# Topological Amenability of Semihypergroups

Choiti Bandyopadhyay  Department of Mathematics & Statistics, Indian Institute
of Technology Kanpur, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this article, we introduce and discuss the notion of topological left
amenability in the general setting of semihypergroups. We acquire several
characterizations of the same in terms of certain stationary, ergodic and
operator algebraic properties of the associated measure algebra. Finally, we
discuss and characterize topological amenability of its sub-semihypergroups in
terms of (topological) invariance properties attained on the corresponding
measure algebras of the parent semihypergroup.

###### Key words and phrases:

semihypergroups, hypergroups, topological invariance, topological left
amenability, invariant means

###### 2020 Mathematics Subject Classification:

Primary 43A62, 43A07, 43A10, 43A05, 22A20, 20M12; Secondary 46L05, 46G12,
46E27

##  1\. Introduction

The study of (topological) amenability on a given category of objects, have
been one of the fundamental areas of research in abstract harmonic analysis.
As a result, an extensive research has been carried out on different
equivalent criteria and consequences exhibited by (topological) left/right
amenable groups, semigroups and hypergroups (see [7, 8, 14, 16, 18, 19] for
details). This article extends the concept of topological amenability to the
broader class of (semitopological) semihypergroups, and attains several
characterizations of the same in terms of certain stationary, ergodic,
operator algebraic and hereditary properties exhibited by the corresponding
measure algebras.

A semihypergroup is essentially a locally compact space, where the space of
complex radon measures admits a certain associative convolution-algebra. Of
course, as the name suggests, one can regard a semihypergroup as a
generalization of a locally compact semigroup, where the product of two points
is a compact set, rather than a single point. The study of (semi)hypergroups
was initiated by Dunkl [9], Jewett [10] and Spector [17] independently around
1970’s and has seen substantial progress since then, as intriguing connections
to natural structures arising from different fields of mathematics are being
discovered till date [6].

Other than locally compact semigroups, groups and hypergroups, some important
examples of semihypergroups include certain coset spaces, homogeneous spaces
and certain orbit spaces arising in different areas of research such as Lie
groups, dynamical systems, topological graph theory and partial differential
equations, to name a few (see [2, 6, 10] for examples explaining why these
spaces fail to even have a hypergroup structure). The intrinsically different
nature of the convolution algebra (and hence of the subsequent topology) when
compared to the natural convolution of measures on a locally compact
semigroup, prevents the existing classical results of semigroup theory from
extending readily to this broader class of objects. We have organized the
article as the following.

The next, i.e, second section of the article contains a basic set of
preliminary notations and definitions that we would need for the rest of the
article. In the third section, we first introduce topological invariant means
and amenability in the category of semitopological semihypergroups, followed
by a brief exposition of F-algebras and associated amenability concepts. The
first main theorem (Theorem 3.2) explores several equivalent criteria for the
existence of a topological left invariant mean (TLIM) on a semihypergroup, in
terms of certain stationary, ergodic and operator algebraic criteria.

The rest of this section is dedicated to understanding the relation between a
sub-semihypergroup and its parent semihypergroup, in terms of topological
invariance as well as extensions and restrictions of certain measures and
functions. Theorem 3.4 provides necessary insight into how the restrictions of
convolution of measures interact with the convolutions of the restricted
measures. This in turn provides us with the assertion that these measures
indeed coincide whenever the semihyergroup in question is a hypergroup. Using
this phenomenon, in the final result of this section (Theorem 3.5) we acquire
some useful equivalent criteria for a sub-semihypergroup $H$ to admit a TLIM,
in terms of certain $H$-invariant means of the parent semihypergroup. This
extends a result of Wong [21] even in the case of locally compact
semitopological semigroups.

##  2\. Preliminary

Here we first list some of the preliminary set of basic notations that we will
use throughout the text, followed by a brief introduction to the tools and
concepts needed for the formal definition of a semihypergroup (referred to as
‘semiconvo’ in [10]). All the topologies throughout this text are assumed to
be Hausdorff, unless otherwise specified.

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

Next, we introduce two very important topologies on the positive measure space
and the space of compact subsets for any locally compact topological space
$X$. Unless mentioned otherwise, we will always assume these two topologies on
the respective spaces.

The cone topology on $M^{+}(X)$ is defined as the weakest topology on
$M^{+}(X)$ for which the maps $\mu\mapsto\int_{X}f\ d\mu$ is continuous for
any $f\in C_{c}^{+}(X)\cup\\{1_{X}\\}$ where $1_{X}$ denotes the
characteristic function of $X$. Note that if $X$ is compact then it follows
immediately from the Riesz representation theorem that the cone topology
coincides with the weak*-topology on $M^{+}(X)$ in this case.

We denote by $\mathfrak{C}(X)$ the set of all compact subsets of $X$. The
Michael topology [13] on $\mathfrak{C}(X)$ is defined to be the topology
generated by the sub-basis $\\{\mathcal{C}_{U}(V):U,V\textrm{ are open sets in
}X\\}$, where

$\mathcal{C}_{U}(V)=\\{C\in\mathfrak{C}(X):C\cap U\neq\O,C\subset V\\}.$

Note that $\mathfrak{C}(X)$ actually becomes a locally compact Hausdorff space
with respect to this natural topology. For any locally compact Hausdorff space
$X$ and 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 holds :

  1. (1)

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

  2. (2)

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

Now we are ready to state the formal definition for a semihypergroup. Note
that we follow Jewett’s notion [10] 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.

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

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

It is known [10, 13] that this convolution is a continuous binary operation on
$\mathfrak{C}(K)$ and for any Borel subspace $E\subseteq K$, we have that
$\mathfrak{C}(E)$ is again a Borel subspace of $\mathfrak{C}(K)$, with
$\mathfrak{C}(E)$ being closed (resp. compact) if $E$ is closed (resp.
compact) in $K$.

A pair $(K,*)$ is called a semitopological semihypergroup if if it satisfies
all the conditions of Definition 2.1 with property (A$2$) and (A$4$) replaced
respectively by the following:

(A2′):

The map $(\mu,\nu)\mapsto\mu*\nu$ is positive and separately continuous on
$M^{+}(K)\times M^{+}(K)$.

(A4′):

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

Recall that for any locally compact Hausdorff space $X$, a homeomorphism
$i:X\rightarrow X$ is called a topological involution if $i\circ i(x)=x$ for
each $x\in X$. For a semihypergroup $K$, a topological involution
$i:K\rightarrow K$ given by $i(x):=\tilde{x}$ is called a hypergroup
involution if for any $\mu,\nu\in M(K)$ we have that
$(\mu*\nu)^{\widetilde{}}=\tilde{\nu}*\tilde{\mu}$, where for any measure
$\omega\in M(K)$ we have that
$\tilde{\omega}(B):=\omega(\tilde{B})=\omega(i(B))$ for any Borel measurable
subset $B$ of $K$.

As expected, a hypergroup 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 2.2.

(Hypergroup) A pair $(H,*)$ is called a (topological) hypergroup if it is a
semihypergroup and 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 a hypergroup involution $x\mapsto\tilde{x}$ on $H$ such that
$e\in\mbox{supp}(p_{x}*p_{y})$ if and only if $x=\tilde{y}$.

The element $e$ in the above definition is called the identity of $H$. Note
that the identity and the involution are necessarily unique [10].

######  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 we
can then extend ‘$*$’ linearly to $M_{F}^{+}(K)$. As $M_{F}^{+}(K)$ is dense
in $M^{+}(K)$ [10], we can further extend ‘$*$’ to $M^{+}(K)$ and hence to the
whole of $M(K)$ using linearity.

As mentioned in the introduction above, important examples of semihypergroups
which fail to be hypergroups, include coset spaces and certain orbit spaces
with non-automorphic actions arising from locally compact groups (see [2,
Section 3], [6, 10] for details on such natural structures as well as why most
of these structures fail to retain a hypergroup structure).

For the rest of this article, unless otherwise specified, $K$ will denote a
semitopological semihypergroup. A closed subset $H$ of $K$ is called a sub-
semihypergroup if it is closed under convolution, i.e, if $H*H\subseteq H$.
Similarly, if $K$ is a hypergroup, then a closed subset $Q$ of $K$ is called a
sub-hypergroup if it is closed under convolution and hypergroup involution,
i.e, if $Q*Q\subseteq Q$ and $\widetilde{Q}=Q$.

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

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

The right centre of $K$, denoted as $Z_{r}(K)$, is defined as the following:

$Z_{r}(K):=\\{x\in K:\mbox{supp}(p_{x}*p_{y})\mbox{ is singleton for each
}y\in K\\}.$

We can similarly define the left centre of a semihypergroup (see [4, 5, 15]
for examples and detailed discussions on centres of semihypergroups as well as
hypergroups). Given an element $x\in K$ and a Borel set $E\subseteq K$, we
introduce the following two notions, adhereing to the conventional notions of
‘left inverse sets’ in the theory of topological semigroups.

$\displaystyle\tilde{x}E$ $\displaystyle:=$ $\displaystyle\\{y\in
K:(p_{x}*p_{y})(E)>0\\},$ $\displaystyle x^{-1}E$ $\displaystyle:=$
$\displaystyle\\{y\in K:(p_{x}*p_{y})(E)=1\\}.$

Note that both these sets are Borel subsets of $K$, since
$\tilde{x}E=\mbox{supp}(L_{x}\chi_{{}_{E}})$ and translates of Borel functions
on a semihypergroup are Borel measurable as well [10]. Furthermore,
$x^{-1}E=\tau^{-1}(\mathfrak{C}(E))$ where $\tau:K\rightarrow\mathfrak{C}(E)$
is the continuous map [10, 13] given by $\tau(z):=\\{x\\}*\\{z\\}$. We follow
the same convention and define ‘right inverse sets’ on a semihypergroup $K$ in
the following manner:

$\displaystyle E\tilde{x}$ $\displaystyle:=$ $\displaystyle\\{y\in
K:(p_{y}*p_{x})(E)>0\\},$ $\displaystyle Ex^{-1}$ $\displaystyle:=$
$\displaystyle\\{y\in K:(p_{y}*p_{x})(E)=1\\}.$

######  Remark 2.4.

We immediately see that if $x\in Z_{r}(K)$ or $x\in E$ where $E$ is a right
ideal, i.e, $E*K\subseteq E$, then we have that $\tilde{x}E=x^{-1}E$ (see [2]
for details on ideals of semihypergroups). In fact, if $K$ is a hypergroup,
then we have that $\tilde{x}E\subseteq\\{\tilde{x}\\}*E$, since
$(\\{x\\}*\\{y\\})\cap E\neq\O$ if and only if $y\in\\{\tilde{x}\\}*E$ [10,
Lemma 4.1B]. Moreover, if $E$ has non-empty interior, then we have that
$\tilde{x}E=\\{\tilde{x}\\}*E$. Hence if $K$ is a hypergroup and $Q\subseteq
K$ is any sub-hypergroup, then $\tilde{x}Q=x^{-1}Q=Q$ for any $x\in Q$, as
$\\{\tilde{x}\\}*Q=Q$.

We say that a subset $E$ of a semitopological semihypergroup $K$ is of Type L
(resp. of Type R), if we have that $\tilde{x}E=x^{-1}E$ (resp.
$E\tilde{x}=Ex^{-1}$) for each $x\in E$.

##  3\. Topological Left Amenability

In this section, we first introduce the concept of topological amenability in
the context of a semihypergroup, and then proceed to discuss some hereditary
properties of the same, as well as those regarding (classical) left
amenability.

Whenever appropriate, we identify a bounded measurable function $f$ on $K$
with the function $\phi_{f}\in M(K)^{*}$ defined as $\phi_{f}(\mu)=\int_{K}f\
d\mu$. We know that $M(K)$ is an associative Banach algebra, and hence
adhering to the Arens product structure (see [1, 2, 5] for details) on
$M(K)^{**}$, for each $\mu\in P(K),\phi\in M(K)^{*}$, we define the left
translate of $\phi$ by $\mu$, denoted as $L_{\mu}\phi\in M(K)^{*}$, as

$L_{\mu}\phi(\nu):=\phi(\mu*\nu)$

for each $\nu\in M(K)$. Note that no confusion in notations should arise here,
when compared to the left translation $L_{x}f$ for a function $f$ on $K$, as
it is understood that for each $y\in K$, we have that

$L_{x}f(y)=\phi_{f}(p_{x}*p_{y})=L_{p_{{}_{x}}}\phi_{f}(p_{y}),$

whenever all the quantities exist. In fact, it immediately follows [10, pp 11]
that for each $x\in K$ and bounded measurable function $f$ on $K$, we have
that $\phi_{L_{x}f}=L_{p_{{}_{x}}}\phi_{f}$, since for each $\nu\in M(K)$ we
have that

$\phi_{L_{x}f}(\nu)=\int_{K}\int_{K}f(z)\ d(p_{{}_{x}}*p_{y})(z)\
d\nu(y)=\int_{K}f(z)\ d(p_{x}*\nu)(z)=L_{p_{{}_{x}}}\phi_{f}(\nu).$

We define the right translate $R_{\mu}\phi\in M(K)^{*}$ of $\phi\in M(K)^{*}$
by $\mu\in M(K)$ in a similar manner, as

$R_{\mu}\phi(\nu):=\phi(\nu*\mu),$

for each $\nu\in M(K)$. Recall that a positive linear functional $m$ on
$M(K)^{*}$ with norm $1$ is called a mean, i.e, we have that $m\in M(K)^{**}$
such that $||m||=1$ and $m(\phi)\geq 0$ whenever $\phi\geq 0$ for any $\phi\in
M(K)^{*}$. We denote the set of all means on $M(K)^{*}$ as $\mathcal{M}(K)$.
We know [7] that the following three formulations of a mean $m\in M^{**}(K)$
are equivalent:

  1. (1)

$m$ is positive and $||m||=1$.

  2. (2)

$||m||=m(\mathbb{1})=1$, where $\mathbb{1}$ denotes the function $\chi_{K}\in
M(K)^{*}$ given as $\mu\mapsto\mu(K)$.

  3. (3)

$\inf\\{\phi(\nu):\nu\in P(K)\\}\leq m(\phi)\leq\sup\\{\phi(\nu):\nu\in
P(K)\\}$ for each $\phi\in M(K)^{*}$.

We say that a mean $m\in\mathcal{M}(K)$ is a topological left-invariant mean
(TLIM) if $m(L_{\nu}\phi)=m(\phi)$ for any $\nu\in P(K)$, $\phi\in M(K)^{*}$.
Similarly, we say that a mean $m\in\mathcal{M}(K)$ is a left-invariant mean
(LIM) if $m(L_{p_{{}_{x}}}\phi)=m(\phi)$ for any $x\in K$, $\phi\in M(K)^{*}$.
A semitopological semihypergroup is called topological left amenable (resp.
left amenable), if it admits a topological left-invariant mean (resp. left-
invariant mean).

######  Remark 3.1.

In some previous works [2, 5] we defined the left amenability of a
semitopological semihypergroup $K$ in terms of the existence of a left
invariant mean on the function space $C(K)$. The notion of topological
amenability used in the present article in terms of means on $M(K)^{*}$ is a
more general formulation of amenability, and readily implies amenability in
terms of means on $C(K)$ via the natural inclusion $C(K)\subseteq M(K)^{*}$.

The following theorem offers several characterizations of topological
amenability for a semihypergroup in terms of certain stationary, ergodic and
operator algebraic conditions. Certain equivalences for the category of
locally compact semigroups, were proved by Wong in [20]. We make use of
similar techniques in certain parts of the proof. Before stating the theorem,
we need a quick detour through the basics of Banach algebra amenability and
F-algebras [12, 14].

For a Banach algebra $A$, a Banach space $X$ is called a left Banach
$A$-module if there is a bounded bi-linear map $(a,x)\mapsto a.x:A\times
X\rightarrow X$ such that $a_{1}.(a_{2}.x)=(a_{1}.a_{2}).x$ for each
$a_{1},a_{2}\in A,x\in X$. A right Banach $A$-module is defined likewise,
along with an associative bounded bilinear map $(x,a)\mapsto x.a:X\times
A\rightarrow X$. A Banach $A$-bimodule is a Banach space $X$ which is both
left and right Banach $A$-module such that

$a_{1}.(x.a_{2})=(a_{1}.x).a_{2}$

for each $a_{1},a_{2}\in A,x\in X$. Recall [14] that for every Banach
$A$-bimodule $X$, we have that $X^{*}$ is naturally a Banach $A$-bimodule as
well, via the following operations for each $a\in A,\phi\in X^{*},x\in X$.

$(a.\phi)(x):=\phi(x.a),\ \ (\phi.a)(x):=\phi(a.x).$

Given a Banach $A$-bimodule $X$, a derivation from $A$ into $X$ is a linear
map $D:A\rightarrow X$ such that

$D(a.b)=D(a).b+a.D(b)$

for each $a,b\in A$. We immediately see that for each $x\in X$, the map
$D_{x}(a):=a.x-x.a$ is a bounded derivation from $A$ into $X$. A derivation
$D$ from $A$ into $X$ is called an inner derivation if $D=D_{x}$ for some
$x\in X$. We know that a Banach algebra $A$ is called amenable if for any
Banach $A$-bimodule $X$ we have that every bounded derivation of $A$ into
$X^{*}$ is inner. We know that [11] a locally compact group $G$ is amenable if
and only if $L^{1}(G)$ is amenable as a Banach algebra, although the
implication fails to be true for the broader categories of locally compact
semigroups and hypergroups (see [8, 11, 16] for details on this subject).

Recall that as introduced by Lau in [12], a Banach algebra $A$ is called an
F-algebra (referred to as Lau algebra by Pier in [14]) if $A$ is the (unique)
predual of a W∗-algebra $M$ such that the identity $e_{M}$ of $M$ is a
multiplicative linear functional on $A$. We denote the predual of $M$ as
$M_{*}=A$.

For any (semitopological) semihypergroup $K$, we know that $C(K)^{*}=M(K)$.
Since $C(K)$ is a commutative C∗-algebra with the point-wise product, we
immediately see that $M(K)^{*}$ is a W∗-algebra. In addition, since
$(\mu*\nu)(K)=\mu(K)\nu(K)$ for any $\mu,\nu\in M(K)$, we have that $M(K)$ is
indeed an F-algebra. An F-algebra $A$ with $A=M_{*}$ (i.e, $A^{*}=M$) is
called left amenable if for each Banach $A$-bimodule $X$ with the left module
action given by

$a.x=\langle a,e_{M}\rangle x\ \ \mbox{ for each }\ \ a\in A,x\in X,$

we have that every bounded derivation from $A$ into $X^{*}$ is inner.

Finally, recall [7, 20] that the study of (topological) stationary properties
is of pivotal importance in the study of amenability properties of locally
compact groups and semigroups in general. We introduce the concept in a
similar manner for the category of (semitopological) semihypergroups. We say
that a (semitopological) semihypergroup $K$ is topological right stationary if
there exists a net $\\{\nu_{\alpha}\\}\subset P(K)$ such that for each
$\phi\in M(K)^{*}$ there exists some scalar $c_{0}\in\mathbb{C}$ such that
$R_{\nu_{\alpha}}\phi\longrightarrow^{w^{*}}\ c_{0}\mathbb{1}$ in $M(K)^{*}$.

######  Theorem 3.2.

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

  1. (1)

$K$ admits a TLIM.

  2. (2)

$\exists$ a net $\\{\nu_{\alpha}\\}$ in $P(K)$ such that for each $\nu\in
P(K)$, the net $\\{(\nu*\nu_{\alpha}-\nu_{\alpha})\\}$ converges to $0$ in
norm topology on $M(K)$.

  3. (3)

For each $\mu\in M(K)$, we have $|\mu(K)|=\inf\\{||\mu*\nu||:\nu\in P(K)\\}$.

  4. (4)

$K$ is topological right stationary.

  5. (5)

$M(K)$ is left amenable as an F-algebra.

###### Proof.

$(2)\Rightarrow(1)$ is trivial by setting $m\in\mathcal{M}(K)$ to be the
weak∗-limit of $\mu_{\alpha}$’s.

$(1)\Rightarrow(2)$ follows in a similar manner as in the case for locally
compact semigroups [18] (a similar construction was given in the setting of
semihypergroups in [5, Section 3]).

$(3)\Rightarrow(2)$: We first fix some $\nu_{0}\in P(K)$, and construct a net
$\\{\nu_{\alpha}\\}_{\alpha\in J}\subseteq P(K)$ that satisfies the condition
in $(2)$ for each $\nu\in P(K)$. For any $n\in\mathbb{N}$, here $P(K)^{n}$
denotes the product space $\\{(\mu_{1},\mu_{2},\ldots,\mu_{n}):\mu_{k}\in
P(K)\mbox{ for }1\leq k\leq n\\}$. We define the index set $J$ as the
following.

$J:=\\{\alpha:\alpha=(n_{\alpha},P_{\alpha},\varepsilon_{\alpha})\mbox{ where
}n_{\alpha}\in\mathbb{N},P_{\alpha}=(\mu_{1}^{\alpha},\mu_{2}^{\alpha},\ldots,\mu_{n_{\alpha}}^{\alpha})\in
P(K)^{n_{\alpha}},0<\varepsilon_{\alpha}<1\\}.$

For any $\alpha,\beta\in J$ we say that $\alpha\geq\beta$ if $n_{\alpha}\geq
n_{\beta}$, $P_{\alpha}\supseteq P_{\beta}$ and
$\varepsilon_{\alpha}\leq\varepsilon_{\beta}$. It is easy to see that
$(J,\leq)$ is a directed set since given $\alpha,\beta\in J$, we will always
get an upper bound $\gamma=(n_{\gamma},P_{\gamma},\varepsilon_{\gamma})\in J$
where $P_{\gamma}=P_{\alpha}\cup P_{\beta}$, $n_{\gamma}=|P_{\gamma}|$ and
$\varepsilon_{\gamma}=\min(\varepsilon_{\alpha},\varepsilon_{\beta})$.

Now pick any $\alpha\in J$. Since $P(K)*P(K)\subseteq P(K)$, for each
$k=1,2,\ldots,n_{\alpha}$ we have that
$(\mu_{k}^{\alpha}*\mu_{0}-\mu_{0})(K)=0$, and hence for any $\omega\in P(K)$
we have that

$\big{(}(\mu_{k}^{\alpha}*\mu_{0}-\mu_{0})*\omega\big{)}(K)=0.$

In particular, our assumption in $(3)$ implies that

$\inf\\{||(\mu_{1}^{\alpha}*\mu_{0}-\mu_{0})*\nu||:\nu\in P(K)\\}=0.$

Pick $\nu_{1}^{\alpha}\in P(K)$ such that
$||(\mu_{1}^{\alpha}*\mu_{0}-\mu_{0})*\nu_{1}^{\alpha}||\leq\varepsilon_{\alpha}$.
Next, since
$\big{(}(\mu_{2}^{\alpha}*\mu_{0}-\mu_{0})*\nu_{1}^{\alpha}\big{)}(K)=0$ as
well, using the condition in $(3)$ again we get some $\nu_{2}^{\alpha}\in
P(K)$ such that
$||(\mu_{2}^{\alpha}*\mu_{0}-\mu_{0})*\nu_{1}^{\alpha}*\nu_{2}^{\alpha}||\leq\varepsilon_{\alpha}$.
Proceeding this way inductively, for each $1\leq k\leq n_{\alpha}$ we get some
$\nu_{k}^{\alpha}\in P(K)$ such that

$||(\mu_{k}^{\alpha}*\mu_{0}-\mu_{0})*\nu_{1}^{\alpha}*\nu_{2}^{\alpha}\cdots*\nu_{k}^{\alpha}||\leq\varepsilon_{\alpha}.$

For each $\alpha\in J$ we now set
$\nu_{\alpha}:=\mu_{0}*\nu_{1}^{\alpha}*\nu_{2}^{\alpha}\cdots*\nu_{n_{\alpha}}^{\alpha}\in
P(K)$. Given any $\nu\in P(K)$ and $\varepsilon>0$, pick $\alpha_{0}\in J$
such that $\nu\in P_{\alpha_{0}}$ and $\varepsilon_{\alpha_{0}}<\varepsilon$.
Then for each $\alpha\geq\alpha_{0}$ there exists some
$j_{\alpha}\in\\{1,2,\ldots,n_{\alpha}\\}$ such that
$\nu=\mu_{j_{\alpha}}^{\alpha}\in P_{\alpha}\supseteq P_{\alpha_{0}}$. Hence
we have that

$\displaystyle||\nu*\nu_{\alpha}-\nu_{\alpha}||$ $\displaystyle=$
$\displaystyle||(\nu*\mu_{0}-\mu_{0})*\nu_{1}^{\alpha}*\nu_{2}^{\alpha}\cdots*\nu_{n_{\alpha}}^{\alpha}||$
$\displaystyle\leq$
$\displaystyle||(\nu*\mu_{0}-\mu_{0})*\nu_{1}^{\alpha}*\nu_{2}^{\alpha}\cdots*\nu_{j_{\alpha}}^{\alpha}||\>||\nu_{j_{\alpha}+1}^{\alpha}*\cdots*\nu_{n_{\alpha}}^{\alpha}||$
$\displaystyle=$
$\displaystyle||(\mu_{j_{\alpha}}^{\alpha}*\mu_{0}-\mu_{0})*\nu_{1}^{\alpha}*\nu_{2}^{\alpha}\cdots*\nu_{j_{\alpha}}^{\alpha}||\
\leq\ \varepsilon_{\alpha}\leq\varepsilon_{\alpha_{0}}<\varepsilon,$

as required.

$(5)\Leftrightarrow(1)$: It was shown in [12, Theorem 4.1] that an F-algebra
$A$ is left amenable if and only if there exists an element $m\in
P_{1}(A^{**}):=\\{m\in A^{**}:m\geq 0,\ m(e_{M})=1\\}$ such that
$m(x.\phi)=m(x)$ for each $\phi\in P_{1}(A^{**})\cap A$, $x\in A^{*}$, where
$x.\phi\in A^{*}$ is defined as

$(x.\phi)(\gamma):=x(\phi.\gamma),$

for each $\gamma\in A$. In particular, letting $(A,\cdot)=(M(K),*)$ we
immediately see that $P_{1}(A^{**})\cap A=P(K)$ and hence for each $\nu\in
P(K)$, $\phi\in M(K)^{*}$ we have that $(\phi.\nu)=L_{\nu}\phi$, since

$(\phi.\nu)(\mu)=\phi(\nu*\mu)=L_{\nu}\phi(\mu)$

for each $\mu\in M(K)$. Thus we readily have that $M(K)$ is a left amenable
F-algebra if and only if $K$ admits a TLIM.

$(5)\Leftrightarrow(4)\Leftrightarrow(2)$: For any F-algebra $A$, for each
$\phi\in P_{1}(A^{**})\cap A$ and $x\in A^{*}$, the element $\phi.x\in A^{*}$
is defined as

$(\phi.x)(\gamma):=x(\gamma.\phi),$

for each $\gamma\in A$. Hence again, setting $(A,\cdot)=(M(K),*)$ in [12,
Theorem 4.6] and observing that $(\nu.\phi)=R_{\nu}\phi$ for each $\nu\in
P(K)=P_{1}(A^{**})\cap A$, $\phi\in M(K)^{*}$, we have the desired
implications.

$(4)\Rightarrow(3)$: Suppose that $K$ is topological right stationary. Pick
any $\mu\in M(K)$ and set $\zeta_{\mu}:=\inf\\{||\mu*\nu||:\nu\in P(K)\\}$. We
first observe that $|\mu(K)|\leq\zeta_{\mu}$ since for each $\nu\in P(K)$ we
have that

$|\mu(K)|=|\mu(K)\nu(K)|=|(\mu*\nu)(K)|\leq|(\mu*\nu)|(K)=||\mu*\nu||.$

Now consider the convex set $S_{\mu}:=\\{\mu*\nu:\nu\in P(K)\\}$. Since
$S_{\mu}$ is bounded below by $\zeta_{\mu}$, by an application of the Hahn-
Banach extension theorem, there exists $\psi_{\mu}\in M(K)^{*}$ such that
$||\psi_{\mu}||=1$ and $|\psi_{\mu}(\mu*\nu)|\geq\zeta_{\mu}$ for each $\nu\in
P(K)$. For each $\nu,\omega\in P(K)$ we have that

$\displaystyle|R_{\nu}\psi_{\mu}(\mu*\omega)|$ $\displaystyle=$
$\displaystyle|\psi_{\mu}(\mu*\omega*\nu)|$ $\displaystyle=$
$\displaystyle|\psi_{\mu}(\mu*(\omega*\nu))|\geq\zeta_{\mu},$

where the inequality follows since $\omega*\nu\in P(K)$. But $K$ is
topological right stationary, and hence there exists a scalar
$c_{0}\in\mathbb{C}$ such that $c_{0}\mathbb{1}$ lies in the weak∗-closure of
the convex set $\\{R_{\nu}\psi_{\mu}:\nu\in P(K)\\}\subseteq M(K)^{*}$. In
particular, we have that $|c_{0}\mathbb{1}(\mu*\omega)|\geq\zeta_{\mu}$ for
each $\omega\in P(K)$. But $c_{0}\mathbb{1}\equiv
c_{0}\mu(K)=c_{0}\mathbb{1}(\mu)$ on $S_{\mu}$. Hence in particular, we have
the following set of inequalities.

$\displaystyle\zeta_{\mu}\ \leq\ |c_{0}\mu(K)|$ $\displaystyle=$
$\displaystyle\inf\\{|c_{0}\mathbb{1}(\mu*\omega)|:\omega\in P(K)\\}$
$\displaystyle\leq$ $\displaystyle|c_{0}|\inf\\{||\mu*\omega||:\omega\in
P(K)\\}$ $\displaystyle=$ $\displaystyle|c_{0}|\zeta_{\mu}.$

Since $\zeta_{\mu}\neq 0$ we must have that $|c_{0}|=1$ and

$\zeta_{\mu}=|c_{0}|\>|\mu(K)|=|\mu(K)|,$

as required.

∎

Next, we proceed towards investigating the relation between the topological
amenability of a semihypergroup and its sub-semihypergroups, in terms of the
sub-semihypergroup invariance of a mean on $M(K)^{*}$. Let $H\subseteq K$ be a
sub-semihypergroup of $K$. A mean $m\in\mathcal{M}(K)$ is called topological
left $H$-invariant if for any $\nu\in P(K)$ such that
$\mbox{supp}(\nu)\subseteq H$ and $\phi\in M(K)^{*}$ we have that
$m(L_{\nu}\phi)=m(\phi)$. Similarly, a mean $m\in\mathcal{M}(K)$ is called
left $H$-invariant if for any $y\in H$ and $\phi\in M(K)^{*}$ we have that
$m(L_{p_{{}_{y}}}\phi)=m(\phi)$.

For any Borel set $E\subseteq K$ and $\mu\in M(K)$, we denote the restriction
of $\mu$ on $E$ as $\mu_{{}_{E}}$, i.e, we have that $\mu_{{}_{E}}\in M(E)$
where $\mu_{{}_{E}}(B)=\mu(B)$ for any Borel set $B\subseteq E$. Similarly,
the restriction $h|_{{}_{E}}$ of a Borel function $h\in\mathcal{B}(K)$ on $E$
is denoted by $h_{{}_{E}}$. On the other hand, given a measure $\mu\in M(E)$,
we denote the natural extension of $\mu$ to $M(K)$ as $\mu^{e}$, i.e, we have
that $\mu^{e}\in M(K)$ where $\mu^{e}(B):=\mu(B\cap E)$ for any Borel set
$B\subseteq K$. Similarly, the extension of a Borel function
$h\in\mathcal{B}(E)$ to $K$ is denoted by $h^{e}$, and is defined as
$h^{e}(x)=h(x)$ if $x\in E$ and $h^{e}(x)=0$ for any $x\in E^{c}$.

######  Remark 3.3.

It follows readily from the definitions and notations above, that for any
Borel set $E\subseteq K$ and a measure $\omega\in M(K)$ such that
$\mbox{supp}(\omega)\subseteq E$, we have that $\omega_{H}^{e}=\omega$.
Moreover, we have that $M(E)=\\{\mu_{{}_{E}}:\mu\in M(K)\\}$ and that

$\int_{E}h^{e}\ d\mu=\int_{K}h^{e}\ d\mu=\int_{E}h\ d\mu_{{}_{E}},$

for any $h\in\mathcal{B}(E)$, $\mu\in M(K)$, whenever the above quantities
exist.

The following theorem provides insight into how the convolution operation of
the measure algebra $M(K)$ is distributed among the restrictions of measures
to $M(H)$ for a sub-semihypergroup $H\subseteq K$.

######  Theorem 3.4.

Let $K$ be a semitopological semihypergroup and $H\subseteq K$ is a sub-
semihypergroup. Then the following statements hold true for any $\mu,\nu\in
M(K)$.

  1. (1)

$||(\mu*\nu)_{{}_{H}}-(\mu_{{}_{H}}*\nu_{{}_{H}})||\leq\int_{K}|\mu|\big{(}(H\tilde{y}\cap
H^{c})\ d|\nu|(y)+\int_{K}|\nu|(\tilde{x}H\cap H^{c})\ d|\mu|(x)$.

  2. (2)

If $\mbox{supp}(\mu)\subseteq H$, then we have that

$||(\mu*\nu)_{{}_{H}}-(\mu_{{}_{H}}*\nu_{{}_{H}})||\leq\int_{H}|\nu|\>(\tilde{y}H\cap
H^{c})\ d\mu(y).$

  3. (3)

If $\mbox{supp}(\mu),\mbox{supp}(\nu)\subseteq H$, then
$(\mu*\nu)_{{}_{H}}=(\mu_{{}_{H}}*\nu_{{}_{H}})$.

  4. (4)

If $K$ is a hypergroup and $H$ a sub-hypergroup of $K$, then
$(\mu*\nu)_{{}_{H}}=(\mu_{{}_{H}}*\nu_{{}_{H}})$ whenever
$\mbox{supp}(\mu)\subseteq H$.

###### Proof.

First to prove $(1)$, pick and fix some $f\in C(H)$. Although the functions
$(R_{y}f)^{e}$ and $R_{y}f^{e}$ do not coincide everywhere on $K$, for any
$x,y\in H$, we immediately have that

$(R_{y}f)^{e}(x)=R_{y}f(x)=\int_{H}f\ d(p_{x}*p_{y})=\int_{K}f^{e}\
d(p_{x}*p_{y})=R_{y}f^{e}(x).$

Hence for each $y\in H$ we have the following.

$\displaystyle\int_{H}f\ d(\mu_{{}_{H}}*\nu_{{}_{H}})$ $\displaystyle=$
$\displaystyle\int_{H}\int_{H}f(x*y)\ d\mu_{{}_{H}}(x)\ d\nu_{{}_{H}}(y)$
$\displaystyle=$ $\displaystyle\int_{H}\int_{H}R_{y}f(x)\ d\mu_{{}_{H}}(x)\
d\nu_{{}_{H}}(y)$ $\displaystyle=$
$\displaystyle\int_{H}\int_{H}(R_{y}f)^{e}(x)\ d\mu(x)\ d\nu_{{}_{H}}(y)$
$\displaystyle=$ $\displaystyle\int_{H}\int_{H}R_{y}f^{e}(x)\ d\mu(x)\
d\nu_{{}_{H}}(y)$ $\displaystyle=$ $\displaystyle\int_{H}\int_{H}f^{e}(x*y)\
d\mu(x)\ d\nu(y).$

Now consider the set $R_{H}:=\\{(x,y)\in(H^{c}\times K)\cup(K\times
H^{c}):(p_{x}*p_{y})(H)>0\\}\subseteq K\times K$. Then for each $f\in C(H)$ we
have the following inequality.

$\displaystyle\Big{|}\int_{H}fd(\mu*\nu)_{{}_{H}}-\int_{H}fd(\mu_{{}_{H}}*\nu_{{}_{H}})\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\int_{K}f^{e}d(\mu*\nu)-\int_{H}\int_{H}f^{e}(x*y)d\mu(x)d\nu(y)\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\int_{K}\int_{K}f^{e}(x*y)d\mu(x)d\nu(y)-\int_{H}\int_{H}f^{e}(x*y)d\mu(x)d\nu(y)\Big{|}$
$\displaystyle=$
$\displaystyle\Big{|}\iint_{{R_{H}}}f^{e}(x*y)d\mu(x)d\nu(y)\Big{|}$
$\displaystyle\leq$
$\displaystyle\iint_{{R_{H}}}\int_{K}\big{|}f^{e}(z)\big{|}d(p_{x}*p_{y})(z)d|\mu|(x)d|\nu|(y)$
$\displaystyle\leq$
$\displaystyle||f||_{\infty}\iint_{R_{H}}d|\mu|(x)d|\nu|(y).$

For any $y\in K$ if $(x,y)\in(H^{c}\times K)\cap{R_{H}}$, then following the
notations introduced in previous section, we must have that $x\in H\tilde{y}$.
Similarly, for any $x\in K$ if $(x,y)\in(K\times H^{c})\cap{R_{H}}$, then we
have that $y\in\tilde{x}H$ as well. Hence finally we have that

$\displaystyle\Big{|}\int_{H}fd(\mu*\nu)_{{}_{H}}-\int_{H}fd(\mu_{{}_{H}}*\nu_{{}_{H}})\Big{|}$
$\displaystyle\leq$ $\displaystyle||f||_{\infty}\Big{(}\iint_{(H^{c}\times
K)\cap{R_{H}}}d|\mu|(x)d|\nu|(y)+\iint_{(K\times
H^{c})\cap{R_{H}}}d|\nu|(y)d|\mu|(x)\Big{)}$ $\displaystyle=$
$\displaystyle||f||_{\infty}\Big{(}\int_{K}\int_{(H\tilde{y}\cap
H^{c})}d|\mu|(x)d|\nu|(y)+\int_{K}\int_{(\tilde{x}H\cap
H^{c})}d|\nu|(y)d|\mu|(x)\Big{)}$ $\displaystyle=$
$\displaystyle||f||_{\infty}\Big{(}\int_{K}|\mu|\big{(}(H\tilde{y}\cap
H^{c})d|\nu|(y)+\int_{K}|\nu|(\tilde{x}H\cap H^{c})d|\mu|(x)\Big{)},$

as required.

The statements $(2)$ and $(3)$ follow immediately from $(1)$. To prove $(4)$,
notice that it follows immediately from Remark 2.4 that for each $y\in H$, we
have that $\tilde{y}H=\\{\tilde{y}\\}*H=H$. Hence $\tilde{y}H\cap H^{c}=\O$
for each $y\in\mbox{supp}(\mu)$, and the conclusion follows from $(2)$. ∎

We now prove the following equivalence between the existence of a TLIM on a
semihypergroup and one of its sub-semihypergroups. The first two equivalences
were proved in the setting of locally compact semigroups in [21]. We follow
similar ideas in proving certain sections of the following theorem. In
addition to the intrinsically different nature of the measure algebra
constructions between semihypergroups and locally compact (semi)groups, one of
the main challenges here is the fact that for a given sub-semihypergroup $H$
of $K$ and $x\in K$, we do not necessarily have that
$L_{x}\chi_{{}_{H}}=\chi_{{}_{\tilde{x}H}}$, unlike locally compact
semigroups, and even hypergroups when $x\in H$.

######  Theorem 3.5.

Let $K$ be a semitopological semihypergroup and $H\subseteq K$ is a sub-
semihypergroup. Then the following properties are equivalent.

  1. (1)

$H$ admits a TLIM.

  2. (2)

$\exists$ a topological left $H$-invariant mean $m\in\mathcal{M}(K)$ such that
$m(\chi_{{}_{H}})=1$.

Moreover, if $H$ is of Type L, then the above properties are equivalent to the
following:

  1. (3)

$\exists$ a topological left $H$-invariant mean $m\in\mathcal{M}(K)$ such that
$m(\chi_{{}_{H}})>0$.

###### Proof.

$(1)\Rightarrow(2)$ : Let $m_{0}\in M(H)^{**}$ be a TLIM of $H$. Consider the
restriction map $r_{H}:M(K)^{*}\rightarrow M(H)^{*}$ defined as

$r_{H}(\phi)(\mu):=\phi(\mu^{e}),$

for each $\phi\in M(K)^{*},\ \mu\in M(H)$. Since $\mu^{e}\geq 0$ whenever
$\mu\geq 0$ in $M(H)$, we immediately see that $r_{H}$ is a positive linear
map on $M(K)^{*}$. The natural extension of $m_{0}$ to $M(K)^{**}$ is the
functional $m\in M(K)^{**}$ defined as $m:=m_{0}\circ r_{H}$.

Now observe that for each $\mu\in P(K)$ where $\mbox{supp}(\mu)\subseteq H$,
we have that $r_{H}(L_{\mu}\phi)=L_{\mu_{{}_{H}}}r_{H}(\phi)$, since for each
$\nu\in M(H)$ and $\phi\in M(K)^{*}$ we have the following equality.

$\displaystyle r_{H}(L_{\mu}\phi)(\nu)$ $\displaystyle=$ $\displaystyle
L_{\mu}\phi(\nu^{e})$ $\displaystyle=$ $\displaystyle\phi(\mu*\nu^{e})$
$\displaystyle=$ $\displaystyle\phi((\mu*\nu^{e})_{H}^{e})$ $\displaystyle=$
$\displaystyle r_{H}(\phi)(\mu*\nu^{e})_{H}$ $\displaystyle=$ $\displaystyle
r_{H}(\phi)(\mu_{H}*\nu_{H}^{e})\ =\ r_{H}(\phi)(\mu_{H}*\nu)\
=L_{\mu_{{}_{H}}}r_{H}(\phi)(\nu),$

where the third and sixth equalities follow from Remark 3.3 and since

$\mbox{supp}(\mu*\nu^{e})=\overline{\mbox{supp}(\mu)*\mbox{supp}(\nu^{e})}\subseteq\overline{H*H}\subseteq
H$

and the fifth equality follows from Theorem 3.4. Hence for any $\nu\in P(K)$
such that $\mbox{supp}(\nu)\subseteq H$ and $\phi\in M(K)^{*}$ we have that

$\displaystyle m(L_{\nu}\phi)\ =\ m_{0}\circ r_{H}(L_{\nu}\phi)$
$\displaystyle=$ $\displaystyle
m_{0}\big{(}L_{\nu_{{}_{H}}}r_{H}(\phi)\big{)}$ $\displaystyle=$
$\displaystyle m_{0}\big{(}r_{H}(\phi)\big{)}\ =\ m(\phi),$

where the last equality follows since $m_{0}$ is a TLIM. Thus we see that $m$
is topologically $H$-invariant. Moreover, since $r_{H}$ is a positive linear
map, we have that $m$ is a positive linear functional on $M(K)^{*}$ as well,
such that $||m||\leq 1$. Finally, since

$m(\chi_{{}_{H}})=m_{0}\circ r_{H}(\chi_{H})=m_{0}(\chi_{{}_{H}})=1,$

we see that $m$ is indeed a mean on $K$, as needed.

$(2)\Rightarrow(1)$ : On the other hand, let $m$ be a topological left
$H$-invariant mean in $\mathcal{M}(K)$ such that $m(\chi_{{}_{H}})=1$.
Consider the extension map $e_{H}:M(H)^{*}\rightarrow M(K)^{*}$ defined as

$e_{H}(\phi)(\mu):=\phi(\mu_{{}_{H}}),$

for each $\phi\in M(H)^{*},\ \mu\in M(K)$. Hence the natural restriction of
$m$ to $M(H)^{**}$ is the functional $m_{0}\in M(H)^{**}$ defined as
$m_{0}:=m\circ e_{H}$. Similarly as before, we immediately see that $e_{H}$ is
a positive linear map on $M(K)^{*}$, and hence $m_{0}\in\mathcal{M}(H)$.

Now to see that $m_{0}$ is indeed topological left invariant, first note that
for each $\phi\in M(H)^{*}$, $\nu\in P(H)$ and $\mu\in M^{+}(K)$ we have the
following:

$\displaystyle\big{|}e_{H}(L_{\nu}\phi)(\mu)-L_{{\nu^{e}}}e_{H}(\phi)(\mu)\big{|}$
$\displaystyle=$
$\displaystyle\big{|}L_{\nu}\phi(\mu_{{}_{H}})-e_{H}(\phi)(\nu^{e}*\mu)\big{|}$
$\displaystyle=$
$\displaystyle\big{|}\phi(\nu*\mu_{{}_{H}})-\phi\big{(}(\nu^{e}*\mu)_{{}_{H}}\big{)}\big{|}$
$\displaystyle\leq$
$\displaystyle||\phi||\>||((\nu^{e})_{{}_{H}}*\mu_{{}_{H}})-(\nu^{e}*\mu)_{{}_{H}}||$
$\displaystyle\leq$ $\displaystyle||\phi||\>|\mu|(H^{c})\>\nu^{e}(H)\ =\
||\phi||\>\mu(H^{c})\ =\ ||\phi||\>\chi_{{}_{H^{c}}}(\mu),$

where the first inequality follows since $\mbox{supp}(\nu^{e})\subseteq H$ and
the second inequality follows from Theorem 3.4. Hence for each $\phi\in
M(H)^{*}$ and $\nu\in P(H)$ we have the following invariance, as required.

$\displaystyle\big{|}m_{0}(L_{\nu}\phi)-m_{0}(\phi)\big{|}$ $\displaystyle=$
$\displaystyle\big{|}m(e_{H}(L_{\nu}\phi))-m(e_{H}(\phi))\big{|}$
$\displaystyle=$
$\displaystyle\big{|}m\big{(}e_{H}(L_{\nu}\phi)-L_{\nu^{e}}e_{H}(\phi)\big{)}\big{|}$
$\displaystyle\leq$ $\displaystyle||\phi||\>m\big{(}\chi_{{}_{H^{c}}}\big{)}\
=\ 0,$

where the second equality follows since $m\in\mathcal{M}(K)$ is topological
$H$-invariant, the inequality follows since $m$ is a positive linear
functional and the last equality holds true since we have that

$m(\chi_{{}_{H^{c}}})=m(\chi_{{}_{K}}-\chi_{{}_{H}})=m(\chi_{{}_{K}})-m(\chi_{{}_{H}})=0.$

Now we further assume that $H$ is of Type L, i.e, $\tilde{x}H=x^{-1}H$ for
each $x\in H$. Since it is sufficient to show that $(3)\Rightarrow(1)$, let
$m$ be a topological left $H$-invariant mean in $\mathcal{M}(K)$ such that
$m(\chi_{{}_{H}})=:\alpha>0$. Set $m_{0}:=\alpha^{-1}(m\circ e_{H})$. It then
follows readily as above that $m_{0}\in\mathcal{M}(H)$. Now proceeding as
above to check topological invariance of $m_{0}$, we first note that for each
$\phi\in M(H)^{*}$, $\nu\in P(H)$ and $\mu\in M^{+}(K)$ we have the following:

$\displaystyle\big{|}e_{H}(L_{\nu}\phi)(\mu)-L_{{\nu^{e}}}e_{H}(\phi)(\mu)\big{|}$
$\displaystyle=$
$\displaystyle\big{|}L_{\nu}\phi(\mu_{{}_{H}})-e_{H}(\phi)(\nu^{e}*\mu)\big{|}$
$\displaystyle=$
$\displaystyle\big{|}\phi(\nu*\mu_{{}_{H}})-\phi\big{(}(\nu^{e}*\mu)_{{}_{H}}\big{)}\big{|}$
$\displaystyle\leq$
$\displaystyle||\phi||\>||((\nu^{e})_{{}_{H}}*\mu_{{}_{H}})-(\nu^{e}*\mu)_{{}_{H}}||$
$\displaystyle\leq$ $\displaystyle||\phi||\int_{H}\mu(\tilde{y}H\cap H^{c})\
d\nu^{e}(y)$ $\displaystyle=$
$\displaystyle||\phi||\int_{H}\Big{(}\mu(\tilde{y}H\setminus H)\Big{)}\
d\nu^{e}(y)$ $\displaystyle=$
$\displaystyle||\phi||\Big{(}\int_{H}\mu(\tilde{y}H)\
d\nu^{e}(y)-\mu(H)\Big{)}$ $\displaystyle=$
$\displaystyle||\phi||\Big{(}\int_{H}\int_{K}\chi_{{}_{\tilde{y}H}}(z)\
d\mu(z)\ d\nu^{e}(y)-\mu(H)\Big{)}$ $\displaystyle=$
$\displaystyle||\phi||\Big{(}\int_{H}\int_{K}\chi_{H}(y*z)\ d\mu(z)\
d\nu^{e}(y)-\mu(H)\Big{)}$ $\displaystyle=$
$\displaystyle||\phi||\Big{(}\int_{K}\int_{K}\chi_{H}(y*z)\ d\nu^{e}(y)\
d\mu(z)-\mu(H)\Big{)}$ $\displaystyle=$
$\displaystyle||\phi||\big{(}\chi_{H}(\nu^{e}*\mu)-\mu(H)\big{)}\ =\
||\phi||\big{(}L_{\nu^{e}}\chi_{H}-\chi_{H}\big{)}(\mu),$

where the third equality follows since $H\subseteq\tilde{y}H$ as $y\in H$ and
$H$ is a sub-semihypergroup, the sixth equality holds true since $H$ is of
Type L and the second inequality follows from Theorem 3.4. Hence for each
$\phi\in M(H)^{*}$ and $\nu\in P(H)$ we have the following invariance, as
required.

$\displaystyle\big{|}m_{0}(L_{\nu}\phi)-m_{0}(\phi)\big{|}$ $\displaystyle=$
$\displaystyle\alpha^{-1}\big{|}m(e_{H}(L_{\nu}\phi))-m(e_{H}(\phi))\big{|}$
$\displaystyle=$
$\displaystyle\alpha^{-1}\big{|}m\big{(}e_{H}(L_{\nu}\phi)-L_{\nu^{e}}e_{H}(\phi)\big{)}\big{|}$
$\displaystyle\leq$
$\displaystyle\alpha^{-1}||\phi||\>m\big{(}L_{\nu^{e}}\chi_{{}_{H}}-\chi_{{}_{H}}\big{)}\
=\ 0,$

where the second and last equalities follow since $m\in\mathcal{M}(K)$ is
topological $H$-invariant and the inequality follows since $m$ is a positive
linear functional on $M(K)^{*}$. ∎

######  Remark 3.6.

Since any commutative F-algebra is amenable [12], we immediately see that
commutative semihypergroups are topological left amenable. In addition, it
follows trivially that any non-commutative semihypergroup $(K,*)$ where the
convolution is given as $p_{x}*p_{y}=p_{y}$ for each $x,y\in K$, is
topological left amenable .

As mentioned at the beginning of this section, although topological left
amenability of a semihypergroup immediately implies left amenability, it is
still an open question whether the converse is true for general
semitopological semihypergroups. Moreover, although Theorem 3.5 extends the
results of [21] even for the case of locally compact (semitopological)
semigroups, it is not known whether in the setting of a semihypergroup, the
restriction that $H$ be a Type L sub-semihypergroup, can be lifted.

## Acknowledgement

The author would like to gratefully acknowledge the financial support provided
by the Indian Institute of Technology Kanpur, India, throughout the course of
this research.

## References

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2021 TBA tba1 tba2

#  Quasinormal modes of charged fermions in linear dilaton black hole
spacetime: Exact frequencies

İzzet SAKALLI1 , Gülni̇hal TOKGÖZ HYUSEIN1,2

1Physics Department<EMAIL_ADDRESS>Eastern Mediterranean University
Famagusta  North Cyprus
99628 via Mersin 10  Turkey
2Güzeloba Mahallesi  Örnekköy Caddesi  Örnekköy Sitesi  2993 Ada  11KA Blok
No: 2  Muratpaşa
07230 Antalya  Turkey

Corresponding Author: İzzet SAKALLI1
.201 .201 ..201

###### Abstract

We study charged massless fermionic perturbations in the background of
$4$-dimensional linear dilaton black holes in Einstein-Maxwell-dilaton theory
with double Liouville-type potentials. We present the analytical fermionic
quasinormal modes, whose Dirac equations are solved in terms of hypergeometric
functions. We also discuss the stability of these black holes under the
charged fermionic perturbations.

###### doi:

10.3906/fiz-2012-6

###### keywords:

Quasinormal modes, fermionic perturbations, dilaton, Dirac equation, Newman-
Penrose, Liouville-type potential, hypergeometric functions.

##  1 Introduction

At high enough energies, it is quite possible that gravity is not described by
the action of Einstein’s general relativity theory. Today, the literature has
a compelling evidence that in string theory (ST), gravity becomes a scalar-
tensor [1]. The low-energy limit of the ST corresponds to Einstein’s
gravitational attraction, which is non-minimally coupled to a scalar dilaton
field [2]. When the Einstein-Maxwell theory is combined with a dilaton field,
the resulting spacetime solutions have important consequences. There have been
significant studies in the literature to find exact solutions of the Einstein-
Maxwell-dilaton (EMD) theory. For example, in the absence of a dilaton
potential, EMD gravity’s charged dilaton black hole (BH) solutions were found
by many researchers [3, 4, 5, 6, 7, 8, 9, 10, 11]. Essentially, the presence
of the dilaton alters the arbitrary structure of spacetime and causes to
curvature singularities with finite radii. The obtained dilaton BHs in general
have non-asymptotically flat (NAF) structure. In recent years, even in the
various extensions of general relativity, there has been an active interest in
the NAF spacetimes. Among them, probably the most important ones are the
asymptotic anti-de Sitter (AdS) BHs [12], which play a vital role in string
and quantum gravity theories as well as for the AdS/CFT (conformal field
theory) correspondence, which is sometimes called Maldacena duality or
gauge/gravity duality [13].

The BH solutions of EMD theory were studied by many researchers: the uncharged
EMD BH solutions were found in [14, 15, 16], while the charged EMD BH
solutions were considered in [17, 18]. With the inclusion of the Liouville-
type potentials, the static charged BH solutions were found by [19, 20, 21].
The generalization to dyonic (having both electric and magnetic charges) BH
solutions in $4$-dimensional and higher-dimensional EMD gravity with single
and double Liouville-type potentials were also found in [22]. In fact, similar
to the Higgs potential, the double Liouville-type potentials have the ability
to admit local extremes and critical points. However, a single Liouville
potential lacks from these features. Besides, the double Liouville-type
potentials also appear when the higher-dimensional theories are compressed
into a $4$-dimensional spacetime. Many of the BH solutions with the double
Liouville-type potentials to date can be seen from [23] and references
therein. Among them Mazharimousavi et al’s BH [24] in the EMD theory with the
double Liouville-type potentials has some unique features. First of all, those
BHs cover Reissner-Nordström (RN) type BHs and Bertotti-Robinson (BR)
spacetimes [25] interpolated within the same metric. Remarkably, in between
the two spacetimes, there exists a linear dilaton BH (LDBH) [26] solution for
the specific values of the dilaton, double Liouville-type potentials, and EMD
parameters. The particular motivation of this work is to compute the
quasinormal modes (QNMs) [27, 28] for the charged fermionic field
perturbations [29, 30, 31, 32] in the spacetime of those LDBHs. Since the QNMs
can signal information about the stability of BHs, we shall also analyze the
stability of the LDBHs under the charged fermionic perturbations.

Meanwhile, we would like to remind the reader for information purposes that
QNMs are nothing but the energy dissipation of a perturbed BH. Similar to an
ordinary object, when a BH is perturbed it begins to ring with its natural
frequencies, which are the modes of its energy dissipation. These
characteristic modes are called QNMs. The amplitudes of their oscillations
decay in time. The amplitude of the oscillation can be approximated by

$\displaystyle\Psi$ $\displaystyle\approx e^{i\omega t}\text{
}=e^{-\omega_{I}t}\cos(\omega_{R}t),$ (1.1)

where $\omega=\omega_{R}+i\omega_{I}$ is the frequency of the QNM [28]. Here,
$\omega_{R}$ and $\omega_{I}$ are the frequencies of oscillatory and
exponential modes, respectively. After LIGO’s great successes about the
gravitational wave measurements [33], the subject of QNMs has gained
considerable importance in the direct identification of BHs. Today, there are
numerous well-known studies which show that the surrounding geometry of a BH
experiences QNMs under perturbations (see for example [34, 35, 36, 37, 38, 39,
40, 41] and references therein). One of the most important features to know
about QNMs is that they allow us to analyze the quantum entropy/area spectrum
of the BHs. In this regard, the reader is referred to [42, 43, 44, 45, 46, 47,
48] and references therein.

The organization of the present paper is as follows: In Sec. 2, we review the
LDBH solution, which is obtained from the $4$-dimensional action of the EMD
theory having the double Liouville-type potentials. In Sec. 3, we first derive
the spin-$\frac{1}{2}$ field equations by using $4$-dimensional charged
massless Dirac equations within the framework of Newman-Penrose (NP)
formalism. Solution procedure to the obtained Dirac equations is given in Sec.
4. Section 5 is devoted to the computations of the QNM frequencies of the
charged fermions. We summarize our results in Sec. 6. (Throughout the paper,
we use the geometrized units of $G=c=\hbar=1$.)

##  2 LDBH spacetime in EMD theory with double Liouville-type potential

$4$-dimensional action of the EMD theory having the double Liouville-type
potentials is given by [24]

$S=\int\mathrm{d}^{4}x\sqrt{-g}\left(\frac{1}{2}R-\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-V(\varphi)-\frac{1}{2}W(\varphi)\left(F_{\lambda\sigma}F^{\lambda\sigma}\right)\right),$
(2.1)

where

$V(\varphi)=V_{1}\mathrm{e}^{\beta_{1}\varphi}+V_{2}\mathrm{e}^{\beta_{2}\varphi},\text{
\ \ \ \
}W(\varphi)=\lambda_{1}\mathrm{e}^{-2\gamma_{1}\varphi}+\lambda_{2}\mathrm{e}^{-2\gamma_{2}\varphi},$
(2.2)

where $\varphi$ is the dilaton, $V_{1,2}$ are the double Liouville-type
potentials, $\gamma_{1,2}$ denote the dilaton parameter, and
$\lambda_{1,2},\thinspace\beta_{1,2}$ are constants. Moreover, $R$ is the
Ricci scalar and the Maxwell 2-form is given by

$F=d\mathcal{A}.$ (2.3)

For the choice of pure magnetic potential

$\mathcal{A}=-Q\cos\theta d\varphi,$ (2.4)

where $Q$ denotes the charge, we have

$F=Q\sin\theta d\theta\wedge d\varphi.$ (2.5)

It is worth noting that with the current choice of $W(\varphi)$, the magnetic-
electric symmetry, which exists in the standard dilatonic coupling, namely
$\lambda_{1}\left(\lambda_{2}\right)=0,$ is not valid anymore. Thus, as
highlighted in [24], the charged LDBH spacetime is pure magnetic. Variations
of the action (2.1) with respect to the gravitational field $g_{\mu\nu}$ and
the dilaton $\varphi$ yield the following EMD field equations

$R_{\mu\nu}=\partial_{\mu}\varphi\partial_{\nu}\varphi+Vg_{\mu\nu}+W\left(2F_{\mu\lambda}F_{\nu}^{\lambda}-\frac{1}{2}F_{\lambda\sigma}F^{\lambda\sigma}g_{\mu\nu}\right),$
(2.6)
$\nabla^{2}\varphi-V^{\prime}-\frac{1}{2}W^{\prime}\left(F_{\lambda\sigma}F^{\lambda\sigma}\right)=0,$
(2.7)

where $R_{\mu\nu}$ is the Ricci tensor and the prime symbol (′) denotes the
derivative with respect to $\varphi$. Furthermore, the Maxwell equation is
obtained with the variation with respect to $\mathcal{A}$ as

$d\left(W^{\star}F\right)=0$ (2.8)

in which the Hodge star (⋆) refers to duality. After substituting the
following metric (ansatz):

$ds^{2}=B(r)dt^{2}-\frac{dr^{2}}{B(r)}-R(r)d\Omega_{2}^{2},$ (2.9)

into Eqs.(2.6) and (2.7) and in the sequel of compelling calculations as made
in [24], the field equations result in the metric functions of the LDBH as
follows

$B(r)=b\frac{(r-r_{2})(r-r_{1})}{r}\text{ \ \ and \ \ }R(r)=A^{2}r\text{,}$
(2.10)

where

$b=\frac{1}{A^{2}}-2(V_{1}+V_{2})>0,\text{ \ \ \ and \ \
}A^{2}=2\lambda_{2}Q^{2},\text{ \ \ \ }(A:Real\medspace constant).$ (2.11)

The inner and outer horizons of the LDBH are given by [24]

$r_{1}=\frac{1}{2b}\left(c-\sqrt{c^{2}-4ab}\right)\text{ \ and \ \ \
}r_{2}=\frac{1}{2b}\left(c+\sqrt{c^{2}-4ab}\right),$ (2.12)

in which the physical parameters read

$c=4M\text{ \ \ \ \ and \ \ }a=\frac{\lambda_{1}}{\lambda_{2}A^{2}},$ (2.13)

where $M$ denotes the quasilocal mass [49] and

$\varphi\left(r\right)=-\frac{1}{\sqrt{2}}\ln\left(r\right),\text{
}\beta_{1}=\beta_{2}=\sqrt{2},\text{\ \ }\gamma_{1}=-\frac{1}{\sqrt{2}}\text{
},\text{ }\gamma_{2}=\frac{1}{\sqrt{2}},\text{\ \ \ }$ (2.14)

$V=\frac{V_{1}+V_{2}}{r},\text{ \ \ and \ \ \
}W=\frac{\lambda_{1}}{r}+\lambda_{2}r.$ (2.15)

The above spacetime corresponds to a phase transition geometry, which changes
the structure of spacetime from RN to BR [24]. The case of $c^{2}=4ab$ gives
us the extremal ($r_{2}=r_{1}$) LDBHs whose congenerics can be seen in [26,
50].

##  3
Charged Dirac equation in LDBH geometry

In the NP formalism [51], massless Dirac equations with charge coupling are
given as follows [29, 52]

$\displaystyle\left[D+iql^{j}\mathcal{A}_{j}+\varepsilon-\rho\right]F_{1}+\left[\overline{\delta}+iq\overline{m}^{j}\mathcal{A}_{j}+\pi-\alpha\right]F_{2}$
$\displaystyle=0,$ $\displaystyle\left[\delta+iqm^{j}\mathcal{A}_{j}^{\text{ \
\ }}+\beta-\tau\right]F_{1}+\left[\Delta+iqn^{j}\mathcal{A}_{j}^{\text{ \ \
}}+\mu-\gamma\right]F_{2}$ $\displaystyle=0,$
$\displaystyle\left[D+iql^{j}\mathcal{A}_{j}+\overline{\varepsilon}-\overline{\rho}\right]\widetilde{G}_{2}-\left[\delta+iqm^{j}\mathcal{A}_{j}+\overline{\pi}-\overline{\alpha}\right]\widetilde{G}_{1}$
$\displaystyle=0,$ $\displaystyle\left[\Delta+iqn^{j}\mathcal{A}_{j}^{\text{ \
\
}}+\overline{\mu}-\overline{\gamma}\right]\widetilde{G}_{1}-\left[\overline{\delta}+iq\overline{m}^{j}\mathcal{A}_{j}+\overline{\beta}-\overline{\tau}\right]\widetilde{G}_{2}$
$\displaystyle=0,$ (3.1)

where $q$ is the charge of the fermion and $A_{j}$ represents the $j^{th}$
component of the vector potential of the background electromagnetic field
(2.4). The wave functions $F_{1},F_{2},\widetilde{G}_{1},\widetilde{G}_{2}$
represent the Dirac spinors while
$\alpha,\beta,\gamma,\varepsilon,\mu,\pi,\rho,\tau$ are the spin (Ricci
rotation) coefficients. The directional derivatives for NP tetrads are defined
as

$D=l^{j}\nabla_{j},\medspace\Delta=n^{j}\nabla_{j},\medspace\delta=m^{j}\nabla_{j},\medspace\bar{\delta}=\bar{m}^{j}\nabla_{j},$
(3.2)

In the meantime, a bar over a quantity stands for the complex conjugation. We
choose a complex null tetrad $\left\\{l,n,m,\overline{m}\right\\}$ (the
covariant one-forms) for the LDBH geometry as

$\displaystyle l_{j}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\sqrt{B(r)},-\frac{1}{\sqrt{B(r)}},0,0\right],$
$\displaystyle n_{j}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\sqrt{B(r)},\frac{1}{\sqrt{B(r)}},0,0\right],$
$\displaystyle m_{j}$
$\displaystyle=A\sqrt{\frac{r}{2}}\left[0,0,1,i\sin\theta\right],$
$\displaystyle\overline{m}_{j}$
$\displaystyle=A\sqrt{\frac{r}{2}}\left[0,0,1,-i\sin\theta\right],$ (3.3)

which all together satisfy the orthogonality conditions:
$l.n=-m.\overline{m}=1$. The non-zero spin coefficients are obtained as

$\displaystyle\rho$
$\displaystyle=\mu=-\frac{-1}{2\sqrt{2}}\frac{\sqrt{B(r)}}{r},$
$\displaystyle\epsilon$
$\displaystyle=\gamma=\frac{b}{4\sqrt{2B(r)}}\left(1-\frac{r_{2}r_{1}}{r^{2}}\right),$
$\displaystyle\alpha$ $\displaystyle=-\beta=\frac{\cot\theta}{2A\sqrt{2r}}.$
(3.4)

The form of the Dirac equations (3.1) suggests that the spinors can be chosen
as follows

$\displaystyle F_{1}$
$\displaystyle=f_{1}\left(r\right)A_{1}\left(\theta\right)e^{i\left(kt+m\varphi\right)},$
$\displaystyle\widetilde{G}_{1}$
$\displaystyle=g_{1}\left(r\right)A_{2}\left(\theta\right)e^{i\left(kt+m\varphi\right)},$
$\displaystyle F_{2}$
$\displaystyle=f_{2}\left(r\right)A_{3}\left(\theta\right)e^{i\left(kt+m\varphi\right)},$
$\displaystyle\widetilde{G}_{2}$
$\displaystyle=g_{2}\left(r\right)A_{4}\left(\theta\right)e^{i\left(kt+m\varphi\right)},$
(3.5)

where $k$ is the frequency of the wave corresponding to the Dirac particle and
$m$ is the azimuthal quantum number.

##  4
Solution of charged Dirac equation in LDBH geometry

After substituting the spin coefficients (3.4) and the spinors (3.5) into the
Dirac equations (3.1), one gets

$\displaystyle\frac{1}{f_{2}}\widetilde{Z}f_{1}-\frac{\left(LA_{3}\right)}{A_{1}}$
$\displaystyle=0,$
$\displaystyle\frac{1}{f_{1}}\overline{\widetilde{Z}}f_{2}+\frac{\left(L^{{\dagger}}A_{1}\right)}{A_{3}}$
$\displaystyle=0,$
$\displaystyle\frac{1}{g_{1}}\widetilde{Z}g_{2}-\frac{\left(L^{{\dagger}}A_{2}\right)}{A_{4}}$
$\displaystyle=0,$
$\displaystyle\frac{1}{g_{2}}\overline{\widetilde{Z}}g_{1}+\frac{\left(LA_{4}\right)}{A_{2}}$
$\displaystyle=0.$ (4.1)

The radial operators appear in the above simplified Dirac equations are

$\displaystyle\widetilde{Z}$
$\displaystyle=A\sqrt{\Lambda}\partial_{r}+H+\frac{iAkr}{\sqrt{\Lambda}},$
$\displaystyle\overline{\widetilde{Z}}$
$\displaystyle=A\sqrt{\Lambda}\partial_{r}+H-\frac{iAkr}{\sqrt{\Lambda}},$
(4.2)

in which

$\Lambda=b(r-r_{2})(r-r_{1}),$ (4.3)

$H=\frac{A}{2r}\left[\frac{b(r^{2}-r_{2}r_{1})}{2\sqrt{\Lambda}}+\sqrt{\Lambda}\right].$
(4.4)

The angular operators are

$\displaystyle L$
$\displaystyle=\partial_{\theta}+\frac{m}{\sin\theta}+\left(\frac{1}{2}-p\right)\cot\theta,$
$\displaystyle L^{{\dagger}}$
$\displaystyle=\partial_{\theta}-\frac{m}{\sin\theta}+\left(\frac{1}{2}+p\right)\cot\theta,$
(4.5)

where $p=qQ$. Further, choosing

$f_{1}=g_{2},\medspace f_{2}=g_{1},\medspace A_{1}=A_{2},\medspace
A_{3}=A_{4},$ (4.6)

and introducing a real eigenvalue $\lambda$ which is the separation constant
of the complete Dirac equations, one can separate the Dirac equations into two
sets of master equations. The radial master equations read

$\displaystyle\overline{\widetilde{Z}}g_{1}$ $\displaystyle=-\lambda g_{2},$
$\displaystyle\widetilde{Z}g_{2}$ $\displaystyle=-\lambda g_{1},$ (4.7)

and the angular master equations become

$\displaystyle L^{{\dagger}}A_{2}$ $\displaystyle=\lambda A_{4},$
$\displaystyle LA_{4}$ $\displaystyle=-\lambda A_{2}.$ (4.8)

###  4.1
Angular equation

The laddering operators $L$ and $L^{{\dagger}}$ [53] govern the spin-weighted
spheroidal harmonics as

$\displaystyle\left(\partial_{\theta}-\frac{m}{\sin\theta}-s\cot\theta\right)\left({}_{s}Y_{l}^{m}(\theta)\right)$
$\displaystyle=-\sqrt{\left(l-s\right)\left(l+s+1\right)}_{s+1}Y_{l}^{m}\left(\theta\right),$
$\displaystyle\left(\partial_{\theta}+\frac{m}{\sin\theta}+s\cot\theta\right)\left({}_{s}Y_{l}^{m}(\theta)\right)$
$\displaystyle=\sqrt{\left(l+s\right)\left(l-s+1\right)}_{s-1}Y_{l}^{m}\left(\theta\right).$
(4.9)

Eigenfunctions ${}_{s}Y_{l}^{m}(\theta)$, called the spin-weighted spheroidal
harmonics, are complete and orthogonal, and have an explicit form [54]:

${}_{s}Y_{l}^{m}(\theta)$
$\displaystyle=\sqrt{\frac{2l+1}{4\pi}\frac{\left(l+m\right)!\left(l-m\right)!}{\left(l+s\right)!\left(l-s\right)!}}\left(\sin\frac{\theta}{2}\right)^{2l}$
$\displaystyle\times\overset{l}{\underset{r=-l}{\sum\left(-1\right)}}^{l+m-r}\binom{l-s}{r-s}\binom{l+s}{r-m}\left(\cot\frac{\theta}{2}\right)^{2r-m-s},$
(4.10)

where $l$ and $s$ are the angular quantum number and the spin-weight,
respectively. They satisfy the following expressions:

$l=\left|s\right|,\left|s\right|+1,\left|s\right|+2,....\medspace\text{and
}-l<m<+l.$ (4.11)

Comparison between the angular master equations (4.8) and (4.9) leads us to
identify

$\displaystyle A_{2}$ $\displaystyle=_{-(\frac{1}{2}+p)}Y_{l}^{m},$
$\displaystyle A_{4}$ $\displaystyle=_{(\frac{1}{2}-p)}Y_{l}^{m}.$ (4.12)

Since $l$ and $s$ both must be integers or half-integers, we impose the ”Dirac
quantization condition” [55]:

$2qQ=2p=n,\text{\ \ \ \ \ \ }n=0,\pm 1,\pm 2....,$ (4.13)

and obtain the eigenvalue of the spin-weighted spheroidal harmonic equation as

$\displaystyle\lambda$
$\displaystyle=-\sqrt{\left(l+\frac{1}{2}\right)^{2}-p^{2}},\text{\ \ \
\bigg{(}Real: $\left(l+\frac{1}{2}\right)\geq p$\bigg{)},}$
$\displaystyle\therefore\medspace\medspace\lambda^{2}$
$\displaystyle=\left(l+\frac{1}{2}\right)^{2}-p^{2}.$ (4.14)

###  4.2
Radial equation and Zerilli potential

Letting

$\displaystyle g_{1}$ $\displaystyle=\frac{G_{1}}{A(r\Lambda)^{\frac{1}{4}}},$
$\displaystyle\ g_{2}$
$\displaystyle=\frac{G_{2}}{A(r\Lambda)^{\frac{1}{4}}},$ (4.15)

and substituting them into the radial master equations (4.7), we obtain

$\displaystyle G_{1}^{\prime}-\frac{ik}{B}$ $\displaystyle=-\frac{\lambda
G_{2}}{A\sqrt{\Lambda}},$ $\displaystyle G_{2}^{\prime}+\frac{ik}{B}$
$\displaystyle=-\frac{\lambda G_{1}}{A\sqrt{\Lambda}}.$ (4.16)

Introducing the tortoise coordinate [26]

$dr_{\ast}=\frac{dr}{B}\text{ \ }\longrightarrow\text{\ \ \
}r_{\ast}=\frac{1}{b(r_{2}-r_{1})}\ln\left[\frac{\left(r-r_{2}\right)^{r_{2}}}{\left(r-r_{1}\right)^{r_{1}}}\right],$
(4.17)

one can rewrite Eq. (4.16) as

$\displaystyle G_{1,r_{\ast}}-ik$
$\displaystyle=-\frac{\lambda\sqrt{\Lambda}G_{2}}{Ar},$ $\displaystyle
G_{2,r\ast}+ik$ $\displaystyle=-\frac{\lambda\sqrt{\Lambda}G_{1}}{Ar}.$ (4.18)

Setting the solutions of the above equations into the following forms

$\displaystyle G_{1}$ $\displaystyle=P_{1}+P_{2},$ $\displaystyle G_{2}$
$\displaystyle=P_{1}-P_{2},$ (4.19)

we get two decoupled radial equations, which correspond to $1$-dimensional
Schrödinger equation or the so-called Zerilli equation [26] :

$P_{j,r\ast r\ast}+k^{2}P_{j}=V_{j}P_{j},\text{ \ \ \ \ \ }j=1,2$ (4.20)

where the effective potentials are given by

$V_{j}=\lambda^{2}\frac{B}{R}+(-1)^{j}\frac{\lambda
b}{2r^{2}}\sqrt{\frac{B}{R}}\left[r(r_{1}+r_{2})-2r_{2}r_{1}\right].$ (4.21)

The near-horizon and asymptotic limits of the potentials are as follows

$\displaystyle\vskip 12.0pt plus 4.0pt minus 4.0pt\lim_{r\rightarrow
r_{2}}V_{j}$ $\displaystyle\equiv\lim_{r^{\ast}\rightarrow-\infty}V_{j}=0,$
$\displaystyle\lim_{r\rightarrow\infty}V_{j}$
$\displaystyle\equiv\lim_{r^{\ast}\rightarrow\infty}V_{j}=\frac{\lambda^{2}b}{A^{2}}.$
(4.22)

For the outer region of LDBH, the behaviors of potentials $V_{j=1,2}$
according to the various angular quantum numbers are depicted in Figures 1 and
2, respectively.

Figure 1: $V_{1}$ versus $r$ graph. The physical parameters are chosen as
$A=b=r_{2}=2r_{1}=2p=1$. Plot is governed by Eq. (4.21). Figure 2: $V_{2}$
versus $r$ graph. The physical parameters are chosen as
$A=b=r_{2}=2r_{1}=2p=1$. Plot is governed by Eq. (4.21).

Near-horizon ($r\rightarrow r_{2}$) solutions of the radial equations (4.20)
read

$P_{j}=C_{1j}e^{ikr_{\ast}}+C_{2j}e^{-ikr\ast},$ (4.23)

and thus we have

$\displaystyle G_{1}$ $\displaystyle=P_{1}+P_{2}\text{ \ \
}\longrightarrow\text{\ \ \ }G_{1}=D_{1}e^{-ikr\ast}+D_{3}e^{ikr_{\ast}},$
(4.24) $\displaystyle G_{2}$ $\displaystyle=P_{1}-P_{2}\text{ \ \
}\longrightarrow\text{\ \ \ }G_{2}=D_{2}e^{-ikr\ast}+D_{4}e^{ikr_{\ast}}.$

We impose one of the QNM conditions that the wave should be purely ingoing at
the horizon by letting $D_{3}=D_{4}=0$; thence

$G_{j}=D_{j}e^{-ikr\ast}.$ (4.25)

Asymptotic ($r\rightarrow\infty$) solutions of the radial equations (4.20) are
found as

$P_{j}=\widetilde{C}_{1j}e^{i\eta r_{\ast}}+\widetilde{C}_{2j}e^{-i\eta
r\ast},$ (4.26)

where

$\eta=\frac{1}{A}\sqrt{k^{2}A^{2}-\lambda^{2}b}.$ (4.27)

For $k^{2}>\frac{\lambda^{2}b}{A^{2}}$, the Dirac waves can propagate to
spatial infinity without fading. The asymptotic solutions to the radial
equations are then found to be

$\displaystyle G_{1}$ $\displaystyle=P_{1}+P_{2}\text{ \ \
}\longrightarrow\text{\ \ \ }G_{1}=\widetilde{D}_{1}e^{i\eta
r_{\ast}}+\widetilde{D}_{3}e^{-i\eta r\ast},$ $\displaystyle G_{2}$
$\displaystyle=P_{1}-P_{2}\text{ \ \ }\longrightarrow\text{\ \ \
}G_{2}=\widetilde{D}_{2}e^{i\eta r_{\ast}}+\widetilde{D}_{4}e^{-i\eta r\ast}.$
(4.28)

We impose the other QNM condition which requires the propagating waves to be
purely outgoing at spatial infinity. To this end, we simply set
$\widetilde{D}_{3}=\widetilde{D}_{4}=0$ in Eq. (4.28) and obtain

$G_{j}=\widetilde{D}_{j}e^{i\eta r_{\ast}}.$ (4.29)

At the asymptotic region, the tortoise coordinate becomes

$r_{\ast}=\frac{1}{b(r_{2}-r_{1})}\ln\left[\frac{\left(r-r_{2}\right)^{r_{2}}}{\left(r-r_{1}\right)^{r_{1}}}\right]\approx\frac{1}{b(r_{2}-r_{1})}\ln
r^{r_{2}-r_{1}}.$ (4.30)

Therefore, one can see that

$\eta r_{\ast}\approx\frac{\widetilde{\alpha}}{r_{2}-r_{1}}\ln
r^{r_{2}-r_{1}}\text{, \ \ \ \ }$ (4.31)

where $\widetilde{\alpha}=\frac{\eta}{b}$ and $e^{i\eta
r_{\ast}}=r^{i\widetilde{\alpha}}$. By this way, the asymptotic solutions
(4.29) can be expressed in terms of the radial coordinate as follows

$G_{j}=\widetilde{D}_{j}r^{i\widetilde{\alpha}}.$ (4.32)

An alternative way of getting the latter result is the decoupling of $G_{j}$
from Eq. (4.16):

$\Lambda
G_{j}^{\prime\prime}-\frac{b(r_{2}+r_{1}-2r)}{2}G_{j}^{\prime}+\left\\{(-1)^{j}ik\left[\frac{b(r_{2}r_{1}-r^{2})}{\Lambda}-\frac{b(r_{2}+r_{1}-2r)}{2\Lambda}\right]+\frac{k^{2}r^{2}}{\Lambda}-\frac{\lambda^{2}}{A^{2}}\right\\}G_{j}=0.$
(4.33)

As $r\rightarrow\infty$, Eq. (4.33) takes the following form

$r^{2}G_{j}^{\prime\prime}+rG_{j}^{\prime}+(\frac{k^{2}A^{2}-\lambda^{2}b}{A^{2}b^{2}}=\widetilde{\alpha}^{2})G_{j}=0.$
(4.34)

whose solutions are found to be

$G_{j}=\widetilde{D}_{j}r^{i\widetilde{\alpha}}+\widehat{C}_{j}r^{-i\widetilde{\alpha}}.$
(4.35)

Since $e^{i\eta r_{\ast}}=r^{i\widetilde{\alpha}}$ and the wave should be
purely outgoing at the spatial infinity, we set $\widehat{C}_{j}=0$ and thus
find $G_{j}=\widetilde{D}_{j}r^{i\widetilde{\alpha}}$ which is nothing but the
same result that was obtained in Eq. (4.32).

##  5 Exact QNM frequencies

To find the analytical solutions of $G_{j}$, we first introduce

$G_{j}=H_{j}e^{(-1)^{j+1}ikr_{\ast}},$ (5.1)

and change the variable to

$z=-x=-\frac{r-r_{2}}{r_{2}-r_{1}}.$ (5.2)

After inserting Eq. (5.2) in Eq. (4.33), we obtain

$z(1-z)H_{j}^{\prime\prime}+\left[\boldsymbol{c}-(1+\boldsymbol{a}+\boldsymbol{b})z\right]H_{j}^{\prime}-\boldsymbol{a}\boldsymbol{b}H_{j}=0,$
(5.3)

which is the Euler’s hypergeometric differential equation [56] with

$\displaystyle\boldsymbol{a}$
$\displaystyle=i\left[(-1)^{j+1}\frac{k}{b}+\widetilde{\alpha}\right],$
$\displaystyle\boldsymbol{b}$
$\displaystyle=i\left[(-1)^{j+1}\frac{k}{b}-\widetilde{\alpha}\right],$
$\displaystyle\boldsymbol{c}$
$\displaystyle=\frac{1}{2}+2ik(-1)^{j+1}\frac{r_{2}}{b(r_{2}-r_{1})}.$ (5.4)

The general solution of Eq. (5.3) is given by [56]

$H_{j}=C_{2}F(\boldsymbol{a},\boldsymbol{b};\boldsymbol{c};z)+C_{1}z^{1-\boldsymbol{c}}F(\widetilde{\boldsymbol{a}},\widetilde{\boldsymbol{b}};\widetilde{\boldsymbol{c}};z),$
(5.5)

where $C_{1}$ and $C_{2}$ are the integration constants and
$F(\boldsymbol{a},\boldsymbol{b};\boldsymbol{c};z)$ and
$F(\widetilde{\boldsymbol{a}},\widetilde{\boldsymbol{b}};\widetilde{\boldsymbol{c}};z)$
are the hypergeometric (or Gaussian) functions with

$\displaystyle\widetilde{\boldsymbol{a}}$
$\displaystyle=\boldsymbol{a}-\boldsymbol{c}+1=\frac{1}{2}-i\left[k(-1)^{j+1}\frac{r_{2}+r_{1}}{b(r_{2}-r_{1})}+\widetilde{\alpha}\right],$
$\displaystyle\widetilde{\boldsymbol{b}}$
$\displaystyle=\boldsymbol{b}-\boldsymbol{c}+1=\frac{1}{2}-i\left[k(-1)^{j+1}\frac{r_{2}+r_{1}}{b(r_{2}-r_{1})}-\widetilde{\alpha}\right],$
$\displaystyle\widetilde{\boldsymbol{c}}$
$\displaystyle=2-\boldsymbol{c}=\frac{3}{2}-2ik(-1)^{j+1}\frac{r_{2}}{b(r_{2}-r_{1})}.$
(5.6)

In the near-horizon region: $r\rightarrow r_{2},$ $r_{\ast}\rightarrow-\infty$
$\Rightarrow$ $z\approx e^{r_{\ast}}\rightarrow 0$. By recalling that
$F(\boldsymbol{a},\boldsymbol{b};\boldsymbol{c};0)=1$ [56], the physical
solutions (5.1), which fulfill the QNM condition that only ingoing waves can
propagate near the horizon, are found to be as follows

$\displaystyle G_{2}$
$\displaystyle=H_{2}e^{-ikr_{\ast}}=C_{2}e^{-ikr_{\ast}}F(\boldsymbol{a},\boldsymbol{b};\boldsymbol{c};z),$
$\displaystyle G_{1}$
$\displaystyle=H_{1}e^{ikr_{\ast}}=C_{1}z^{1-\boldsymbol{c}}e^{+ikr_{\ast}}F(\widetilde{\boldsymbol{a}},\widetilde{\boldsymbol{b}};\widetilde{\boldsymbol{c}};z).$
(5.7)

By using one of the special features of the hypergeometric functions [56]:

$\displaystyle F(\boldsymbol{a},\mathbf{b};\boldsymbol{c};y)$
$\displaystyle=\frac{\Gamma(\boldsymbol{c})\Gamma(\boldsymbol{b}-\boldsymbol{a})}{\Gamma(\boldsymbol{b})\Gamma(\boldsymbol{c}-\boldsymbol{a})}(-y)^{-\boldsymbol{a}}F(\boldsymbol{a},\boldsymbol{a}+1-\boldsymbol{c};\boldsymbol{a}+1-\boldsymbol{b};1/y)$
$\displaystyle+\frac{\Gamma(\boldsymbol{c})\Gamma(\boldsymbol{a}-\boldsymbol{b})}{\Gamma(\boldsymbol{a})\Gamma(\boldsymbol{c}-\boldsymbol{b})}(-y)^{-\boldsymbol{b}}F(\boldsymbol{b},\boldsymbol{b}+1-\boldsymbol{c};\boldsymbol{b}+1-\boldsymbol{a};1/y),$
(5.8)

one can extend the solutions (5.7) to spatial infinity ($r\rightarrow\infty,$
$r_{\ast}\rightarrow\infty$ $\Rightarrow\ \frac{1}{z}\rightarrow 0$) as
follows

$\displaystyle G_{1}$
$\displaystyle=C_{1}z^{1-\boldsymbol{c}}e^{ikr_{\ast}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bigg{[}}\frac{\Gamma(\widetilde{\boldsymbol{c}})\Gamma(\widetilde{\boldsymbol{b}}-\widetilde{\boldsymbol{a}})}{\Gamma(\widetilde{\boldsymbol{b}})\Gamma(\boldsymbol{c}-\widetilde{\boldsymbol{a}})}(x)^{-\widetilde{\boldsymbol{a}}}F(\widetilde{\boldsymbol{a}},\widetilde{\boldsymbol{a}}+1-\widetilde{\boldsymbol{c}};\widetilde{\boldsymbol{a}}+1-\widetilde{\boldsymbol{b}};1/z)$
$\displaystyle+\frac{\Gamma(\widetilde{\boldsymbol{c}})\Gamma(\widetilde{\boldsymbol{a}}-\widetilde{\boldsymbol{b}})}{\Gamma(\widetilde{\boldsymbol{a}})\Gamma(\widetilde{\boldsymbol{c}}-\widetilde{\boldsymbol{b}})}(x)^{-\widetilde{\boldsymbol{b}}}F(\widetilde{\boldsymbol{b}},\widetilde{\boldsymbol{b}}+1-\widetilde{\boldsymbol{c}};\widetilde{\boldsymbol{b}}+1-\widetilde{\boldsymbol{a}};1/z){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bigg{]},}$
$\displaystyle G_{2}$
$\displaystyle=C_{2}e^{-ikr_{\ast}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bigg{[}}\frac{\Gamma(\boldsymbol{c})\Gamma(\boldsymbol{b}-\boldsymbol{a})}{\Gamma(\boldsymbol{b})\Gamma(\boldsymbol{c}-\boldsymbol{a})}(x)^{-\boldsymbol{a}}F(\boldsymbol{a},\boldsymbol{a}+1-\boldsymbol{c};\boldsymbol{a}+1-\boldsymbol{b};1/z)$
$\displaystyle+\frac{\Gamma(\boldsymbol{c})\Gamma(\boldsymbol{a}-\boldsymbol{b})}{\Gamma\boldsymbol{a})\Gamma(\boldsymbol{c}-\boldsymbol{b})}(x)^{-\boldsymbol{b}}F(\boldsymbol{b},\boldsymbol{b}+1-\boldsymbol{c};\boldsymbol{b}+1-\boldsymbol{a};1/z){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(5.9)

Since $x=\frac{r-r_{2}}{r_{2}-r_{1}},$ and thus $x\simeq r\simeq
e^{br_{\ast}}$ at the infinity, we have

$\displaystyle G_{1}$ $\displaystyle\approx
C_{1}\frac{\Gamma(\widetilde{\boldsymbol{c}})\Gamma(\widetilde{\boldsymbol{b}}-\widetilde{\boldsymbol{a}})}{\Gamma(\widetilde{\boldsymbol{b}})\Gamma(\boldsymbol{c}-\widetilde{\boldsymbol{a}})}r^{i\widetilde{\alpha}}+C_{1}\frac{\Gamma(\widetilde{\boldsymbol{c}})\Gamma(\widetilde{\boldsymbol{a}}-\widetilde{\boldsymbol{b}})}{\Gamma(\widetilde{\boldsymbol{a}})\Gamma(\widetilde{\boldsymbol{c}}-\widetilde{\boldsymbol{b}})}r^{-i\widetilde{\alpha}}.$
(5.10) $\displaystyle G_{2}$ $\displaystyle\approx
C_{2}\frac{\Gamma(\boldsymbol{c})\Gamma(\boldsymbol{b}-\boldsymbol{a})}{\Gamma(\boldsymbol{b})\Gamma(\boldsymbol{c}-\boldsymbol{a})}r^{i\widetilde{\alpha}}+C_{2}\frac{\Gamma(\boldsymbol{c})\Gamma(\boldsymbol{a}-\boldsymbol{b})}{\Gamma(\boldsymbol{a})\Gamma(\boldsymbol{c}-\boldsymbol{b})}r^{-i\widetilde{\alpha}}.$
(5.11)

The correspondence between the asymptotic solutions (4.28) and the above
solutions (5.11) yields

$\displaystyle\widetilde{D}_{1}$
$\displaystyle=C_{1}\frac{\Gamma(\widetilde{\boldsymbol{c}})\Gamma(\widetilde{\boldsymbol{b}}-\widetilde{\boldsymbol{a}})}{\Gamma(\widetilde{\boldsymbol{b}})\Gamma(\boldsymbol{c}-\widetilde{\boldsymbol{a}})},$
$\displaystyle\widetilde{D}_{2}$
$\displaystyle=C_{2}\frac{\Gamma(\boldsymbol{c})\Gamma(\boldsymbol{b}-\boldsymbol{a})}{\Gamma(\boldsymbol{b})\Gamma(\boldsymbol{c}-\boldsymbol{a})},$
$\displaystyle\widetilde{D}_{3}$
$\displaystyle=C_{1}\frac{\Gamma(\widetilde{\boldsymbol{c}})\Gamma(\widetilde{\boldsymbol{a}}-\widetilde{\boldsymbol{b}})}{\Gamma(\widetilde{\boldsymbol{a}})\Gamma(\widetilde{\boldsymbol{c}}-\widetilde{\boldsymbol{b}})},$
$\displaystyle\widetilde{D}_{4}$
$\displaystyle=C_{2}\frac{\Gamma(\boldsymbol{c})\Gamma(\boldsymbol{a}-\boldsymbol{b})}{\Gamma(\boldsymbol{a})\Gamma(\boldsymbol{c}-\boldsymbol{b})}.$
(5.12)

Therefore, to have only pure outgoing waves at the spatial infinity (i.e.,
$\widetilde{D}_{3},\widetilde{D}_{4}=0$), we appeal to the pole structure of
the Gamma functions. The Gamma functions $\Gamma(\widetilde{x})$ have poles at
$\widetilde{x}=-n$ for $n=0,1,2$…. [56]. Hence, we obtain the QNMs by using
the following relations:

$\displaystyle\boldsymbol{a}$
$\displaystyle=i\left[-\frac{k}{b}+\widetilde{\alpha}\right]=-n,$
$\displaystyle\boldsymbol{c}-\boldsymbol{b}$
$\displaystyle=\frac{1}{2}+i\left[-k\frac{r_{2}+r_{1}}{b(r_{2}-r_{1})}-\widetilde{\alpha}\right]=-n,$
$\displaystyle\widetilde{\boldsymbol{a}}$
$\displaystyle=\frac{1}{2}-i\left[k\frac{r_{2}+r_{1}}{b(r_{2}-r_{1})}+\widetilde{\alpha}\right]=-n,$
$\displaystyle\widetilde{\boldsymbol{c}}-\widetilde{\boldsymbol{b}}$
$\displaystyle=1+i\left[k\frac{r_{2}+r_{1}}{b(r_{2}-r_{1})}-\widetilde{\alpha}\right]=-n.$
(5.13)

Recall that the time dependence of the QNMs is governed by $e^{ikt}$ (see
Eq.(3.5)). Therefore, having QNMs is conditional on $Im(k)>0$, which
guaranties the stability [40]. The conditions under which frequencies obtained
from $G_{1}$ and $G_{2}$ spinor solutions produce stable modes (i.e., QNMs)
and which ones generate the unstable modes are summarized in Tables 1 and 2:

Frequencies | Stable Modes (QNMs) | Unstable Modes
---|---|---
| $k=-\frac{I}{2A(\beta^{2}-1)}\left[A\beta b(2n+1)+\right.$
---
$\left.2\sqrt{b}\sqrt{A^{2}b(n+\frac{1}{2})^{2}+\lambda^{2}(\beta^{2}-1))}\right]$
if $A<0$ and $A<\frac{\lambda}{\sqrt{b}(n+\frac{1}{2})}$ |  | if $A>0$ or
---
if $A<0$ and $A>\frac{\lambda}{\sqrt{b}(n+\frac{1}{2})}$
| $k=\frac{I}{2A(\beta^{2}-1)}\left[A\beta b(n+1)+\right.$
---
$\left.\sqrt{b}\sqrt{A^{2}b(n+1)^{2}+\lambda^{2}(\beta^{2}-1))}\right]$
| if $A>0$
---
if $A<0$ and $A<\frac{\lambda}{\sqrt{b}(n+1)}$
if $A<0$ and $A>\frac{\lambda}{\sqrt{b}(n+1)}$
Table 1: QNM frequencies for $G_{1}$. The results are obtained from Eqs. (5.10) and (5.13). Frequencies | Stable Modes (QNMs) | Unstable Modes
---|---|---
$k=\frac{I}{2A^{2}n}(\lambda^{2}-n^{2}A^{2}b)$ |  | if $A>0$ and $A<\frac{-\lambda}{\sqrt{b}n}$
---
if $A<0$ and $A>\frac{\lambda}{\sqrt{b}n}$
| if $A>0$ and $A>\frac{-\lambda}{\sqrt{b}n}$
---
if $A<0$ and $A<\frac{\lambda}{\sqrt{b}(n}$
| $k=-\frac{I}{2A(\beta^{2}-1)}\left[A\beta b(2n+1)+\right.$
---
$\left.2\sqrt{b}\sqrt{A^{2}b(n+\frac{1}{2})^{2}+\lambda^{2}(\beta^{2}-1))}\right]$
if $A<0$ and $A<\frac{\lambda}{\sqrt{b}(n+\frac{1}{2})}$ |  | if $A>0$ or
---
if $A<0$ and $A>\frac{\lambda}{\sqrt{b}(n+\frac{1}{2})}$
Table 2: QNM frequencies for $G_{2}$. The results are obtained from Eqs.
(5.11) and (5.13).

In Tables 1 and 2, the parameter of $\beta$ is given by

$1<\beta=\frac{r_{2}+r_{1}}{r_{2}-r_{1}}<\infty.$ (5.14)

As can be seen from above, depending on the values of $A,\lambda,b,n$ some QNM
frequencies can have positive/negative imaginary part, and therefore the LDBHs
can be stable/unstable under charged fermionic perturbations.

##  6
Conclusion

In this paper, we have analytically computed the QNMs of charged fermionic
perturbations for the $4$-dimensional LDBHs which are the solutions to the EMD
theory with the double Lioville-type potentials. The fermionic fields are the
solutions of the Dirac equation. For the LDBH geometry, we have found that the
massless but charged fermionic fields are governed by the Gaussian
hypergeometric functions. We matched the exact solutions obtained with the
solutions found for near-horizon and asymptotic regions. Thus, we have shown
that there are two sets of frequencies for each component ($G_{1}$ and
$G_{2}$) of the fermionic fields. As being summarized in Tables 1 and 2, in
each set the frequencies obtained can belong to either the stable modes (QNMs)
or the unstable modes depending on the relations between the parameters of
$A,\lambda,b,$ and $n$. Meanwhile, the reader may question what the
advantage/disadvantage of perturbing the LDBH with fermionic fields instead of
the bosonic fields is. If precise measurements of QNM frequencies become
possible in the future and if the fermionic QNM data obtained for spin-up and
spin-down fields (i.e., $G_{1}$ and $G_{2}$) coincide with (a kind of double
check) the analytical expressions that we have derived, those results will
likely reveal the presence of LDBHs more accurately than the bosonic QNM ones.

As is known, rotating BH solutions are more considerable for testing relevant
theories with the astrophysical observations. For this reason, in the near
future, we plan to extend our study to the rotating LDBHs. To this end, we
shall apply the Newman–Janis algorithm [57] to the metric (2.9) and perturb
the obtained rotating LDBH to reveal the effect of rotation on their QNMs.

## Acknowledgements

The authors are grateful to the Editor and anonymous Referees for their
valuable comments and suggestions to improve the paper.

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