# Fixed Points and Continuity of Semihypergroup Actions

Choiti Bandyopadhyay  Department of Mathematical and Statistical Sciences,
University of Alberta, Canada  Current Address: Department of Mathematics,
Indian Institute of Technology Kanpur, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In a couple of previous papers [2, 3], 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 this article, we
introduce and examine (separately) continuous actions on the category of
semihypergroups. In particular, we discuss the continuity properties of such
actions and explore the equivalence relations between different fixed-point
properties of certain actions and the existence of left-invariant mean(s) on
the space of almost periodic functions on a semihypergroup.

###### 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

Part of this work is included in the author’s PhD thesis at the University of
Alberta

##  1\. Introduction

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 a prominent area of
research since its inception. 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 [7, 10, 11, 13, 14, 15]
for example, among others). In this article, we initiate the study of
semihypergroup actions and pursue similar results in the more general setting
of semihypergroups, in order to thereby realise where and why this theory
deviates from the classical theory of locally compact semigroups.

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[9],
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, 9] 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 couple of previous
papers [2, 3] 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. In the
first two installments of the study, we introduced and developed the concepts
and theories of homomorphisms, ideals, kernels, almost periodic and weakly
almost periodic functions and free products on a semihypergroup.

In this article, we advance the theory by introducing and investigating
actions on the category of semihypergroups. In particular, we introduce and
discuss the relation between several possible natural definitions of
semihypergroup actions, investigate their continuity properties, and finally
proceed to examine the equivalence between amenability properties on the space
of almost periodic functions of a semihypergroup, and the existence of certain
natural fixed point properties of semihypergroup actions. We see that the
classical equivalences in the category of semigroups, do not always extend to
semihypergroups for a number of obstacles and differences in the product-
structure of these two classes of objects, as discussed before. For example,
unlike topological semigroups, the spaces of almost periodic functions on a
semihypergroup do not necessarily form an algebra. 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 [9], and introduce some new
definitions required for further development. We conclude the section with
listing some important examples of semihypergroups and hypergroups.

In the third section, we initiate the study of semihypergroup actions. We
first introduce a ‘general action’ of a semihypergroup $K$ on a Hausdorff
topological space $X$, and investigate when separate continuity of such an
action forces joint continuity on the underlying product space. Later we
introduce a couple of natural definitions of semihypergroup actions on a
locally convex space and discuss their structural equivalence, as well as
their relation to general semihypergroup actions, and the immediate
implications to the continuity conditions.

In the final, i.e, fourth section of the article, we investigate the existence
and equivalence-relations between two different fixed-point properties and
amenability on the space $AP(K)$ of almost periodic functions of a
(semitoplogical) semihypergroup. We first show that any continuous affine
action of a commutative semihypergroup must have a fixed point. On the other
hand, we see that for any semihypergroup $K$, if any continuous affine action
of $K$ on a compact convex space has a fixed point, then $AP(K)$ must admit a
left-invariant mean. Next we discuss the existing Arens product structure [1,
2] on $AP(K)^{*}$, and thereby provide some necessary and sufficient
conditions for $K$ to admit a left invariant mean on $AP(K)$. Now finally, we
pursue similar equivalence conditions in terms of the measure algebra $M(K)$
of $K$. We consider a different natural fixed point property in terms of
$M(K)$, and thereby acquire a complete characterization of amenability
properties on $AP(K)$.

##  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 [9], 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
[12] 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 [9] 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
$AP(K)\subseteq UC(K)$.

Now recall [9] that for any locally compact Hausdorff space $X$, a map
$i:X\rightarrow X$ is called a topological involution if $i$ is a
homeomorphism and $(i\circ i)(x)=x$ for each $x\in X$. On a semitopological
semihypergroup $(K,*)$, a topological involution $i:K\rightarrow K$ given by
$i(x):=x^{-}$ is called a (semihypergroup) involution if
$(\mu*\nu)^{-}=\nu^{-}*\mu^{-}$ for any $\mu,\nu\in M(K)$. For any measure
$\omega\in M(K)$, we have that

$\omega^{-}(B):=\omega(B^{-})=\omega(i(B)),$

for any Borel measurable subset $B$ of $K$. As expected, an involution on a
semihypergroup is analogous to the inverse function on a semigroup. Hence a
semihypergroup with an identity and an involution of the following
characteristic is a hypergroup.

######  Definition 2.3 (Hypergroup).

A semihypergroup $(H,*)$ is called a hypergroup if it satisfies the following
conditions :

(A5):

There exists an element $e\in H$ such that $p_{x}*p_{e}=p_{e}*p_{x}=p_{x}$ for
any $x\in H$.

(A6):

There exists an involution $x\mapsto x^{-}$ on $H$ such that
$e\in\mbox{supp}(p_{x}*p_{y})$ if and only if $x=y^{-}$.

The element $e$ in the above definition is called the identity of $H$. Note
that the identity and involution of a hypergroup are necessarily unique [9].

######  Remark 2.4.

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 [9], one can
further achieve a continuous extension of ‘$*$’ to $M^{+}(K)$ and hence to the
whole of $M(K)$ using bilinearity.

Finally, the centre $Z(K)$ of a (semitopological) semihypergroup $(K,*)$ is
defined to be the largest semigroup included in $(K,*)$. In other words, we
have that

$Z(K):=\\{x\in K:\mbox{ supp}(p_{x}*p_{y})\mbox{ and supp}(p_{y}*p_{x})\mbox{
are singleton for any }y\in K\\}.$

Note that it is possible that $Z(K)=\O$, and even if $x\in Z(K)$, then
$(p_{x}*p_{y})$ and $(p_{y}*p_{x})$ need not be supported on the same element
in $K$, for each $y\in K$.

But if $K$ is a hypergroup, then we immediately see that $e\in Z(K)$. In fact,
if $K$ is a hypergroup, then the centre $Z(K)$ is indeed the largest group
included in $K$, and it can easily be checked [5] that the following
equivalence holds true.

$Z(K):=\\{x\in K:p_{x}*p_{x^{-}}=p_{x^{-}}*p_{x}=p_{e}\\}.$

Hence the definition for center of a hypergroup[9] defined as above, coincides
with that of the center of a semihypergroup. The following is an example of a
hypergroup with a non-trivial centre.

######  Example 2.5 (Zeuner [18]).

Consider the hypergroup $(H,*)$ where $H=[0,1]$ and the convolution is defined
as

$p_{s}*p_{t}=\frac{p_{|s-t|}+p_{1-|1-s-t|}}{2}.$

We immediately see that $Z(H)=\\{0,1\\}$.

Now we list some well known examples [9, 18] 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 2.6.

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 2.7.

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 2.8.

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 2.9.

Let $G$ be a locally compact topological group and $H$ be any compact group
with normalized Haar measure $\sigma$. For any continuous affine action[9]
$\pi$ 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 [5, 9] 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\. General Semihypergroup Action and Continuity

We first provide a general definition of action of a semihypergroup on a
Hausdorff topological space, analogous to its definition for the classical
case of topological semigroups. Hence in particular, whenever we take the
semihypergroup to be a locally compact semigroup as outlined in Example 2.6,
the definition coincides with that of topological semigroups.

######  Definition 3.1.

Let $(K,*)$ be a semihypergroup and $X$ be a Hausdorff topological space. A
map $\sigma:M^{+}(K)\times X\rightarrow X$ is called a general action of $K$
on $X$ if the following two conditions hold true:

  1. (1)

For each $\omega\in M^{+}(K)$ the map $\sigma_{\omega}:X\rightarrow X$ given
by $\sigma_{\omega}(x):=\sigma(\omega,x)$ is continuous.

  2. (2)

For any $\mu,\nu\in M^{+}(K)$, $x\in X$ we have that
$\sigma(\mu*\nu,x)=\sigma(\mu,\sigma(\nu,x))$

We can define general actions of a left/ right/ semitopological semihypergroup
$K$ on a Hausdorff space $X$ in the same manner as above.

Let $\sigma:M^{+}(K)\times X\rightarrow X$ be a general action of $(K,*)$ on
$X$. Then $\sigma$ is called a seperately continuous general action, if for
each $x\in X$ the map $\mu\mapsto\sigma(\mu,x):M^{+}(K)\rightarrow X$ is also
continuous on $M^{+}(K)$ in the cone topology. Similarly, we say that $\sigma$
is a continuous general action, if $\sigma$ is jointly continuous on
$M^{+}(K)\times X$. Finally, for any two subsets $N\subseteq M^{+}(K)$ and
$V\subseteq X$, we define

$N.V:=\\{\sigma(\mu,x):\mu\in N,x\in V\\}.$

We see that if a a left (right) topological semihypergroup $(K,*)$ has an
identity $e$, then given any two distinct points in a Hausdorff space $X$, a
certain general action $\sigma$ of $K$ on $X$ will always map each point away
from the other. In particular, we have the following assertion.

######  Proposition 3.2.

Let $(K,*)$ be a compact left topological semihypergroup, and $\sigma$ be a
separately continuous general action of $K$ on a Hausdorff space $X$.
Furthermore, let $K$ has an identity $e$ and $\sigma_{p_{{}_{e}}}$ is the
identity map on $X$. Then for any two distinct points $x,y$ in $X$, there
exist neighborhoods $N$ of $p_{{}_{e}}$, $U$ of $x$ and $V$ of $y$ such that
$N.U\cap V=\O$.

###### Proof.

Using Urysohn’s Lemma, we first choose a continuous function
$f:X\rightarrow[-1,1]$ such that $f(x)\neq f(y)=0$. Set $g:=f\circ\sigma$.

Since $K$ is compact, the cone toplogy coincides with the weak∗-topology, and
hence $M^{+}(K)$ is a locally compact Hausdorff space [8] here. Also since
$\sigma$ is separately continuous, the function $g:M^{+}(K)\times
X\rightarrow[-1,1]$ is separately continuous. Hence we can find [16] a dense
$G_{\delta}$ subset $M$ of $M^{+}(K)$ such that $g$ is continuous at each
point $(\mu,x)$ in $M\times X$. Now set

$S:=\\{\mu\in M^{+}(K):g(\mu,x)\neq g(\mu,y)\\}.$

Note that $S$ is non-empty since $\sigma_{p_{e}}$ is identity, and hence we
have

$\displaystyle g(p_{e},x)$ $\displaystyle=$ $\displaystyle f(\sigma(p_{e},x))$
$\displaystyle=$ $\displaystyle f(x)\ \neq\ f(y)\ =\ f(\sigma(p_{e},y))\ =\
g(p_{e},y).$

Also, $S$ is open in $M^{+}(K)$ since the map $\mu\mapsto
g(\mu,x)-g(\mu,y):M^{+}(K)\rightarrow\mathbb{R}$ is continuous. Hence $S\cap
M\neq\O$. Pick any $\mu_{0}\in S\cap M$, and in particular, set
$\varepsilon_{0}:=|g(\mu_{0},x)-g(\mu_{0},y)|>0$.

Since $g$ is jointly continuous at $(\mu_{0},x)$, there exist open
neighborhoods $N_{0}$ of $\mu_{0}$ and $U$ of $x$ such that

$|g(\nu,z)-g(\mu_{0},x)|<\varepsilon_{0}/4\mbox{ \ \ \ for any \ \ }\nu\in
N_{0},\ z\in U.$

We further set $V:=\\{z\in
X:|g(\mu_{0},z)-g(\mu_{0},y)|<\varepsilon_{0}/4\\}$. Hence $x\notin V$, and
since $g$ is separately continuous, $V$ is an open neighborhood of $y$.

But $K$ is left topological, and hence the map
$L_{\mu_{0}}:M^{+}(K)\rightarrow M^{+}(K)$ is continuous. In particular, since
$L_{\mu_{0}}(p_{{}_{e}})=\mu_{0}$ and $N_{0}$ is an open neighborhood of
$\mu_{0}$, we can find an open neighborhood $N$ of $p_{{}_{e}}$ in $M^{+}(K)$
such that $\\{\mu_{0}\\}*N=L_{\mu_{0}}(N)\subseteq N_{0}$.

Now if possible, assume that $N.U\cap V\neq\O$. Pick some $\nu\in N$, $u\in
U$, $v\in V$ such that $\sigma(\nu,u)=v\in V$. Then we have

$\displaystyle\varepsilon_{0}=|g(\mu_{0},x)-g(\mu_{0},y)|$ $\displaystyle\leq$
$\displaystyle|g(\mu_{0},x)-g(\mu_{0}*\nu,u)|+|g(\mu_{0}*\nu,u)-g(\mu_{0},y)|$
$\displaystyle<$
$\displaystyle\varepsilon_{0}/4+|g(\mu_{0},\sigma(\nu,u))-g(\mu_{0},y)|$
$\displaystyle<$
$\displaystyle\varepsilon_{0}/4+\varepsilon_{0}/4<\varepsilon_{0},$

where the third inequality follows from the choice of $N$ and $U$, since
$\sigma(\nu,u)\in V$. Thus we arrive at a contradiction, and must have that
$N.U\cap V=\O$. ∎

Next we proceed towards the main result of this section, that investigates the
instances when separate continuity forces joint continuity of the general
action of a left topological semihypergroup on a compact Hausdorff space. The
proof of the result follows similar ideas as outlined in [6], for the category
of compact topological groups and semigroups. We show, with the help of
Proposition 3.2 and some important additional details, that for the general
category of compact left topological semihypergroups, separate continuity of a
general action forces joint continuity on a certain set of points. In
addition, if the general action is linear in $M^{+}(K)$, then the result can
be extended further to the whole measure space of the said set.

We first prove the result for the case of all general actions $\sigma$ of $K$
on $X$, for which $\sigma_{p_{{}_{e}}}$ is the identity map on $X$, where $e$
is the identity of $K$.

######  Theorem 3.3.

Let $(K,*)$ be a compact left topological semihypergroup with identity $e$ and
$\sigma$ be a separately continuous general action of $K$ on a compact
Hausdorff space $X$ such that $\sigma(p_{e},x)=x$ for each $x\in X$. Let
$H\subseteq K$ be a hypergroup. Then $\sigma$ is continuous at each point
$(p_{{}_{g}},x)$, for any $g\in Z(H)$, $x\in X$.

###### Proof.

Pick and fix some $x\in X$ and let $W_{x}$ be an open neighborhood of
$x=\sigma(p_{e},x)$ in $X$.

Pick any $y\in X\setminus W_{x}$. Using Proposition 3.2 we can find open
neighborhoods $N_{y}$ of $p_{e}$, $U_{x}^{{}^{y}}$ of $x$ and $V_{y}$ of $y$
such that $N_{y}.U_{x}^{{}^{y}}\cap V_{y}=\O$. We can find such a set of open
neighborhoods for each $y\in X\setminus W_{x}$. Since $X\setminus W_{x}$ is
compact, there exists $y_{1},y_{2},\ldots,y_{n}\in X\setminus W_{x}$ such that

$X\setminus W_{x}\subseteq\Big{(}\cup_{i=1}^{n}V_{y_{i}}\Big{)}=:V.$

Set $N:=\cap_{i=1}^{n}N_{y_{i}}$ and $U:=\cap_{i=1}^{n}U_{x}^{{}^{y_{i}}}.$
Then clearly $N.U\cap V=\O$. Thus we get open neighborhoods $N$ of $p_{e}$,
$U$ of $x$ such that $N.U\subseteq W_{x}$. Since this is true for any $x\in
X$, we get that $\sigma$ is continuous at $(p_{e},x)$ for each $x\in X$.

Next pick any $g\in Z(H)$, $x\in X$. Let $\\{(\mu_{\alpha},x_{\alpha})\\}$ be
a net in $M^{+}(K)\times X$ that converges to $(p_{g},x)$. Since $K$ is left
topological, $L_{p_{g^{-}}}$ is continuous and hence
$p_{g^{-}}*\mu_{\alpha}\rightarrow p_{g^{-}}*p_{g}=p_{e}$ in $M^{+}(K)$, as
$g\in Z(H)$. But $\sigma$ is continuous at $(p_{e},x)$, and hence
$\sigma(p_{g^{-}}*\mu_{\alpha},x_{\alpha})\rightarrow\sigma(p_{e},x)=x$ in
$X$. Thus finally we have that

$\displaystyle\sigma(\mu_{\alpha},x_{\alpha})$ $\displaystyle=$
$\displaystyle\sigma(p_{e}*\mu_{\alpha},x_{\alpha})$ $\displaystyle=$
$\displaystyle\sigma\big{(}(p_{g}*p_{g^{-}})*\mu_{\alpha},x_{\alpha}\big{)}$
$\displaystyle=$
$\displaystyle\sigma(p_{g},\sigma(p_{g^{-}}*\mu_{\alpha},x_{\alpha}))\longrightarrow\sigma(p_{g},x),$

where the third equality follows since $(M(K),*)$ is associative, and the
convergence follows from the continuity of the map
$\sigma_{p_{{}_{g}}}:X\rightarrow X$. ∎

Now we use Theorem 3.3 to show that a similar conclusion holds true for any
separately continuous general action $\sigma$ of $K$ on $X$.

######  Theorem 3.4.

Let $(K,*)$ be a compact left topological semihypergroup with identity $e$ and
$\sigma$ be a separately continuous general action of $K$ on a compact
Hausdorff space $X$. Let $H\subseteq K$ be a hypergroup. Then $\sigma$ is
continuous at each point $(p_{{}_{g}},x)$, for any $g\in Z(H)$, $x\in X$.

###### Proof.

For each $x\in X$, we define $\tilde{x}=\sigma(p_{e},x)$, and set
$L:=\\{\tilde{x}:x\in X\\}=\\{p_{e}\\}.X$.

For any $\mu\in M^{+}(K)$, $x\in X$ we have that $M^{+}(K).L\subseteq L$,
since

$\displaystyle\sigma(\mu,\tilde{x})\ =\ \sigma(\mu,\sigma(p_{e},x))$
$\displaystyle=$ $\displaystyle\sigma(\mu*p_{e},x)$ $\displaystyle=$
$\displaystyle\sigma(p_{e}*\mu,x)\ =\ \sigma(p_{e},\sigma(\mu,x))\in L.$

In particular, for each $x\in X$ we have that

$\sigma(p_{e},\tilde{x})=\sigma(p_{e},\sigma(p_{e},x))=\sigma(p_{e}*p_{e},x)=\tilde{x}.$

Hence the restricted general action $\tilde{\sigma}:=\sigma|_{M^{+}(K)\times
L}:M^{+}(K)\times L\rightarrow L$ is well defined and
$\tilde{\sigma}_{p_{{}_{e}}}$ is the identity map on $L$. Note that $L$ is
compact in $X$ since the map $\sigma_{p_{{}_{e}}}:X\rightarrow X$ is
continuous and $\sigma_{p_{{}_{e}}}(X)=L$. Moreover, since $\tilde{\sigma}$ is
separately continuous when $L$ is equipped with the subspace topology induced
from $X$, using Theorem 3.3 we have that $\tilde{\sigma}$ is continuous at
each point $(p_{g},\tilde{x})$, for any $g\in Z(H)$, $x\in X$.

Now pick any $g\in Z(H)$, $x\in X$ and let $\\{(\mu_{\alpha},x_{\alpha})\\}$
be any net in $M^{+}(K)\times X$ that converges to $(p_{g},x)$. Since
$\sigma_{p_{{}_{e}}}$ is continuous, the net
$\\{\tilde{x}_{\alpha}\\}:=\\{\sigma(p_{e},x_{\alpha})\\}$ converges to
$\tilde{x}$ in $X$, and hence in $L$. Hence
$\\{(\mu_{\alpha},\tilde{x}_{\alpha})\\}$ converges to $(p_{g},\tilde{x})$ in
$M^{+}(K)\times L$. Thus finally, we have that

$\displaystyle\sigma(\mu_{\alpha},x_{\alpha})$ $\displaystyle=$
$\displaystyle\sigma(\mu_{\alpha}*p_{e},x_{\alpha})$ $\displaystyle=$
$\displaystyle\tilde{\sigma}(\mu_{\alpha},\tilde{x}_{\alpha})\longrightarrow\tilde{\sigma}(p_{g},\tilde{x})=\sigma(p_{g}*p_{e},x)=\sigma(p_{g},x).$

∎

######  Remark 3.5.

Observe that if the general action $\sigma$ in Theorem 3.4 is linear in
$M^{+}(K)$ as well, then using the density of $M_{F}^{+}(K)$ in $M^{+}(K)$ we
can conclude that separate continuity of $\sigma$ forces joint continuity of
$\sigma$ on $M^{+}(Z(H))\times X$.

Keeping this in mind, note that the action of a (semitopological)
semihypergroup $K$ can also be defined in the following manner if the space
$X$ on which $K$ acts, is a locally convex space. A locally convex Hausdorff
topological vector space $E$ with a family $Q$ of seminorms is simply denoted
by $(E,Q)$ or $(E,\tau_{Q})$. In rest of the article, the topology assumed on
$E$ (or on any Borel subset $X$ of $E$), is the (induced) topology $\tau_{Q}$
generated by $Q$.

######  Definition 3.6.

Let $(K,*)$ be a (semitopological) semihypergroup and $X$ be a compact convex
subset of $(E,Q)$. A map $\pi:K\times X\rightarrow X$ is called a
(semihypergroup) action if the following conditions are satisfied:

  1. (1)

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

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

  2. (2)

Whenever $K$ has a two-sided identity $e$, we have $\pi(e,x)=x$ for each $x\in
X$.

Such an action $\pi$ of $K$ on $X$ is called (separately) continuous, if the
map $\pi$ is (separately) continuous on $K\times X$. For each $s\in K$, we
denote the map $x\mapsto\pi(s,x):X\rightarrow X$ by $\pi_{s}$.

######  Remark 3.7.

The above definition of a semihypergroup action is a particular case of
Definition 3.1. This is true since given a (separately) continuous action
$\pi$ of $K$ on a compact convex subset $X$ of a locally convex Hausdorff
vector space $(E,Q)$ in the sense of Definition 3.6, one can simply define

$\sigma_{\pi}(\mu,x):=\int_{K}\pi(\zeta,x)\ d\mu(\zeta).$

Since $\sigma_{\pi}(p_{s},x):=\pi(s,x)$ for any $s\in K,x\in X$, and
$M_{F}^{+}(K)$ is dense in $M^{+}(K)$ in the cone topology, we see that
$\sigma_{\pi}$ is a (separately) continuous general action of $K$ on $X$
(Definition 3.1), since $\mu*\nu=\int_{K}\int_{K}\ (p_{s}*p_{t})\ d\mu(s)\
d\nu(t)$ for each $\mu,\nu\in M^{+}(K)$ [9].

Note that the action $\sigma_{\pi}$ induced by $\pi$ is linear by
construction. But a general action of a semihypergroup $K$ as defined in
Definition 3.1 need not be linear.

In fact, it follows immediately that Definition 3.6 is structurally equivalent
to the following definition of a semihypergroup action, induced by Jewett’s
definition [9] of hypergroup actions.

######  Definition 3.8.

Let $K$ be a semihypergroup, and $(E,Q)$ be a separated locally convex space.
Then a (semihypergroup) action $\pi$ of $K$ on $E$ is a homomorphism from the
associative algebra $M(K)$ to the algebra $L(E)$ of linear operators on $E$.

In other words, a semihypergroup action $\pi$ of $K$ on $E$ is a bilinear map
$(\mu,s)\mapsto\pi_{\mu}(s):M(K)\times E\rightarrow E$ such that
$\pi_{{}_{(\mu*\nu)}}=\pi_{\mu}\circ\pi_{\nu}$ on $E$ for each $\mu,\nu\in
M(K)$.

We say that a semihypergroup action $\pi:M(K)\times E\rightarrow E$ is
(separately) $\tau$-continuous if the map $\pi$ is (separately) continuous
when $M(K)$ is equipped with a certain topology $\tau$ and $E$ is given the
usual topology $\tau_{Q}$ induced by the associated family of seminorms $Q$.

##  4\. Fixed-Point Properties and Amenability

For the first part of this section, we use Definition 3.6 for semihypergroup
actions, unless otherwise specified. Before proceeding to the main results of
this section, we first briefly recall some definitions and results regarding
amenability on function spaces of a semitopological semihypergroup.

Let $K$ be any (semitopological) semihypergroup and $\mathcal{F}$ a linear
subspace of $C(K)$ containing constant functions. A function
$m\in\mathcal{F}^{*}$ is called a mean of $\mathcal{F}$ if we have that
$||m||=1=m(1)$, where $1$ denotes the constant function $\equiv 1$ on $K$. We
denote the set of all means on $\mathcal{F}$ as $\mathcal{M}(\mathcal{F})$. If
$\mathcal{F}$ is a left translation-invariant linear subspace of $C(K)$
containing constant functions, a mean $m$ of $\mathcal{F}$ is called a left
invariant mean (LIM) if $m(L_{x}f)=m(f)$ for any $x\in K$, $f\in\mathcal{F}$.

######  Remark 4.1.

Let $X$ be a convex subset of $(E,Q)$ and $Y$ be any locally convex space.
Then a continuous map $T:X\rightarrow Y$ is called affine if for any
$\alpha\in[0,1]$, $x_{1},x_{2}\in X$ we have that

$T(\alpha x_{1}+(1-\alpha)x_{2})=\alpha T(x_{1})+(1-\alpha)T(x_{2}).$

We say that a (separately) continuous action $\pi$ of $K$ on $X$ is affine if
for each $s\in K$, the map $\pi_{s}:X\rightarrow X$ is affine. We denote by
$A_{f}(X)$ the set of all affine maps $T:X\rightarrow\mathbb{C}$. We know that
$A_{f}(X)$ is a closed linear subspace of $C(X)$, i.e, is itself a convex
space in $C(K)$ [14].

For each $x\in X$, we define the evaluation map
$\varepsilon_{x}:C(X)\rightarrow\mathbb{C}$ as $\varepsilon_{x}(f):=f(x)$ for
each $f\in C(X)$. Since the constant function $1$ is in $A_{f}(X)$, we have
that each $\varepsilon_{x}$ is a mean on $A_{f}(X)$. In fact, it can be shown
[14] that the map
$x\mapsto\varepsilon_{x}|_{A_{f}(X)}:X\rightarrow\mathcal{M}(A_{f}(X))$ is an
affine homeomorphism. The set of all evaluation maps $\\{\varepsilon_{x}:x\in
K\\}$ is denoted by $\varepsilon(K)$.

Similarly, for each $\mu\in M(K)$, we define the evaluation map
$\varepsilon_{\mu}$ at the measure $\mu$ as the functional $f\mapsto\int_{K}f\
d\mu:C(K)\rightarrow\mathbb{C}$. Thus for each $x\in K$, we simply have that
$\varepsilon_{p_{{}_{x}}}=\varepsilon_{x}$ on $C(K)$.

In the above setting, let $\pi$ be an affine action of $K$ on a compact convex
subset $X\subseteq E$. Then $\pi$ naturally induces an affine action of $K$ on
$A_{f}(X)$ when $K$ is commutative. We provide a brief account of this fact
below.

######  Proposition 4.2.

Let $K$ be a commutative semitopological semihypergroup, $X$ be a compact
convex subset of a locally convex Hausdorff space $(E,Q)$ and $\pi$ be an
affine action of $K$ on $X$. Then $\pi$ naturally induces an affine action
$\tilde{\pi}$ of $K$ on $A_{f}(X)$.

###### Proof.

We define $\tilde{\pi}:K\times A_{f}(X)\rightarrow A_{f}(X)$ as

$\tilde{\pi}(s,f)(x):=f(\pi(s,x)),$

for each $s\in K,f\in A_{f}(X),x\in X$. First note that $\tilde{\pi}$ is well-
defined since both $\pi_{s}$ and $f:X\rightarrow\mathbb{C}$ are affine for
each $s\in K$. Moreover, if $K$ has an identity $e$, then for each $f\in
A_{f}(X),x\in X$, we have $\tilde{\pi}_{e}(f)(x)=f(\pi_{e}(x))=f(x)$. Now for
any $s,t\in K$, $f\in A_{f}(X)$, $x\in X$ we have:

$\displaystyle\tilde{\pi}(s,\tilde{\pi}(t,f))(x)\ =\
\tilde{\pi}_{s}(\tilde{\pi}_{t}(f))(x)$ $\displaystyle=$
$\displaystyle\tilde{\pi}_{t}(f)(\pi_{s}(x))$ $\displaystyle=$ $\displaystyle
f(\pi_{t}(\pi_{s}(x)))$ $\displaystyle=$ $\displaystyle
f\Big{(}\int_{K}\pi_{\zeta}(x)\ d(p_{t}*p_{s})(\zeta)\Big{)}$ $\displaystyle=$
$\displaystyle\int_{K}f(\pi_{\zeta}(x))\ d(p_{t}*p_{s})(\zeta)$
$\displaystyle=$ $\displaystyle\int_{K}\tilde{\pi}_{\zeta}(f)(x)\
d(p_{t}*p_{s})(\zeta)\ =\ \int_{K}\tilde{\pi}(\zeta,f)(x)\
d(p_{s}*p_{t})(\zeta).$

where the third last equality follows since $f$ is affine on $X$. Also, since
$A_{f}(X)\subset C(X)$, we have that $\tilde{\pi}$ is (separately) continuous
if $\pi$ is so. ∎

Now we proceed to derive a sufficient condition for a semihypergroup action to
attain a fixed point. We see that a (separately) continuous action of a
commutative semihypergroup $K$ on a compact convex subset $X$ of $(E,Q)$ will
always attain a fixed point. This is a direct consequence of the fact we
showed previously in [2], that any commutative semihypergroup $K$ is left
amenable. Hence in particular, $K$ will admit a left-invariant mean on the
spaces of (weakly) right uniformly continuous functions and almost periodic
functions on $K$. The proof is inspired by techniques used in Rickert’s Fixed
Point Theorem [15], proved for topological groups.

######  Theorem 4.3.

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

###### Proof.

Let $\pi$ be a separately continuous affine action of $K$ on $X$. Recall from
Proposition 4.2 that $\pi$ naturally induces an affine action $\tilde{\pi}$ of
$K$ on $A_{f}(X)$ where for each $s\in K,f\in A_{f}(X),x\in X$ we have

$\tilde{\pi}_{s}(f)(x)=f(\pi_{s}(x)).$

Now for each $x\in X$, we define a function $\phi_{x}:A_{f}(X)\rightarrow
C(K)$ as

$\phi_{x}(f)(s):=f(\pi_{s}(x)),$

for each $f\in A_{f}(X),s\in K$. Since $\pi$ is separately continuous on
$K\times X$, for any fixed $x\in X$ and a net $(s_{\alpha})$ in $K$ such that
$s_{\alpha}\rightarrow s$ for some $s\in K$, we have that
$\pi_{s_{\alpha}}(x)\rightarrow\pi_{s}(x)$ in $X$. Since $f$ is continuous on
$X$, we have that $\phi_{x}(f)(s_{\alpha})=f(\pi_{s_{\alpha}}(x))\rightarrow
f(\pi_{s}(x))=\phi_{x}(f)(s)$, i.e, $\phi_{x}(f)$ is indeed continuous on $K$,
which is also bounded as $f\in C(X)$.

For each $s\in K,f\in A_{f}(K),x\in X$ we further have that:

$\displaystyle L_{s}\phi_{x}(f)(t)$ $\displaystyle=$
$\displaystyle\phi_{x}(f)(s*t)$ $\displaystyle=$
$\displaystyle\int_{K}\phi_{x}(f)(\zeta)\ d(p_{s}*p_{t})(\zeta)$
$\displaystyle=$ $\displaystyle\int_{K}f(\pi_{\zeta}(x))\
d(p_{s}*p_{t})(\zeta)$ $\displaystyle=$ $\displaystyle
f\Big{(}\int_{K}\pi_{\zeta}(x)\ d(p_{s}*p_{t})(\zeta)\Big{)}$ $\displaystyle=$
$\displaystyle f(\pi_{s}(\pi_{t}(x)))\ =\ \tilde{\pi}_{s}(f)(\pi_{t}(x))\ =\
\phi_{x}(\tilde{\pi}_{s}(f))(t),$

for each $t\in K$, where the fourth equality holds true since $f$ is affine.
Hence we have that

(1) $L_{s}\phi_{x}(f)=\phi_{x}(\tilde{\pi}_{s}(f))$

for each $s\in K$, i.e, the translation-orbit of the function $\phi_{x}(f)\in
C(K)$ coincides with the image of the $\tilde{\pi}$-orbit of $f$, under the
map $\phi_{x}$.

Since $K$ is commutative, we know [2] that $K$ is amenable, i.e, there exists
a translation invariant mean $m$ on $C(K)$. Fix any $x_{0}\in X$, and consider
the map $m\circ\phi_{x_{0}}:A_{f}(X)\rightarrow\mathbb{C}$. Note that
$m\circ\phi_{x_{0}}$ is also a mean on $A_{f}(X)$ since $\phi_{x_{0}}$ is a
bounded linear map and $A_{f}(X)\subset C(X)$. Hence it follows from Remark
4.1 that there exists some $z_{0}\in X$ such that
$m\circ\phi_{x_{0}}=\varepsilon_{z_{0}}$.

In particular, since $\phi_{x_{0}}(f)\in C(K)$ for any $f\in A_{f}(X)$, we
have that $m(L_{s}\phi_{x_{0}}(f))=m(\phi_{x_{0}}(f))$ for each $s\in K$.
Hence using the interplay between orbits derived in Equation 1 we have that

$\displaystyle\ \ \ \ \ m(\phi_{x_{0}}(\tilde{\pi}_{s}(f)))$ $\displaystyle=$
$\displaystyle m(\phi_{x_{0}}(f)),$ $\displaystyle\Rightarrow\
(m\circ\phi_{x_{0}})(\tilde{\pi}_{s}(f))$ $\displaystyle=$
$\displaystyle(m\circ\phi_{x_{0}})(f),$ $\displaystyle\Rightarrow\hskip
36.135pt\varepsilon_{z_{0}}(\tilde{\pi}_{s}(f))$ $\displaystyle=$
$\displaystyle\varepsilon_{z_{0}}(f),$ $\displaystyle\Rightarrow\ \hskip
36.135ptf(\pi_{s}(z_{0}))$ $\displaystyle=$ $\displaystyle f(z_{0}),$

for each $f\in A_{f}(X)$, $s\in K$. Hence in particular, we must have that
$\pi(s,z_{0})=z_{0}$ for each $s\in K$, as required. ∎

On the other hand, recall [2] that the space of almost periodic functions
$AP(K)$ is a translation-invariant, norm-closed linear subspace of $C(K)$ such
that $AP(K)\subseteq UC(K)$. In the following result, we see that the fixed
point property outlined in the statement of Theorem 4.3 is a sufficient
condition for any semihypergroup $K$ to admit a left invariant mean on
$AP(K)$. The idea of the proof is similar to the standard techniques used in
[13] for the case of locally compact semigroups, although the structures of
the function spaces used are substantially different.

######  Theorem 4.4.

Let $K$ be a (semitopological) semihypergroup such that any jointly continuous
affine action $\pi$ of $K$ on a compact convex subset $X$ of a locally convex
vector space $(E,Q)$ has a fixed point. Then there exists a LIM on $AP(K)$.

###### Proof.

In particular, set $E:=AP(K)^{*}$ and equip it with the weak∗-topology. Set
$X:=\mathcal{M}(AP(K))$. Since $X\subset\mathcal{B}_{1}(AP(K)^{*})$, we have
that $X$ is compact in the induced weak∗-topology. Now consider the map
$\pi:K\times X\rightarrow X$ given by

$\pi(s,m)(f)=\pi_{s}(m)(f):=m(L_{s}f)$

for each $s\in K,m\in X,f\in AP(K)$. Note that for any $s,t\in K$, we have

$\displaystyle\int_{K}L_{\zeta}f(y)\ d(p_{s}*p_{t})(\zeta)$ $\displaystyle=$
$\displaystyle\int_{K}R_{y}f(\zeta)\ d(p_{s}*p_{t})(\zeta)$ $\displaystyle=$
$\displaystyle\int_{K}f\ d(p_{s}*p_{t}*p_{y})$ $\displaystyle=$ $\displaystyle
f(s*t*y)$ $\displaystyle=$ $\displaystyle L_{s}f(t*y)\ =\ (L_{t}\circ
L_{s})(y),$

for each $y\in K$. Hence $\pi$ is indeed an action of $K$ on $X$ since for
each $s,t\in K,m\in X$, $f\in AP(K)$ we have that

$\displaystyle\pi(s,\pi(t,m))(f)$ $\displaystyle=$
$\displaystyle\pi_{t}(m)(L_{s}f)$ $\displaystyle=$ $\displaystyle m(L_{t}\circ
L_{s})(f)$ $\displaystyle=$ $\displaystyle m\Big{(}\int_{K}L_{\zeta}f\
d(p_{s}*p_{t})(\zeta)\Big{)}$ $\displaystyle=$
$\displaystyle\int_{K}m(L_{\zeta}f)\ d(p_{s}*p_{t})(\zeta)\ =\
\int_{K}\pi(\zeta,m)(f)\ d(p_{s}*p_{t})(\zeta),$

where the fourth equality follows since $m\in AP(K)^{*}$. Now pick any $s\in
K,m\in X$. Let $(s_{\alpha})$ and $(m_{\beta})$ be sequences in $K$ and $X$
respectively, such that $s_{\alpha}\rightarrow s$ and $m_{\alpha}\rightarrow
m$ in $K$ and $X$ respectively. For each $f\in AP(K)\subseteq UC(K)$ we have
that $||(L_{s_{\alpha}}f-L_{s}f)||_{u}\rightarrow 0$. Hence for each $f\in
AP(K)$ we have that

$\displaystyle\lim_{\alpha,\beta}|\pi(s_{\alpha},m_{\alpha})(f)-\pi(s,m)|$
$\displaystyle=$
$\displaystyle\lim_{\alpha,\beta}|m_{\beta}(L_{s_{\alpha}}f)-m(L_{s}f)|$
$\displaystyle\leq$
$\displaystyle\lim_{\alpha,\beta}|m_{\beta}(L_{s_{\alpha}}f-L_{s}f)|+\lim_{\alpha,\beta}|(m_{\beta}-m)(L_{s}f)|$
$\displaystyle\leq$
$\displaystyle\lim_{\alpha,\beta}||m_{\beta}||\>||(L_{s_{\alpha}}f-L_{s}f)||_{u}+\lim_{\beta}||(m_{\beta}-m)||\>||L_{s}f||_{u}$
$\displaystyle=$
$\displaystyle\lim_{\alpha}||(L_{s_{\alpha}}f-L_{s}f)||+\lim_{\beta}||(m_{\beta}-m)||\>||f||_{u}\
\longrightarrow\ 0,$

since $f$ is bounded on $K$. Hence we have that the action $\pi$ is jointly
continuous on $K\times X$. Since $\pi$ is also affine by construction, we have
that $\pi$ admits a fixed point, i.e, there exists some $m_{0}\in X$ such that
$\pi(s,m_{0})=m_{0}$ for each $s\in K$. Thus we have that

$m_{0}(f)=\pi(s,m_{0})(f)=m_{0}(L_{s}f),$

for each $s\in K,f\in AP(K)$, i.e,  $m_{0}$ is a left invariant mean on
$AP(K)$. ∎

Next, in the final part of the section, we consider a different fixed-point
property involving measure algebra-homomorphisms, that uses Definition 3.8 for
semihypergroup actions. In particular, we consider the topology
$\tau_{{}_{AP(K)}}:=\sigma(M(K),AP(K))$ on $M(K)$ induced by $AP(K)$, and
investigate the fixed points of $\tau_{{}_{AP(K)}}$-continuous actions of $K$
on a separated locally convex space $(E,Q)$, with respect to the space of
probability measures $P(K)$. Before we proceed further along this line of
investigation, we first discuss a couple of lemmas regarding the density
properties of evaluation maps $\varepsilon_{x}$, $x\in K$, in the closed unit
ball of a norm-closed subspace of $C(K)$, which are imperative to the proof of
the next main results.

Recall [4] that a subset $E$ of a locally convex space $V$ is called balanced
or circled if for any $x\in E$, we have that $\alpha x\in E$ for each
$\alpha\in\mathbb{F}$ such that $|\alpha|\leq 1$, where
$\mathbb{F}=\mathbb{C}\ (\mbox{or }\mathbb{R})$ is the associated field of
scalars. Hence the balanced convex hull or circled convex hull of $E$ is the
smallest balanced, convex set in $V$ that contains $E$, and is denoted as
$cco(E)$. It can be shown[4] easily that

$cco(E)=\Big{\\{}\sum_{i=1}^{n}\alpha_{i}x_{i}:x_{i}\in
E,\alpha_{i}\in\mathbb{F}\mbox{ for }1\leq i\leq n;\ n\in\mathbb{N}\mbox{ and
}\sum_{i=1}^{n}|\alpha_{i}|\leq 1\Big{\\}}.$

Note that although $cco(E)$ equals the convex hull of the smallest balanced
set in $V$ containing $E$, it may strictly contain the smallest balanced set
in $V$ containing the convex hull of $E$ [4].

The following couple of results hold true even when the semihypergroup $K$ is
substituted by any locally compact Hausdorff space, in general. We include
brief proofs here for convenience of readers.

######  Lemma 4.5.

Let $\mathcal{F}$ be a norm-closed, conjugate-closed linear subspace of
$C(K)$. Then $cco(\varepsilon(K))$ is weak∗-dense in
$\mathcal{B}_{1}(\mathcal{F}^{*})$.

###### Proof.

Let $A:=\overline{cco}^{w^{*}}(\varepsilon(K))$. Now if possible, let there
exists some $\phi\in\mathcal{B}_{1}(\mathcal{F}^{*})\setminus A$. Let $V$
denote the locally convex topological vector space $\mathcal{F}^{*}$ equipped
with the weak∗-topology. Since $A\subset\mathcal{B}_{1}(\mathcal{F}^{*})$, $A$
is a compact convex subset of $V$. Hence using the strong separation theorem
for locally convex spaces [4, Theorem 7.8.6], we can find a weak∗-continuous
linear functional $\omega$ on $\mathcal{F}^{*}$ such that

$\alpha:=\sup\\{Re(\omega(\psi)):\psi\in A\\}<Re(\omega(\phi)).$

Since $0\in A$ and $A$ is balanced, we know that $Re(\omega(\phi))>\alpha>0$.
Now, we define a weak∗-continuous map
$\rho:\mathcal{F}^{*}\rightarrow\mathbb{C}$ as

$\rho(\psi):=\frac{1}{\alpha}Re(\omega(\psi))-\frac{i}{\alpha}Re(\omega(i\psi)),$

for each $\psi\in\mathcal{F}^{*}$. Note that $\rho$ is a linear functional
since for any $\psi\in\mathcal{F}^{*}$, we have

$\rho(i\psi)=\frac{i}{\alpha}Re(\omega(\psi))+\frac{1}{\alpha}Re(\omega(i\psi))=i\rho(\psi).$

Since $(V,\mathcal{F})$ is a dual pair, there exists [4] an unique
$f_{0}\in\mathcal{F}$ such that

$\rho(\psi)=\langle\psi,f_{0}\rangle=\psi(f_{0})$

for each $\psi\in\mathcal{F}^{*}$. Hence in particular, we have that

$\displaystyle|\phi(f_{0})|\ =\ |\langle\phi,f_{0}\rangle|$ $\displaystyle=$
$\displaystyle|\rho(\phi)|$ $\displaystyle\geq$
$\displaystyle\frac{1}{\alpha}Re(\omega(\phi))\ >\ 1.$

But this poses a contradiction to the fact that $||\phi||<1$, since we have
that $||f_{0}||\leq 1$ as well, which can be verified as the following:

$\displaystyle Re(\langle\psi,f_{0}\rangle)$ $\displaystyle=$ $\displaystyle
Re(\rho(\psi))$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}Re(\omega(\psi))\ \leq 1,$

for each $\psi\in A$. Since $A$ is balanced, we have that $Re(\langle
e^{i\theta}\psi,f_{0}\rangle)\leq 1$ for each $\theta\in\mathbb{R}$, and hence
$|\langle\psi,f_{0}\rangle|\leq 1$ for each $\psi\in A$. Thus in particular,
for each $x\in K$ we have that

$|f_{0}(x)|=|\langle\varepsilon_{x},f_{0}\rangle|\leq 1.$

∎

######  Lemma 4.6.

Let $\mathcal{F}$ be a norm-closed, conjugate-closed linear subspace of
$C(K)$. Then $\\{\varepsilon_{\mu}:\mu\in P(K)\\}$ is weak∗-dense in the set
$\mathcal{M}(\mathcal{F})$ of all means on $\mathcal{F}$.

###### Proof.

Pick any
$m\in\mathcal{M}(\mathcal{F})\subseteq\mathcal{B}_{1}(\mathcal{F}^{*})$. Using
Lemma 4.5 and Remark 4.1, we get a net $\\{\mu_{\alpha}\\}$ in $M(K)$ such
that $\varepsilon_{\mu_{\alpha}}\overset{w^{*}}{\longrightarrow}m$ on
$\mathcal{F}^{*}$. But we have that $m(f)\geq 0$ for any $f\geq 0$ in $C(K)$
and $m(1)=1$, since $m\in\mathcal{M}(\mathcal{F})$. Hence we must have that
$\mu_{\alpha}\in M^{+}(K)$ eventually, and hence
$||\mu_{\alpha}||=\mu_{\alpha}(K)$ must converge to $1$.

Hence without loss of generality, we may assume that each $\mu_{\alpha}\in
M^{+}(K)$. Setting $\nu_{\alpha}:=\mu_{\alpha}/||\mu_{\alpha}||$ we get that
$\varepsilon_{{\nu_{\alpha}}}\overset{w^{*}}{\longrightarrow}m$, where
$\nu_{\alpha}\in P(K)$ for each $\alpha$. ∎

Next, we recall some properties of the dual space $AP(K)^{*}$, in order to use
its algebraic structure to provide certain necessary and sufficient conditions
for the existence of a left-invariant mean on the space of almost periodic
functions.

Let $K$ be a semitopological semihypergroup, and $\mathcal{F}$ be a left
(resp. right) translation-invariant linear subspace of $C(K)$. For each
$\omega\in\mathcal{F}^{*}$, the left (resp. right) introversion operator
$T_{\omega}$ (resp. $U_{\omega}$) determined by $\omega$ is the map
$T_{\omega}:\mathcal{F}\rightarrow B(K)$ (resp.
$U_{\omega}:\mathcal{F}\rightarrow B(K)$) defined as
$T_{\omega}f(x):=\omega(L_{x}f)$ (resp. $U_{\omega}f(x):=\omega(R_{x}f)$) for
each $f\in\mathcal{F},x\in K$. We say that a left (resp. right) translation-
invariant linear subspace $\mathcal{F}$ is left-introverted (resp. right-
introverted) if $T_{\omega}f\in\mathcal{F}$ (resp.
$U_{\omega}f\in\mathcal{F}$) for each $\omega\in\mathcal{F}^{*}$,
$f\in\mathcal{F}$. A translation-invariant subspace $\mathcal{F}$ is called
introverted if it is both left and right introverted.

In a previous work [2] we discussed the basic properties of introversion
operators for a semitopological semihypergroup, and in turn showed that the
function-space $AP(K)$ is translation-invariant and introverted. Hence
although the topological space $K$ lacks an algebraic structure on itself, we
can define both left and right Arens product [1] on the dual space
$AP(K)^{*}$. It can be seen easily[2] that the left Arens product $\lozenge$
on $AP(K)^{*}$ coincides with the definition

$(m\lozenge n)(f):=m(T_{n}(f)),$

for each $m,n\in AP(K)^{*}$, $f\in AP(K)$. Similarly, for each $m,n\in
AP(K)^{*}$ the right Arens product $\square$ on $AP(K)^{*}$ coincides with the
definition

$(m\square n)(f):=n(U_{m}(f)),$

for each $f\in AP(K)$. In fact, it can be shown [2, Theorem 4.27] that
$AP(K)^{*}$ is Arens-regular, i.e, $(m\lozenge n)=(m\square n)$ for any
$m,n\in AP(K)^{*}$. Hence from here onwards, without any loss of generality,
we consider the algebra $(AP(K)^{*},\star)$ where $(m\star n)(f):=m(T_{n}(f))$
for each $f\in AP(K)$.

######  Remark 4.7.

For any $\mu,\nu\in M(K)$, we have that
$\varepsilon_{\mu}\star\varepsilon_{\nu}=\varepsilon_{{\mu*\nu}}$ on $AP(K)$,
since for any $f\in AP(K)$, we have the following:

$\displaystyle(\varepsilon_{\mu}\star\varepsilon_{\nu})(f)$ $\displaystyle=$
$\displaystyle\varepsilon_{\mu}(T_{\varepsilon_{\nu}}(f))$ $\displaystyle=$
$\displaystyle\int_{K}T_{\varepsilon_{\nu}}(f)(x)\ d\mu(x)$ $\displaystyle=$
$\displaystyle\int_{K}\varepsilon_{\nu}(L_{x}f)\ d\mu(x)$ $\displaystyle=$
$\displaystyle\int_{K}\int_{K}L_{x}f(y)\ d\nu(y)\ d\mu(x)$ $\displaystyle=$
$\displaystyle\int_{K}\int_{K}f(x*y)\ d\mu(x)d\nu(y)\ =\int_{K}f\ d(\mu*\nu)\
=\ \varepsilon_{\mu*\nu}(f),$

where the fifth equality holds true since $f$ is bounded on $K$, and both
$||\mu||,||\nu||$ are finite.

The following results are inspired by the techniques and results proved in
[17], for locally compact semigroups. Note that unlike locally compact groups
and semigroups, here the spaces $LUC(K),RUC(K),UC(K)$ and $AP(K)$ do not
necessarily form sub-algebras of $C(K)$, with respect to the point-wise
product.

######  Theorem 4.8.

Let $K$ be a semitopological semihypergroup. Then the following statements are
equivalent.

  1. (1)

$AP(K)$ admits a LIM.

  2. (2)

$\exists$ a mean $m$ on $AP(K)$ such that $\varepsilon_{\mu}\star m=m$ for
each $\mu\in P(K)$.

  3. (3)

$\exists$ a net $\\{\mu_{\alpha}\\}$ in $P(K)$ such that for each $\mu\in
P(K)$, the net $\\{(\mu*\mu_{\alpha}-\mu_{\alpha})\\}$ converges to $0$ in
$(M(K),\sigma(M(K),AP(K)))$.

Moreover, if $m$ is a LIM on $AP(K)$, then $\varepsilon_{\mu}\star m=m$ for
each $\mu\in P(K)$.

###### Proof.

$(1)\Rightarrow(2)$: Let $m$ be a LIM on $AP(K)$, and $\mu\in P(K)$. Since
$||\varepsilon_{\mu}||=1=\varepsilon_{\mu}(1)$, it immediately follows from
Lemma 4.6 that there exists a net $\\{\phi_{\alpha}\\}$ in
$\\{\varepsilon_{\nu}:\nu\in P(K)\\}$ such that
$\phi_{\alpha}\overset{w^{*}}{\longrightarrow}\varepsilon_{\mu}$ on
$AP(K)^{*}$. But $\\{\varepsilon_{\nu}:\nu\in P(K)\\}=co(\varepsilon(K))$, and
hence for each $\alpha$ there exists some $n_{\alpha}\in\mathbb{N}$ such that
$\phi_{\alpha}=\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}\varepsilon_{x_{i}^{\alpha}}$,
where each $x_{i}^{\alpha}\in K$, $0\leq\lambda_{i}^{\alpha}\leq 1$ for $1\leq
i\leq n_{\alpha}$ such that $\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}=1$.
Hence $\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}p_{x_{i}^{\alpha}}$
converges to $\mu$ in $(M(K),\tau_{{}_{AP(K)}})$, as for each $f\in AP(K)$, we
have that

$\displaystyle\int_{K}f\
d\Big{(}\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}p_{x_{i}^{\alpha}}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}f({x_{i}^{\alpha}})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}\varepsilon_{(}{x_{i}^{\alpha}})(f)\
=\phi_{\alpha}(f)\ \rightarrow\ \int_{K}f\ d\mu.$

Now, let $F$ be a norm-bounded subset of $AP(K)^{*}$. Consider the map
$\Omega_{F}:M(K)\times F\rightarrow AP(K)^{*}$ given by

$\Omega_{F}(\mu,\omega):=\varepsilon_{\mu}\star\omega,$

for each $(\mu,\omega)\in M(K)\times F$. Pick $M>0$ such that $||\rho||\leq M$
for each $\rho\in F$.

let $\mu_{\alpha}\longrightarrow\mu$ in $(M(K),\sigma(M(K),AP(K)))$ and
$\omega\in AP(K)^{*}$. Then for each $f\in AP(K)$ we have that

$\displaystyle\Omega_{F}(\mu_{\alpha},\omega)(f)$ $\displaystyle=$
$\displaystyle(\varepsilon_{\mu_{\alpha}}\star\omega)(f)$ $\displaystyle=$
$\displaystyle\int_{K}T_{\omega}f(x)\ d\mu_{\alpha}(x)$
$\displaystyle\longrightarrow$ $\displaystyle\int_{K}T_{\omega}f(x)\ d\mu(x)\
=\ \varepsilon_{\mu}(T_{\omega}f)\ =\ \Omega_{F}(\mu,\omega),$

since $AP(K)$ is left-introverted [2], and hence $T_{\omega}f\in AP(K)$, as
$f\in AP(K)$.

On the other hand, let $\\{\omega_{\alpha}\\}$ be any net in $AP(K)^{*}$ such
that $\omega_{\alpha}\overset{w^{*}}{\longrightarrow}\omega$ for some
$\omega\in AP(K)^{*}$, and consider the family of functions
$\\{T_{\omega_{\alpha}}f\\}$ for some $f\in AP(K)$. For any $x,y\in K$ we have

$\displaystyle|T_{\omega_{\alpha}}f(x)-T_{\omega_{\alpha}}f(y)|$
$\displaystyle=$
$\displaystyle|{\omega_{\alpha}}(L_{x}f)-{\omega_{\alpha}}(L_{y}f)|$
$\displaystyle=$ $\displaystyle|{\omega_{\alpha}}(L_{x}f-L_{y}f)|$
$\displaystyle\leq$
$\displaystyle||{\omega_{\alpha}}||\>||L_{x}f-L_{y}f||_{u}\ \leq\
M||L_{x}f-L_{y}f||_{u}\rightarrow 0\mbox{ whenever }x\rightarrow y,$

since $AP(K)\subset UC(K)$ [2] and hence the map $x\mapsto
L_{x}f:K\rightarrow(C(K),||\cdot||_{u})$ is continuous for each $f\in AP(K)$.
Hence $\\{T_{\omega_{\alpha}}f\\}$ is an equicontinuous family of functions in
$(C(K),||\cdot||_{u})$. Therefore the topology of pointwise convergence will
coincide with the topology of uniform convergence on compact sets. For each
$x\in K$, we have that

$T_{\omega_{\alpha}}f(x)=\omega_{\alpha}(L_{x}f)\rightarrow\omega(L_{x}f)=T_{\omega}f(x),$

since $L_{x}f\in AP(K)$ and
$\omega_{\alpha}\overset{w^{*}}{\longrightarrow}\omega$. Hence
$T_{\omega_{\alpha}}f\rightarrow T_{\omega}f$ uniformly on compact subsets of
$K$. Thus for any $\mu\in M_{C}(K)$ and $f\in AP(K)$ we have:

$\displaystyle\Omega_{F}(\mu,\omega_{\alpha})(f)$ $\displaystyle=$
$\displaystyle(\varepsilon_{\mu}\star{\omega_{\alpha}})(f)$ $\displaystyle=$
$\displaystyle\varepsilon_{\mu}(T_{\omega_{\alpha}}f)$ $\displaystyle=$
$\displaystyle\int_{K}T_{\omega_{\alpha}}f(x)\ d\mu(x)$
$\displaystyle\longrightarrow$ $\displaystyle\int_{K}T_{\omega}f(x)\ d\mu(x)\
=\ (\varepsilon_{\mu}\star\omega)(f)\ =\ \Omega_{F}(\mu,\omega)(f),$

where the fourth implication holds true since $supp(\mu)$ is compact. But note
that $M_{C}(K)$ is norm-dense in $M(K)$ and for each $f\in AP(K)$ and any
$\alpha$ we have that

$||T_{\omega_{\alpha}}(f)||_{u}=\underset{x\in K}{\sup}\
|\omega_{\alpha}(L_{x}f)|\leq\underset{x\in K}{\sup}\
(||\omega_{\alpha}||\>||L_{x}f||)\leq M||f||.$

Hence for each $\mu\in M(K)$ we have that
$\Omega_{F}(\mu,\omega_{\alpha})\overset{w^{*}}{\longrightarrow}\Omega_{F}(\mu,\omega)$.
Thus we see that $\Omega_{F}$ is separately
$\big{(}\tau_{{}_{AP(K)}}\times\mbox{ weak}^{*}\mbox{-topology}\big{)}-\mbox{
weak}^{*}\mbox{-topology}$ continuous.

In particular taking $F=P(K)$ and $\omega=m$, since
$\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}p_{x_{i}^{\alpha}}$ converges to
$\mu$ in $(M(K),\tau_{{}_{AP(K)}})$, we have that

$(\phi_{\alpha}\star
m)=\Big{(}\varepsilon_{{}_{\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}p_{x_{i}^{\alpha}}}}\star
m\Big{)}\overset{w^{*}}{\longrightarrow}(\varepsilon_{\mu}\star m)\mbox{ on
}AP(K)^{*}.$

But for each $\alpha$, we also have that

$\displaystyle(\phi_{\alpha}\star m)(f)$ $\displaystyle=$
$\displaystyle\phi_{\alpha}(T_{m}f)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}(T_{m}f)({x_{i}^{\alpha}})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}m(L_{x_{i}^{\alpha}}f)\
=\ \sum_{i=1}^{n_{\alpha}}\lambda_{i}^{\alpha}m(f)\ =\ m(f),$

for each $f\in AP(K)$. Since the weak∗-topology is Hausdorff, thus we must
have that $\varepsilon_{\mu}\star m=m$ on $AP(K)$.

$(2)\Rightarrow(3)$: Let $m$ be a mean on $AP(K)$ such that
$\varepsilon_{\mu}\star m=m$ for each $\mu\in P(K)$. It follows from Lemma 4.6
that there exists a net $\\{\mu_{\alpha}\\}$ in $P(K)$ such that
$\varepsilon_{\mu_{\alpha}}\overset{w^{*}}{\longrightarrow}m$ in $AP(K)^{*}$.
Hence in particular, taking $F=P(K)$, we have that

$\varepsilon_{\mu}\star\varepsilon_{\mu_{\alpha}}=\Omega_{F}(\mu,\varepsilon_{\mu_{\alpha}})\overset{w^{*}}{\longrightarrow}\Omega_{F}(\mu,m)=\varepsilon_{\mu}\star
m=m.$

Hence for each $\mu\in P(K)$ and $f\in AP(K)$ we have the following.

$\displaystyle\int_{K}f\ d(\mu*\mu_{\alpha}-\mu_{\alpha})$ $\displaystyle=$
$\displaystyle\int_{K}f\ d(\mu*\mu_{\alpha})-\int_{K}f\ d\mu_{\alpha}$
$\displaystyle=$
$\displaystyle\varepsilon_{(\mu*\mu_{\alpha})}(f)-\varepsilon_{\mu_{\alpha}}(f)$
$\displaystyle=$
$\displaystyle(\varepsilon_{\mu}\star\varepsilon_{\mu_{\alpha}})(f)-\varepsilon_{\mu_{\alpha}}(f)$
$\displaystyle\longrightarrow$ $\displaystyle m-m\ =\ 0,$

where the third equality follows from Remark 4.7.

$(3)\Rightarrow(2)$: Let $\\{\mu_{\alpha}\\}_{\alpha\in I}$ be a net in $P(K)$
such that for each $\mu\in P(K)$, the net
$\\{(\mu*\mu_{\alpha}-\mu_{\alpha})\\}_{\alpha\in I}$ converges to $0$ in
$\tau_{{}_{AP(K)}}$. Since
$\mathcal{M}(AP(K))\subset\mathcal{B}_{1}(AP(K)^{*})$ is weak∗-compact, it
follows from Lemma 4.6 that there exists a weak∗-convergent subnet
$\\{\mu_{\beta}\\}_{\beta\in I^{\prime}}$ of $\\{\mu_{\alpha}\\}_{\alpha\in
I}$, $I^{\prime}\subseteq I$, such that
$\varepsilon_{\mu_{{}_{\beta}}}\overset{w^{*}}{\longrightarrow}m$ for some
$m\in\mathcal{M}(AP(K))$.

Now for each $\mu\in P(K)$, using Remark 4.7 and the continuity of
$\Omega_{F}$ with $F={P(K)}$, we get that

$\varepsilon_{{}_{(\mu*\mu_{{}_{\beta}})}}=(\varepsilon_{\mu}\star\varepsilon_{\mu_{{}_{\beta}}})=\Omega_{F}(\mu,\varepsilon_{\mu_{{}_{\beta}}})\overset{w^{*}}{\longrightarrow}\Omega_{F}(\mu,m)=(\varepsilon_{\mu}\star
m).$

Thus finally, for any $f\in AP(K)$, $\mu\in P(K)$, using the given criterion
we have the following, as required.

$\displaystyle\big{(}(\varepsilon_{\mu}\star m)-m\big{)}(f)$ $\displaystyle=$
$\displaystyle\lim_{\beta}{}^{w^{*}}\varepsilon_{{}_{(\mu*\mu_{{}_{\beta}})}}(f)-\lim_{\beta}{}^{w^{*}}\varepsilon_{\mu_{{}_{\beta}}}(f)$
$\displaystyle=$ $\displaystyle\lim_{\beta}{}^{w^{*}}\int_{K}f\
d(\mu*\mu_{{}_{\beta}})-\lim_{\beta}{}^{w^{*}}\int_{K}f\ d\mu_{{}_{\beta}}$
$\displaystyle=$ $\displaystyle\lim_{\beta}{}^{w^{*}}\int_{K}f\
d(\mu*\mu_{{}_{\beta}}-{\mu_{{}_{\beta}}})\ =\ 0.$

$(2)\Rightarrow(1)$: Let $m$ be a mean on $AP(K)$ such that
$\varepsilon_{\mu}\star m=m$ for each $\mu\in P(K)$. In particular, pick any
$x_{0}\in K$ and consider $\mu=p_{{}_{x_{0}}}$. Then it follows immediately
that $m$ is a LIM on $AP(K)$ since for each $f\in AP(K)$ the following holds
true:

$\displaystyle m(L_{x_{0}}f)$ $\displaystyle=$ $\displaystyle T_{m}f(x_{0})$
$\displaystyle=$ $\displaystyle\varepsilon_{p_{{}_{x_{0}}}}(T_{m}f)\ =\
(\varepsilon_{p_{{}_{x_{0}}}}\star m)(f)\ =\ \mu(f).$

∎

We now proceed to the final result of this section. For convenience, we abuse
notation slightly, and for any Borel subset $\Omega\subseteq M(K)$, we denote
the subspace-topology induced by $\tau_{{}_{AP(K)}}$ on $\Omega$, simply as
$\tau_{{}_{AP(K)}}$. For the rest of this article, we consider
(semihypergroup) action in the form of Definition 3.8, which as discussed
before, is equivalent to the standard definition (Definition 3.6) of a
semihypergroup action. Given a semihypergroup action $\pi$ of $K$ on a
separated locally convex space $(E,Q)$ and any subset $\Omega\subset M(K)$, we
say that a subset $F\subset E$ is $\Omega$-invariant, if we have that
$\pi(\mu,x)=\pi_{\mu}(x)\in F$ for each $\mu\in\Omega,x\in F$. We say that the
action $\pi$ has Property $(FP)$ if it satisfies the following property:

$\displaystyle(FP):$ $\displaystyle\mbox{ For any compact, convex,
}P(K)\mbox{-invariant subset }F\mbox{ of }E,\mbox{ if the induced}$
$\displaystyle\mbox{ action }\tilde{\pi}:=\pi{|}_{{}_{P(K)\times
F}}:P(K)\times F\rightarrow F\mbox{ is separately
}\tau_{{}_{AP(K)}}\mbox{-continuous,}$ $\displaystyle\mbox{ then
}\tilde{\pi}\mbox{ has a fixed point.}$

We show that the fixed-point property $(FP)$ completely characterizes the
existence of a LIM on the space $AP(K)$ of almost periodic functions on any
semitopological semihypergroup $K$.

######  Theorem 4.9.

Let $K$ be a semitopological semihypergroup. Then $AP(K)$ has a LIM if and
only if any semihypergroup action $\pi$ of $K$ on a separated locally convex
space $(E,Q)$ satisfies Property $(FP)$.

###### Proof.

First, let $m$ be a left-invariant mean on $AP(K)$, and $\pi:M(K)\times
E\rightarrow E$ be a semihypergroup action of $K$ on a separated locally
convex space $(E,Q)$. Further, let $F$ be a compact convex subset of $E$ such
that $E$ is $P(K)$-invariant under $\pi$ such that the induced action
$\tilde{\pi}:=\pi{|}_{{}_{P(K)\times F}}:P(K)\times F\rightarrow F$ is
separately $\tau_{{}_{AP(K)}}$-continuous.

Since $m$ is a LIM on $AP(K)$, it follows from Theorem 4.8 that there exists a
net of measures $\\{\mu_{\alpha}\\}_{\alpha\in I}$ in $P(K)$ such that
$\\{(\mu*\mu_{\alpha}-\mu_{\alpha})\\}$ converges to $0$ in
$(M(K),\tau_{{}_{AP(K)}})$ for each $\mu\in P(K)$.

Pick some $x_{0}\in F$, and consider the net
$\\{\tilde{\pi}(\mu_{\alpha},x_{0})\\}_{\alpha\in
I}=\\{\tilde{\pi}_{{}_{\mu_{\alpha}}}(x_{0})\\}_{\alpha\in I}$ in $F$. Since
$F$ is compact, we will get a subnet
$\\{\tilde{\pi}_{{}_{\mu_{\beta}}}(x_{0})\\}_{\beta\in I^{\prime}}$,
$I^{\prime}\subseteq I$, such that
$\tilde{\pi}_{{}_{\mu_{\beta}}}(x_{0})\rightarrow z_{0}$ in $(E,Q)$ for some
$z_{0}\in F$. Thus for each $\mu\in P(K)$ we have the following:

$\displaystyle\tilde{\pi}(\mu,z_{0})$ $\displaystyle=$
$\displaystyle\tilde{\pi}\big{(}\mu,\
\lim_{\beta}\tilde{\pi}(\mu_{\beta},x_{0})\big{)}$ $\displaystyle=$
$\displaystyle\lim_{\beta}\tilde{\pi}(\mu*\mu_{\beta},x_{0})$ $\displaystyle=$
$\displaystyle\lim_{\beta}\tilde{\pi}\big{(}(\mu*\mu_{\beta}-\mu_{\beta})+\mu_{\beta},x_{0})$
$\displaystyle=$
$\displaystyle\lim_{\beta}\tilde{\pi}((\mu*\mu_{\beta}-\mu_{\beta}),x_{0})+\lim_{\beta}\tilde{\pi}\big{(}\mu_{\beta},x_{0})$
$\displaystyle=$
$\displaystyle\tilde{\pi}\big{(}\lim_{\beta}(\mu*\mu_{\beta}-\mu_{\beta}),x_{0}\big{)}+\lim_{\beta}\tilde{\pi}\big{(}\mu_{\beta},x_{0})$
$\displaystyle=$
$\displaystyle\tilde{\pi}(0,x_{0})+\lim_{\beta}\tilde{\pi}_{{}_{\mu_{\beta}}}(x_{0})\
=\ z_{0},$

where the second and fifth equality holds since the action $\tilde{\pi}$ is
$\tau_{{}_{AP(K)}}$-continuous, and the fourth equality follows from the
bilinearity (Definition 3.8) of $\tilde{\pi}$. Hence in other words, we see
that $\pi$ satisfies Property $(FP)$.

Conversely, assume that any semihypergroup action $\pi$ of $K$ on a separated
locally convex space $(E,Q)$ satisfies Property $(FP)$. In particular, we set
$E=AP(K)^{*}$, and equip it with the weak∗-topology. We define a map
$\pi:M(K)\times AP(K)^{*}\rightarrow AP(K)^{*}$ by

$\pi(\mu,\phi)=\pi_{\mu}(\phi):=\varepsilon_{\mu}\star\phi,$

for any $\mu\in M(K),\phi\in AP(K)^{*}$. First note that $\pi$ is bilinear
since $(AP(K)^{*},\star)$ is an algebra, and
$\varepsilon_{\mu+\nu}=\varepsilon_{\mu}+\varepsilon_{\nu}$ for any
$\mu,\nu\in M(K)$ by construction. In addition, $\pi$ is indeed a
semihypergroup action of $K$ on $AP(K)^{*}$ since for any $\mu,\nu\in M(K)$
and $\phi\in AP(K)^{*}$ we have that

$\displaystyle\pi_{{}_{(\mu*\nu)}}(\phi)$ $\displaystyle=$
$\displaystyle\varepsilon_{{}_{(\mu*\nu)}}\star\phi$ $\displaystyle=$
$\displaystyle(\varepsilon_{\mu}\star\varepsilon_{\nu})\star\phi$
$\displaystyle=$
$\displaystyle\varepsilon_{\mu}\star(\varepsilon_{\nu}\star\phi)\ =\
\pi_{\mu}(\varepsilon_{\nu}\star\phi)\ =\ (\pi_{\mu}\circ\pi_{\nu})(\phi),$

where the second and third equalities follow from Remark 4.7 and associativity
of Arens product, respectively. Now, set $F:=\mathcal{M}(AP(K))$. Since
$\mathcal{M}(AP(K))$ is weak∗-closed in $\mathcal{B}_{1}(AP(K)^{*})$, we have
that $F$ is a compact convex subset of $E$. Moreover, for each $\mu\in P(K)$,
$\omega\in F$, since $\pi_{\mu}\in L(E)$, we have that

$||\pi_{\mu}(\omega)||=||\varepsilon_{\mu}\star\omega||\leq||\varepsilon_{\mu}||\>||\omega||=||\mu||\
||\omega||=1$

Since $AP(K)$ contains the constant function $1$, we have that
$\pi_{\mu}(\omega)(1)=1=||\pi_{\mu}(\omega)||$, i.e, $F$ is $P(K)$-invariant
under the action $\pi$. Now consider the induced action
$\tilde{\pi}:=\pi{|}_{{}_{P(K)\times F}}:P(K)\times F\rightarrow F$. Note that
the action $\tilde{pi}$ coincides with the map $\Omega_{P(K)}$ defined in the
proof of Theorem 4.8, and hence is separately
$\big{(}\tau_{{}_{AP(K)}}\times\mbox{ weak}^{*}\mbox{-topology}\big{)}-\mbox{
weak}^{*}\mbox{-topology}$ continuous.

Since $\pi$ satisfies Property $(FP)$, $\tilde{\pi}$ has a fixed point $m\in
F=\mathcal{M}(AP(K))$. Hence for each $\mu\in P(K)$ we have:

$m=\tilde{\pi}_{\mu}(m)=\varepsilon_{\mu}\star m.$

Thus $m$ is a LIM on $AP(K)$ by Theorem 4.8. ∎

## 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, for the course of this work.

## References

  * [1] Richard Arens The Adjoint of a Bilinear Operation. Proc. Amer. Math. Soc., Vol. 2, 1951, 839-848.
  * [2] Bandyopadhyay, Choiti Analysis on semihypergroups: function spaces, homomorphisms and ideals. Semigroup Forum 100 (2020), no. 3, 671-697.
  * [3] Bandyopadhyay, Choiti Free Product of Semihypergroups. Semigroup Forum 102 (2021), no. 1, 28–47.
  * [4] Beckenstein, E, Narici, L Topological Vector Spaces. Chapman & Hall, 2011.
  * [5] Bloom, W, Heyer, H Harmonic Analysis of Probability Measures on Hypergroups. Walter de Gruyter Studies in Mathematics 20, 1995.
  * [6] Berglund, J. F., Junghenn, H. D., Milnes, P. : Analysis on Semigroups. Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley Interscience Publication. 24 (1989)
  * [7] Day, M. M. : Amenable Semigroups. Illinois J. Math. 1, 509-544 (1957)
  * [8] Dunkl, C. F. : The Measure Algebra of a Locally Compact Hypergroup. Trans. Amer. Math. Soc. 179, 331-348 (1973)
  * [9] Jewett, R. I. : Spaces with an Abstract Convolution of Measures. Advances in Mathematics. 18, no. 1, 1-101 (1975)
  * [10] Lau, Anthony To-Ming, Takahashi, Wataru Invariant Means and Fixed Point Properties for Non-expansive Representations of Topological semigroups. Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Centre. Volume 5, 1995, 39-57.
  * [11] Lau, Anthony To-Ming, Zhang, Yong Amenability Properties and Fixed Point Properties of Affine Representations of Semigroups. Proc. Amer. Math. Soc. Volume 149 no. 12, 2021.
  * [12] Michael, E. : Topologies on Spaces of Subsets. Trans. Amer. Math. Soc. 71, 152-182 (1951)
  * [13] Mitchell, T. : Topological semigroups and fixed points. Illinois J. Math. 14, 630–641 (1970)
  * [14] Paterson, Alan L. T. Amenability. Mathematical Surveys and Monographs, 29. American Mathematical Society, Providence, RI (1988).
  * [15] Rickert, Neil W. Amenable groups and groups with the fixed point property. Trans. Amer. Math. Soc. 127 (1967), 221–232.
  * [16] Rudin, W. Functional Analysis. Springer-Verlag 2nd Ed.
  * [17] Wong, J. Invariant Means on Locally Compact Semigroups. Proc. Amer. Math. Soc. 31, no. 1 (1972), 39-45.
  * [18] Zeuner, H. One Dimensional Hypergroups. Advances in Mathematics 76, no. 1 (1989), 1-18.