\nu}f||_{\infty}+||\phi-\psi||\ ||L_{\nu}f||_{\infty}$
$\displaystyle\leq$ $\displaystyle||\phi||\ ||f||_{\infty}\
||\mu-\nu||+||\phi-\psi||\ ||f||_{\infty}\ ||\nu||$ $\displaystyle=$
$\displaystyle\ \Big{(}||\phi||\ ||\mu-\nu||+||\nu||\
||\phi-\psi||\Big{)}||f||_{\infty},$

where the third inequality follows since for each $s\in K$ we have

$|(L_{\mu}f-L_{\nu}f)(s)|=|\int_{K}L_{x}f\
d(\mu-\nu)(x)|\leq\int_{K}||L_{x}f||_{\infty}\
d(|\mu-\nu|)(x)\leq||f||_{\infty}\ |\mu-\nu|(K)$

and similarly

$|L_{\nu}f(s)|\leq\int_{K}||L_{x}f||_{\infty}\ d|\nu|(x)\leq||f||_{\infty}\
|\nu|(K).$

Hence $\Psi$ is jointly continuous.

Next, we choose some $v_{0}\in AP(K)^{*}$ such that $v_{0}(\mathbf{1})=1$.
Then for any $\mu\in M(K)$ we have that

$L_{\mu}^{*}v_{0}(\mathbf{1})=v_{0}(L_{\mu}(\mathbf{1})=v_{0}(||\mu||\mathbf{1})=||\mu||.$

Hence in particular, we have that
$\big{(}L_{p_{s}}^{*}v_{0}-v_{0}\big{)}\in\mathbb{C}\mathbf{1}^{\perp}\simeq
X^{*}$. We consider the map $\pi:K\times X^{*}\rightarrow X^{*}$ given by
$\pi(s,u):=T_{s}u$ for each $s\in K,u\in X^{*}$ where $T_{s}u$ is given by

$T_{s}u:=L_{p_{s}}^{*}(u+v_{0})-v_{0}.$

Now in order to prove that $\pi$ indeed is an action of $K$ on $X^{*}$, we
derive the following equality for any $s,t\in K$, $u\in X^{*}$ and for each
$f\in X$.

$\displaystyle T_{s}(T_{t}u)(f)$ $\displaystyle=$ $\displaystyle
L_{p_{s}}^{*}(T_{t}u)(f)+L_{p_{s}}^{*}v_{0}(f)-v_{0}(f)$ $\displaystyle=$
$\displaystyle T_{t}u(L_{p_{s}}f)+v_{0}(L_{p_{s}}f)-v_{0}(f)$ $\displaystyle=$
$\displaystyle T_{t}u(L_{s}f)+v_{0}(L_{s}f)-v_{0}(f)$ $\displaystyle=$
$\displaystyle\Big{(}L_{p_{t}}^{*}u(L_{s}f)+L_{p_{t}}^{*}v_{0}(L_{s}f)-v_{0}(L_{s}f)\Big{)}+v_{0}(L_{s}f)-v_{0}(f)$
$\displaystyle=$ $\displaystyle(u+v_{0})(L_{t}(L_{s}f))-v_{0}(f)$
$\displaystyle=$ $\displaystyle(u+v_{0})(L_{(p_{s}*p_{t})}f)-v_{0}(f)$
$\displaystyle=$ $\displaystyle(u+v_{0})\Big{(}\int_{K}L_{\zeta}f\
d(p_{s}*p_{t})(\zeta)\Big{)}-v_{0}(f)$ $\displaystyle=$
$\displaystyle\int_{K}\Big{(}u(L_{\zeta}f)+v_{0}(L_{\zeta}f)\Big{)}\
d(p_{s}*p_{t})(\zeta)-v_{0}(f)\int_{K}\ d(p_{s}*p_{t})$ $\displaystyle=$
$\displaystyle\int_{K}\Big{(}u(L_{p_{\zeta}}f)+v_{0}(L_{p_{\zeta}}f)-v_{0}(f)\Big{)}\
d(p_{s}*p_{t})(\zeta)$ $\displaystyle=$
$\displaystyle\int_{K}\Big{(}L_{p_{\zeta}}^{*}u(f)+L_{p_{\zeta}}^{*}v_{0}(f))-v_{0}(f)\Big{)}\
d(p_{s}*p_{t})(\zeta)$ $\displaystyle=$ $\displaystyle\int_{K}T_{\zeta}(f)\
d(p_{s}*p_{t})(\zeta).$

Thus we see that $\pi$ is a jointly continuous affine representation of $K$ on
$X^{*}$. Moreover, fixing any $u_{0}\in X^{*}$ we have that

$\displaystyle|T_{s}u_{0}(f)|$ $\displaystyle=$
$\displaystyle|L_{p_{s}}^{*}(u_{0}+v_{0})(f)-v_{0}(f)|$ $\displaystyle=$
$\displaystyle|(u_{0}+v_{0})(L_{s}f)-v_{0}(f)|$ $\displaystyle\leq$
$\displaystyle\big{(}||u_{0}+v_{0}||+||v_{0}||\big{)}\ ||f||_{\infty},$

for each $s\in K,f\in X$, i.e, the orbit $\mathcal{O}(u_{0})$ is bounded.
Hence the given fixed point property ensures that $\pi$ has a common fixed
point $w_{0}\in X^{*}\simeq\mathbb{C}\mathbf{1}^{\perp}$, i.e,

$w_{0}=T_{s}w_{0}=L_{p_{s}}^{*}(w_{0}+v_{0})-v_{0}$

for each $s\in K$. In particular, setting $m_{0}:=w_{0}+v_{0}$ we have that

$m_{0}(L_{s}f)=m_{0}(L_{p_{s}}f)=(w_{0}+v_{0})(L_{p_{s}}f)=L_{p_{s}}^{*}(w_{0}+v_{0})=w_{0}+v_{0}=m_{0},$

for each $s\in K,f\in AP(K)$. Since $w_{0}(\mathbf{1})=0$, it immediately
follows from the choice of $v_{0}$ that $m_{0}(\mathbb{1})=1$. Finally, it is
immediate from the duality property of functionals that $m_{0}$ is indeed a
mean, and hence a LIM on $AP(K)$.

The converse of the statement follows as an application of the Krein- Milman
theorem, using the boundedness of the orbit. ∎

######  Remark 3.6.

Note that although these characterizations answer certain open questions
affirmatively in terms of the interplay between amenability and fixed points
of actions in the setting of coset spaces, orbit space, hypergroups and in
general semitopological semihypergroups, there are still many open questions
which remain unanswered. For example, it is not known if similar
characterizations of amenability is possible for non-affine actions of a
semihypergroup in this setting. Furthermore, it is not known whether the
existence of a bounded orbit in Theorem 3.5 can be replaced by some other
equivalent condition on the representation, which is easier to check and more
conventional in nature. Finally, it is only natural that the common fixed
points of such actions would be intrinsically connected with the concept of
$F$-algebraic amenability. Other that the interplays investigated in [6, 16]
for certain kinds of general actions, this area remains unexplored so far.

## Acknowledgement

The author would like to sincerely thank her doctoral thesis advisor Dr.
Anthony To-Ming Lau, for suggesting the topic of this study and for the
helpful discussions during the initiation of this work. She would also like to
gratefully acknowledge the financial support provided by the Indian Institute
of Technology Kanpur, India, Harish-Chandra Research Institute, India and SRM
University AP, India during the preparation of this manuscript.

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