# Power boundedness in Fourier and Fourier-Stieltjes algebra of an
ultraspherical hypergroup

Reza Esmailvandi  Reza Esmailvandi
Department of Mathematical Sciences,
Isfahan University of Technology,
Isfahan 84156-83111, Iran<EMAIL_ADDRESS>,  Mehdi Nemati
Mehdi Nemati
Department of Mathematical Sciences,
Isfahan University of Technology,
Isfahan 84156-83111, Iran.
and
School of Mathematics,
Institute for Research in Fundamental Sciences (IPM),
P.O. Box: 19395–5746, Tehran, Iran<EMAIL_ADDRESS>and  N. Shravan kumar
N. Shravan kumar
Department of mathematics,
Indian Institute of Technology Delhi,
Delhi - 110016,
India<EMAIL_ADDRESS>
###### Abstract.

Let $H$ be an ultraspherical hypergroup associated with a locally compact
group $G$ and a spherical projector $\pi$ and let $A(H)$ and $B(H)$ denote the
Fourier and Fourier-Stieltjes algebras, respectively, associated with $H.$ In
this note, we study power boundedness and Cesàro boundedness in $B(H)$. We
also characterize the power bounded property for both $A(H)$ and $B(H).$

###### Key words and phrases:

Fourier algebras, ultraspherical hypergroups, locally compact groups, power
boundedness, Cesàro boundedness.

###### 2010 Mathematics Subject Classification:

43A62, 43A30,; Secondary: 46J10

##  1\. Introduction

Let $G$ be a locally compact abelian group. A classical result of Schreiber
[Sch70] states that the algebras $L^{1}(G)$ and $M(G)$ are power bounded if
and only if $G$ is compact and finite, respectively. A natural generalizations
of the group algebra and the measure algebra are the Fourier and Fourier-
Stieltjes algebras associated to general locally compact groups. These
algebras are introduced by Eymard [Eym64] and since then, they have been a
major object of study in abstract harmonic analysis. In 2011, Lau and Kaniuth
showed that $A(G)$ and $B(G)$ are power bounded if and only if $G$ is discrete
and finite, respectively.

Let $G$ be a locally compact group and let $\pi$ be a spherical projector. Let
$H$ be an ultraspherical hypergroup associated to $G$ and $\pi.$ Let $A(H)$
and $B(H)$ denote the Fourier and Fourier-stieltjes algebras on $H,$
introduced by Muruganandam in 2008 [Mur08]. The purpose of this paper, is to
extend the result of Lau and Kaniuth on groups to the context of
ultraspherical hypergroups. More precisely, we show that $A(H)$ and $B(H)$
possess the power bounded property if and only if $H$ is discrete and finite,
respectively.

In Section 4, we characterize the power boundedness of $A(H).$ In fact, this
is the content of Theorem 4.3. Initially, in Section 3, results from [KLU10,
KLU11] are used to deduce relations among power bounded elements of $B(H)$ and
certain sets from the coset ring. We also study the sequence
$\\{u^{n}\\}_{n\in\mathbb{N}}$ in relation to the same sets considered, where
$u$ is a power bounded element of $B(H).$ Finally, we study Cesàro
boundedness. By making use of the results of Section 3, in Section 4, we
characterize the power bounded property for $B(H).$ This is Theorem 4.9.

We shall begin with some preliminaries that are needed in the sequel.

##  2\. preliminaries

###  2.1. Hypergroups

For the definition and basic properties of hypergroups, we refer the reader
mainly to [BH11, Jew75]. In [Jew75], Jewett calls a hypergroup as convo. For a
hypergroup $H$ with the identity element $e$, the mapping $x\mapsto\check{x}$
will always denote the involution of $H$. For the sake of completeness and the
convenience of the reader, let us recall the definition of an ultraspherical
hypergroup associated with a locally compact group $G$ (see [Mur08, Definition
2.1] and [DSKL14, Definition 2.2]).

######  Definition 2.1.

A linear map $\pi:C_{c}(G)\rightarrow C_{c}(G)$ is called a spherical
projector if it satisfies the following for all $f,g\in C_{c}(G).$

  1. (1)

    1. (a)

$\pi^{2}=\pi$ and $\pi$ is positivity preserving;

    2. (b)

$\pi(\pi(f)g)=\pi(f)\pi(g)$;

    3. (c)

$\langle\pi(f),g\rangle=\langle f,\pi(g)\rangle$;

    4. (d)

$\int_{G}\pi(f)(x)\ dx=\int_{G}f(x)\ dx.$

  2. (2)

$\pi(\pi(f)*\pi(g))=\pi(f)*\pi(g)$

  3. (3)

Let $\pi^{*}:M(G)\rightarrow M(G)$ denote the transpose of $\pi$ and let

$\mathcal{O}_{x}=supp(\pi^{*}(\delta_{x})),\ x\in G.$

Then for all $x,y\in G,$

    1. (a)

either $\mathcal{O}_{x}\cap\mathcal{O}_{y}=\emptyset$ or
$\mathcal{O}_{x}=\mathcal{O}_{y};$

    2. (b)

if $x\in\mathcal{O}_{y}\Rightarrow x^{-1}\in\mathcal{O}_{y^{-1}};$

    3. (c)

if $\mathcal{O}_{xy}=\\{e\\}\Rightarrow\mathcal{O}_{y}=\mathcal{O}_{x^{-1}};$

    4. (d)

the map $x\mapsto\mathcal{O}_{x}$ from $G$ into $\mathcal{K}(G)$ is continuous
with respect to the Michael topology on the space $\mathcal{K}(G)$ of all
nonempty compact subsets of $G$ .

Note that $\pi$ extends to a norm decreasing linear map on various function
spaces, including $L^{p}(G),\ 1\leq p\leq\infty.$ A function $f$ is called
$\pi$-radial if $\pi(f)=f$ and similarly, a measure $\mu$ is called
$\pi$-radial if $\pi^{*}(\mu)=\mu.$

Let $H=\\{\dot{x}=\mathcal{O}x:x\in G\\}$, equipped with the natural quotient
topology under the quotient map $p:G\rightarrow H$. We identify $M(H)$ with
the space of all $\pi^{*}$-radial measures in $M(G).$ Define the product on
$M(H)$ as
$\delta_{\dot{x}}\ast\delta_{\dot{y}}=\pi^{\ast}(\pi^{\ast}(\delta_{x})\ast\pi^{\ast}(\delta_{y}))$
for all $x,y\in G$. With this structure $H$ becomes a locally compact
hypergroup, called a spherical hypergroup [Mur08, Theorem 2.12]. A spherical
hypergroup is further called an ultraspherical hypergroup if the modular
function on $G$ is $\pi$-radial.

One of the most common and interesting example of an ultraspherical hypergroup
is the double coset hypergroup. Let $G$ be a locally compact group containing
a compact subgroup $K.$ Define $\pi:C_{c}(G)\rightarrow C_{c}(G)$ as

$\pi(f)(x)=\int_{K}\int_{K}f(k^{\prime}xk)\ dk\ dk^{\prime}.$

Then $\pi$ defines a spherical projector and the resulting hypergroup is an
ultraspherical hypergroup.

Specific examples of the double coset hypergroups are orbit hypergroups which
are constructed as follows. Let $G$ be a locally compact group and let $K$ be
a compact subgroup of $Aut(G)$, the group of all topological automorphisms of
$G$. For each $x\in G$ denote by $K(x)=\\{\alpha(x):\alpha\in K\\}$ the orbit
of $x$ under $K$. The orbit hypergroup $G_{K}$ is the space of all orbits
$K(x),x\in G$, equipped with the quotient topology and the obvious hypergroup
structure. Now, consider the semidirect product $G\rtimes K$, where the group
product of $(x,\alpha)$ with $(y,\beta)$ in $G\rtimes K$ is given by

$(x,\alpha)(y,\beta)=(x\alpha(y),\alpha\beta).$

Then the map $K(x)\rightarrow K(x,\beta)K$ is a toplogical hypergroup
isomorphism between $G_{K}$ and the double coset hypergroup $(G\rtimes K)//K$
(see [Jew75, Section 8]).

Another interesting class of examples for ultraspherical hypergroups in the
context of Lie groups is due to Damek and Ricci [DR92]. They called the
spherical projectors as average projector.

Unlike the theory of locally compact groups, the existence of a Haar measure
on hypergroups remains an open problem. But for specific cases of hypergroups
including commutative hypergroups, discrete hypergroups, and compact
hypergroups, a unique (up to a scalar multiple) Haar measure always exists,
[BH11]. It is shown in [Mur08, Proposition 2.14] that every ultraspherical
hypergroup admits a Haar measure. More precisely, if $H$ is an ultraspherical
hypergroup associated to a locally compact group $G$, then a left Haar measure
on $H$ is given by

$\int_{H}f(\dot{x})d\dot{x}=\int_{G}f(p(x))dx,\qquad(f\in C_{c}(H)).$

The following lemma is well known in the case of locally compact groups.

######  Lemma 2.2.

Let $H$ be a non-discrete ultraspherical hypergroup with Haar measure
$\lambda.$ Then $H$ possesses a compact nowhere dense subset of positive Haar
measure.

###### Proof.

Suppose that $H$ is second countable. In particular, $H$ is separable. Let
$\\{t_{n}\\}$ be a countable dense subset of $H.$ Now, for each
$n\in\mathbb{N},$ choose an open set $V_{n}$ such that $\dot{e}\in V_{n}$ and
$\lambda(t_{n}*V_{n})\leq\frac{1}{2^{n+1}}.$ Let
$E=\left(\underset{n\in\mathbb{N}}{\cup}t_{n}*V_{n}\right)^{c}.$ Then $E$
satisfies the requirements of the lemma.

Now, suppose that $H$ is not second countable. Using [SK17, Theorem 1.1],
there exists a compact subhypergroup $K$ of $H$ such that $H//K$ is second
countable. Now, by the preceding paragraph, there exists a subset
$\widetilde{E}$ of $H//K$ such that $\widetilde{E}$ is nowhere dense and has
positive Haar measure. Let $E=\textsf{q}^{-1}(\widetilde{E}),$ where q denotes
the canonical quotient map from $H$ onto $H//K.$ This set $E$ satisfies the
requirements of the lemma. ∎

Let $G$ be a locally compact group. The coset ring, denoted
${\mathscr{R}}(G),$ is defined as the Boolean ring generated by all cosets of
subgroups of $G.$ The closed coset ring, denoted ${\mathscr{R}}_{c}(G),$ is
defined to be

${\mathscr{R}}_{c}(G)=\\{E\in{\mathscr{R}}(G):E~{}{\text{is closed
in}}~{}G\\}.$

Let $H$ be an ultraspherical hypergroup associated with $G$. We will write

${\mathscr{R}}(H)=\\{\dot{E}\subseteq
H:p^{-1}(\dot{E})\in{\mathscr{R}}(G)\\},$

and

${\mathscr{R}}_{c}(H)=\\{\dot{E}\subseteq
H:p^{-1}(\dot{E})\in{\mathscr{R}}_{c}(G)\\}.$

###  2.2. Fourier Algebra

Let $B(H)$ denote the Fourier-steiltjes algebra on $H$. Then

$B(H)=B_{\pi}(G)=\\{u\in B(G):u~{}is~{}\pi\text{-}radial\\}.$

Let $C^{*}(H)$ denote the hypergroup C*-algebra associated with $H.$ Then
$B(H)=C^{*}(H)^{*}.$ For each $u\in B(H)$, we denote by $\tilde{u}$ the
element in $B_{\pi}(G)$ associated with $u$; that is,
$u(\dot{x})=\tilde{u}(x)$ for all $x\in G$ and
$\|u\|_{B(H)}=\|\tilde{u}\|_{B(G)}$.

Let $A(H)$ denote the Fourier algebra corresponding to the ultraspherical
hypergroup $H$. Recall that the Fourier algebra $A(H)$ is semisimple, regular
and Tauberian [Mur08, Theorem 3.13]. As in the group case, let $\lambda$
denote the left regular representation of $H$ on $L^{2}(H)$ given by

$\lambda(\dot{x})(f)(\dot{y})=f(\check{\dot{x}}\ast\dot{y})\quad(\dot{x},\dot{y}\in
H,f\in L^{2}(H)).$

This can be extended to $L^{1}(H)$ by $\lambda(f)(g)=f*g$ for all $f\in
L^{1}(H)$ and $g\in L^{2}(H)$. Let $C^{*}_{\lambda}(H)$ denote the completion
of $\lambda(L^{1}(H))$ in $B(L^{2}(H))$ which is called the reduced
$C^{*}$-algebra of $H$. The von Neumann algebra generated by
$\\{\lambda(\dot{x}):\dot{x}\in H\\}$ is called the von Neumann algebra of
$H$, and is denoted by $VN(H)$. Note that $VN(H)$ is isometrically isomorphic
to the dual of $A(H)$. Moreover, $A(H)$ can be considered as an ideal of
$B_{\lambda}(H)$, where $B_{\lambda}(H)$ is the dual of $C_{\lambda}^{*}(H)$.
As an immediate consequence of [Mur08, Theorem 4.2] and [DSKL14, Lemma 3.7] we
obtain the following lemma.

######  Lemma 2.3.

Suppose that $H$ is an ultraspherical hypergroup associated to an amenable
locally compact group $G$. Then

$C^{*}_{\lambda}(H)=C^{*}(H)$ and $B_{\lambda}(H)=B(H)$.

The Fourier algebra $A(H)$ admits a bounded approximate identity. Let
$(e_{\alpha})$ be a bounded approximate identity of $A(H)$. Then, after
passing to a subnet if necessary, $e_{\alpha}\rightarrow 1$ in the
weak∗-topology $\sigma(B_{\lambda}(H),C^{*}_{\lambda}(H))$.

######  Lemma 2.4.

Let $H$ be an ultraspherical hypergroup. Then $C^{*}(H)$ is a Banach
$B(H)$-bimodule with the module action given by

$\langle\varphi\cdot\lambda(f),\psi\rangle=\langle\lambda(f),\varphi\psi\rangle,\qquad(\varphi,\psi\in
B(H),f\in L^{1}(G)).$

In particular, pointwise multiplication in $B(H)$ is weak∗-separately
continuous.

###### Proof.

For each $\varphi\in B(H)$ and $f\in L^{1}(H)$, we have

$\langle\varphi\cdot\lambda(f),\psi\rangle=\langle\lambda(f),\varphi\psi\rangle=\int_{H}f(\dot{x})\varphi(\dot{x})\psi(\dot{x})d\dot{x}=\langle\lambda(\varphi
f),\psi\rangle.$

Thus $\varphi\cdot\lambda(f)\in C^{*}(H).$ Now, the conclusion follows from
the density of $L^{1}(H)$ inside $C^{*}(H)$ ∎

###  2.3. Commutative Banach algebras

Let ${\mathcal{A}}$ be a regular, semisimple, commutative Banach algebra with
the Gelfand structure space $\Delta({\mathcal{A}})$. For a closed ideal $I$ of
${\mathcal{A}}$, the zero set of $I$, denoted $Z(I)$, is a closed subset of
$\Delta({\mathcal{A}})$ given by

$Z(I)=\\{x\in\Delta({\mathcal{A}}):\widehat{a}(x)=0~{}{\text{for all }a\in
I}\\}.$

Associated to each closed subset $E$ of $\Delta({\mathcal{A}}),$ we define the
following distinguished ideals with zero set equal to $E$:

$\displaystyle k_{\mathcal{A}}(E)$
$\displaystyle=\\{a\in{\mathcal{A}}:\widehat{a}(x)=0~{}{\text{for all }x\in
E}\\},$ $\displaystyle j_{\mathcal{A}}(E)$
$\displaystyle=\\{a\in{\mathcal{A}}:\widehat{a}~{}\text{has compact support
disjoint from}~{}E\\},$ $\displaystyle J_{\mathcal{A}}(E)$
$\displaystyle=\overline{j_{\mathcal{A}}(E)}.$

Note that, for each ideal $I$ of ${\mathcal{A}}$ with zero set $E$, we have
$j_{\mathcal{A}}(E)\subseteq I\subseteq k_{\mathcal{A}}(E)$(cf. Kaniuth
[Kan09, Theorem 5.1.6]). Recall that a closed subset $E$ of
$\Delta({\mathcal{A}})$ is a set of synthesis or spectral set  if
$k_{\mathcal{A}}(E)=J_{\mathcal{A}}(E)$; that is, $k_{\mathcal{A}}(E)$ is the
only closed ideal with zero set equal to $E$.

######  Definition 2.5.

An element $a$ of a Banach algebra ${\mathcal{A}}$ is said to be power bounded
if $\sup_{n\in\mathbb{N}}\|a^{n}\|<\infty$. The Banach algebra ${\mathcal{A}}$
is said to have the power boundedness property if any $a\in A$ with
$r(a)=\inf_{n\in\mathbb{N}}\|a^{n}\|^{\frac{1}{n}}\leq 1,$ is power bounded.

It is clear that if $a\in{\mathcal{A}}$ is power bounded, then $r(a)\leq 1.$

##  3\. Power bounded elements and Coset ring

In this section, we study two sets (defined below) associated to an element of
$B(H).$ This section is motivated by some results in [KLU10, KLU11, Mus19].

###  3.1. Coset ring and power bounded elements

We shall begin by defining those two sets.

######  Definition 3.1.

Let $u\in B(H)$. Let $\dot{E}_{u}$ and $\dot{F}_{u}$ denote the sets in $H$
associated to $u$ given by

$\dot{E}_{u}=\\{\dot{x}\in
H:|u(\dot{x})|=1\\}\qquad\text{and}\qquad\dot{F}_{u}=\\{\dot{x}\in
H:u(\dot{x})=1\\}.$

Also, associated to $u$ are the sets $E_{u}$ and $F_{u}$ in $G$ defined by

$E_{u}=p^{-1}(\dot{E}_{u})\qquad\text{and}\qquad F_{u}=p^{-1}(\dot{F}_{u}).$

######  Remark 3.2.

Let $u\in B(H)$ and $\tilde{u}$ be the element in $B(G)$ associated with $u$.
Then

$\\{x\in G:\tilde{u}(x)=1\\}=p^{-1}(\dot{F}_{u})=F_{u},$

and

$\\{x\in G:|\tilde{u}(x)|=1\\}=p^{-1}(\dot{E}_{u})=E_{u}.$

The following theorem generalizes [KLU10, Theorem 4.1] to the context of
ultraspherical hypergroups. Also, the proof of this is a direct consequence of
Remark 3.2 and [KLU10, Theorem 4.1].

######  Lemma 3.3.

Let $H$ be any locally compact ultraspherical hypergroup and let $u\in B(H)$
be power bounded. Then the sets $\dot{E}_{u}$ and $\dot{F}_{u}$ are in
${\mathscr{R}}_{c}(H)$.

The proof of our next result follows the same lines as in [DSKL14, Proposition
3.6] and hence we omit it.

######  Lemma 3.4.

Let $K$ be a closed subhypergroup of an unltraspherical hypergroup $H$. Then
for any $u\in B(K)$, there exists $v\in B(H)$ such that $v|_{K}=u$ and
$\|v\|_{B(H)}=\|u\|_{B(K)}$.

Our next lemma says that we cannot replace in Lemma 3.3, the weaker condition
that $\|u\|=1.$ As the proof of this follows similar to [KLU10, Lemma 4.3], we
omit it.

######  Lemma 3.5.

Let $\dot{E}$ be a closed subset of $H$ such that $H\setminus\dot{E}$ is
$\sigma$-compact. Then there exists $u\in B(H)$ with $\|u\|_{\infty}=1$ and
$\dot{F}_{u}=\dot{E}$.

Here is an analogue of the Host’s idempotent theorem.

######  Lemma 3.6.

Let $u$ be a complex-valued functions on $H$. Then $u$ is an idempotent of
$B(H)$ if and only if $u={\bf 1}_{\dot{E}}$ for some
$\dot{E}\in{\mathscr{R}}_{c}(H)$.

###### Proof.

Let $u$ be an idempotent element of $B(H).$ Then $\tilde{u}$ is an idempotent
element of $B(G)$ and hence by Host’s idempotent theorem [Hos86], there exists
$E\in{\mathscr{R}}_{c}(G)$ such that $\tilde{u}={\bf 1}_{E}.$ Now, it is plain
to show that $E={p^{-1}(\dot{E}_{u})}$. Hence
$\dot{E}_{u}\in{\mathscr{R}}_{c}(H)$ and $u={\bf 1}_{\dot{E}_{u}}$.

For the converse, let $u={\bf 1}_{\dot{E}}$ for some
$\dot{E}\in{\mathscr{R}}_{c}(H)$. Then, by Host’s idempotent theorem [Hos86],
$v={\bf 1}_{p^{-1}(\dot{E})}$ is an idempotent in $B(G)$. It is clear that $v$
is constant on each orbit $\mathcal{O}_{x}$ and hence $v\in B_{\pi}(G)$. Now,
since for each $x\in G$ we have $v(x)=u(\dot{x})$, thus $u$ is an idempotent
in $B(H)$. ∎

######  Remark 3.7.

Since elements of $B(H)$ are continuous functions, it is clear that an
idempotent of $B(H)$ has to be of the form ${\bf 1}_{\dot{F}}$ where $\dot{F}$
is a clopen subset of $H.$

As an immediate consequence of Lemma 3.6 and Lemma 3.3, we obtain the
following corollary.

######  Corollary 3.8.

Let $H$ be an ultraspherical hypergroup. For any subset $\dot{E}$ of $H$, the
following are equivalent.

$\dot{E}\in{\mathscr{R}}_{c}(H)$;

$\dot{E}=\dot{F}_{u}$ for some idempotent $u\in B(H)$;

$\dot{E}=\dot{F}_{u}$ for some power bounded $u\in B(H).$

In the next corollary, we show that the interior and boundary of a set from
the coset ring also belongs to the coset ring.

######  Corollary 3.9.

Let $H$ be an ultraspherical hypergroup associated to an amenable locally
compact group $G$ and let $\dot{E}\in{\mathscr{R}}_{c}(H)$.

The interior $\dot{E}^{\circ}$ and the boundary $\partial\dot{E}$ of $\dot{E}$
both belong to ${\mathscr{R}}_{c}(H).$

$\dot{E}^{\circ}=\dot{F}_{u}$ for some power bounded $u\in B(H).$

$\partial\dot{E}=\dot{F}_{u}$ for some power bounded $u\in B(H)$.

###### Proof.

By Corollary 3.8, we only have to show (i). Since the quotient map
$p:G\rightarrow H$ is open, we have
$p^{-1}(\dot{E}^{\circ})=p^{-1}(\dot{E})^{\circ}$ and
$p^{-1}(\partial\dot{E})=\partial p^{-1}(\dot{E})$. Now since
$p^{-1}(\dot{E})\in{\mathscr{R}}_{c}(G)$, by [KLU14, Lemma 1.1],
$p^{-1}(\dot{E})^{\circ}$ and $\partial p^{-1}(\dot{E})$ both belong to
${\mathscr{R}}_{c}(G).$ In other words, the sets $\dot{E}^{\circ}$ and
$\partial\dot{E}$ are in ${\mathscr{R}}_{c}(H).$ ∎

###  3.2. On the sequence $\\{u^{n}\\}_{n\in\mathbb{N}}$

We now study the sequence $\\{u^{n}\\}_{n\in\mathbb{N}},$ where $u$ is a power
bounded element of $B(H).$

Our first lemma gives the properties of the weak*-limit of the sequence
$\\{u^{n}\\}_{n\in\mathbb{N}},$ whenever it exists.

######  Lemma 3.10.

Let $H$ be an ultraspherical hypergroup associated to a locally compact group
$G$. Let $u\in B(H)$ and suppose that $\theta=\text{weak${}^{*}$-$\lim_{n}$}\
u^{n}$ exists. Then

$\theta$ is an idempotent.

$\theta$ satisfies $\theta u=\theta$ and $\theta={\bf
1}_{\dot{F}_{u}^{\circ}}.$

the set $\dot{F}_{u}^{\circ}$ is closed in $G$ and
$\lambda_{H}(\dot{F}_{u}\setminus\dot{F}_{u}^{\circ})=0$.

###### Proof.

By Lemma 2.4, multiplication in $B(H)$ is separately weak∗ continuous and
hence for each $\varrho\in C^{*}(H)$, we have

$\langle u\theta,\varrho\rangle=\lim_{n}\langle
u^{n+1},\varrho\rangle=\langle\theta,\varrho\rangle,$

i.e., $u\theta=\theta.$ Further,

$\displaystyle\langle\theta^{2},\varrho\rangle$
$\displaystyle=\lim_{n}\lim_{m}\langle
u^{n}u^{m},\varrho\rangle=\lim_{n}\lim_{m}\langle
u^{n+m},\varrho\rangle=\langle\theta,\varrho\rangle,$

showing that $\theta$ is an idempotent. By Lemma 3.6 and Remark 3.7, there
exists a clopen set $\dot{E}\in{\mathscr{R}}(H)$ such that $\theta={\bf
1}_{\dot{E}}.$ The remaining assertions follows as in [KU13, Lemma 3.1] ∎

Our next result gives conditions under which the weak* limit used in Lemma
3.10 holds. As the proof of this follows exactly as in the [KU13, Lemma 3.2],
we omit the proof.

######  Lemma 3.11.

Let $u$ be a power bounded element of $B(H)$ and suppose that
$\lim_{n\rightarrow\infty}\ \|u^{n+1}-u^{n}\|=0$. Then
$\theta=\text{weak${}^{*}$-$\lim_{n}$}\ u^{n}$ exists.

Here is a characterization of the power boundedness of an element $u\in B(H)$
in terms of the set $\dot{F}_{u}.$ For the corresponding result on locally
compact groups, see [KU13, Theorem 3.3].

######  Theorem 3.12.

Let $H$ be an ultraspherical hypergroup associated to a locally compact group
$G$ and let $u\in B(H)$ such that $\underset{n\rightarrow\infty}{\lim}\
\|u^{n+1}-u^{n}\|=0$. Then the following are equivalent.

$u$ is power bounded.

The set $\dot{F}_{u}^{\circ}$ is closed in $H$, belongs to the coset ring
${\mathscr{R}}(H)$ and satisfies $\text{weak${}^{*}$-$\lim_{n}$}\ {\bf
1}_{H\setminus{\dot{F}_{u}^{\circ}}}u^{n}=0$.

###### Proof.

In view of Lemma 2.4, Lemma 3.10 and Lemma 3.11, the proof of this follows
exactly as in [KU13, Theorem 3.3]. ∎

The next lemma is a special case of [KLU10, Corollary 1.3] and hence we omit
the proof.

######  Lemma 3.13.

Let $H$ be an ultraspherical hypergroup. Then for any power bounded element
$u\in B(H)$, the element $\frac{1+u}{2}$ is power bounded and

$\lim_{n\rightarrow\infty}\left\|\left(\frac{1+u}{2}\right)^{n+1}-\left(\frac{1+u}{2}\right)^{n}\right\|=0.$

Combining Lemmas 3.11 and 3.13 and the fact that $F_{u}=F_{\frac{1+u}{2}}$ one
obtains the following description of the power bounded elements of $B(H)$.

######  Corollary 3.14.

Let $H$ be an ultraspherical hypergroup associated to a locally compact group
$G$. Then, for any power bounded $u\in B(H)$,
$\theta=\text{weak${}^{*}$-$\lim_{n}$}(\frac{1+u}{2})^{n}$ exists and
$\theta={\bf 1}_{\dot{F}_{u}^{\circ}}$.

The following is a simple observation.

######  Lemma 3.15.

Let $H$ be an ultraspherical hypergroup associated a locally compact group
$G.$ If $u\in B(H)$ is power bounded, then so is $|u|.$

###### Proof.

Let us first note that $\|u\|=\|\bar{u}\|$ for all $u\in B(H),$ by [Mur07,
Remark 2.9]. Let $u\in B(H)$ be power bounded with $\gamma=\sup_{n}\|u^{n}\|.$
Then for each $n\in\mathbb{N}$, we have

$\||u|^{2n}\|=\|u^{n}\bar{u}^{n}\|\leq\|u^{n}\|\cdot\|\bar{u}^{n}\|=\|u^{n}\|^{2}\leq\gamma^{2}$

and

$\||u|^{2n+1}\|\leq\||u|\|\cdot\||u|^{2n}\|\leq\gamma^{2}\||u|\|.$

Thus, $|u|$ is a power bounded element of $B(H)$. ∎

The next result is an analogue of [KU13, Proposition 3.5].

######  Corollary 3.16.

Let $H$ be an ultraspherical hypergroup associated to a locally compact group
$G$. Then, the set $\dot{E}_{u}^{\circ}$ is closed in $H$ and
$\lambda(\dot{E}_{u}\setminus\dot{E}_{u}^{\circ})=0$.

###### Proof.

This follows from Lemma 3.10, Lemma 3.15 and the fact that
$\dot{E}_{u}=\dot{F}_{|u|}.$ ∎

As an immediate consequence of Corollary 3.14 we obtain the following lemma.

######  Lemma 3.17.

Let $u\in B(H)$ be power bounded and let
$\theta=\text{weak${}^{*}$-$\lim_{n}$}_{\rightarrow\infty}(\frac{1+u}{2})^{n}$.
Then we have

$\langle\theta,f\rangle=\int_{\dot{F}_{u}}f(\dot{x})d\dot{x}$ for all $f\in
L^{1}(H)$;

$\theta=0$ if and only if $\lambda_{H}(\dot{F_{u}})=0$.

######  Lemma 3.18.

Let $u\in B(H)$ be power bounded. Then the following assertions hold:

If $\dot{E}_{u}=\dot{F}_{u}$, then
$\theta=\text{weak${}^{*}$-$\lim_{n}$}_{\rightarrow\infty}u^{n}$ exists and
$\theta={\bf 1}_{\dot{F}_{u}^{\circ}}$.

If $H$ is discrete, then
$\theta=\text{weak${}^{*}$-$\lim_{n}$}_{\rightarrow\infty}u^{n}$ exists if and
only if $\dot{E}_{u}=\dot{F}_{u}$.

$\text{weak${}^{*}$-$\lim_{n}$}_{\rightarrow\infty}|u|^{n}=0$ if and only if
$\dot{E}_{u}^{\circ}=\emptyset.$

If $\text{weak${}^{*}$-$\lim_{n}$}_{\rightarrow\infty}|u|^{n}=0$, then
$\text{weak${}^{*}$-$\lim_{n}$}_{\rightarrow\infty}u^{n}=0$.

###### Proof.

(i). Assume that $\dot{E}_{u}=\dot{F}_{u}$. Then, we can show that the
sequence $\\{u^{n}\\}_{n\in\mathbb{N}}$ converges pointwise to ${\bf
1}_{\dot{F}_{u}}$ and also by Corollary 3.16,
$\lambda(\dot{F}_{u}\setminus\dot{F}_{u}^{\circ})=0$. Using these and the fact
that $\|u\|_{\infty}\leq 1$, we have

$\lim_{n}u^{n}(\dot{x})f(\dot{x})={\bf
1}_{\dot{F}_{u}^{\circ}}(\dot{x})f(\dot{x})\qquad\text{and}\qquad|u^{n}f|\leq|f|,$

almost everywhere on $H$ and for all $f\in L^{1}(H)$. Now, the Lebesgue’s
dominated convergence theorem yields that

$\displaystyle\langle u^{n},\lambda(f)\rangle$
$\displaystyle=\int_{H}u^{n}fd\lambda\rightarrow\int_{H}{\bf
1}_{\dot{F}_{u}^{\circ}}fd\lambda=\langle{\bf
1}_{\dot{F}_{u}^{\circ}},\lambda(f)\rangle$

as $n\rightarrow\infty$. Here, the last equality is a consequence of the fact
that if $u\in B(H)$ is power bounded, then ${\bf 1}_{\dot{F}_{u}^{\circ}}\in
B(H)$. Now, the equality $\theta={\bf 1}_{\dot{F}_{u}^{\circ}}$ follows from a
simple approximation argument.

(ii). If $\theta=\text{weak${}^{*}$-$\lim_{n}$}_{\rightarrow\infty}u^{n}$
exists, then by Lemma 3.10, $\theta={\bf 1}_{\dot{F}_{u}}$. Since $H$ is
discrete, weak∗-topology on $B(H)$ is stronger than topology of pointwise
convergence and thus we have

$\lim_{n\rightarrow\infty}u^{n}(\dot{x})={\bf 1}_{\dot{F}_{u}}(\dot{x}),\
\dot{x}\in H.$

Since $\dot{F}_{u}\subseteq\dot{E}_{u},$ we only need to show the reversed
inclusion. Towards a contradiction, suppose that
$\dot{x}\in\dot{E}_{u}\setminus\dot{F}_{u}$. Then $|u^{n}(\dot{x})|=1$ for all
$n\in\mathbb{N}$ and hence

$|u^{n}(\dot{x})-{\bf 1}_{\dot{F}_{u}}(\dot{x})|=|u^{n}(\dot{x})|\rightarrow
1\neq 0,$

as $n\rightarrow\infty$, a contradiction. The converse follows from (i).

(iii). If $\dot{E}_{u}^{\circ}=\emptyset$, then Corollary 3.16 implies that
$\lambda_{H}(\dot{E})=0$. Since $u\in B(H)$ is power bounded, we have
$|u(\dot{x})|_{\infty}<1$ for all $\dot{x}\in H\setminus\dot{E}$. Thus

$\lim_{n\rightarrow\infty}|u^{n}(\dot{x})|=0,\qquad(\text{a.e. on}~{}H).$

The rest of the proof runs as in (i).

Assume that $\text{weak${}^{*}$-$\lim_{n}$}_{\rightarrow\infty}|u|^{n}=0$.
Then we have

$\lambda_{H}(\dot{E}_{u}^{\circ})\leq\lambda_{H}(\dot{E}_{u})=\lambda_{H}(\dot{F}_{|u|})=0,$

the last equality being a consequence of Lemma 3.17. Hence (iii) holds.

(iv). The same reasoning as in the first part of the proof of (iii) applies to
here. ∎

The assertion of the following corollary is an immediate consequence of Lemma
3.18 and the fact $\dot{E}_{uv}=\dot{E}_{u}\cap\dot{E}_{v}$ for all power
bounded elements $u,v\in B(H)$.

######  Corollary 3.19.

Let $u,v$ be elements of $B(H)$ such that
$\text{weak${}^{*}$-$\lim_{n}$}_{\rightarrow\infty}|u|^{n}=0$ and $v$ is power
bounded. Then $\text{weak${}^{*}$-$\lim_{n}$}_{\rightarrow\infty}(uv)^{n}=0$.

Here is the final result of this section. This is about powers of power
bounded elements and an element from the ideal $J_{A(H)}(C),$ where $C$ is a
closed subset of $H.$ As the proof of this follows exactly as in [Mus19,
Proposition 4.1], we shall omit the proof.

######  Theorem 3.20.

Let $u$ be a power bounded element of $B(H)$ and let $C$ be a closed subset of
$H$. Then the following are equivalent:

$\lim_{n\rightarrow\infty}\left\|\frac{1}{n}\sum_{k=0}^{n-1}|u|^{2k}v\right\|_{A(H)}=0$
, for all $v\in J_{A(H)}(C)$.

$\lim_{n\rightarrow\infty}\|u^{n}v\|_{A(H)}=0$, for all $v\in J_{A(H)}(C)$.

###  3.3. Cesàro boundedness

Here we deal with Cesàro boundedness. This concept was introduced by
Mustafayev [Mus19] in order to study certain non-abelian analogues of the
ergodic theorem. We shall derive relations to Cesàro bounded elements with the
sets $E_{u}$ and $F_{u}$ as done in the previous sections.

We now begin with the definition of Cesàro boundedness.

######  Definition 3.21.

Let $\mathcal{A}$ be a complex Banach algebra with the unit element $e$. An
element $a\in\mathcal{A}$ is said to be Cesàro bounded if

$\sup_{n\in\mathbb{N}}\left\|\frac{1}{n}\sum_{k=0}^{n-1}a^{k}\right\|<\infty.$

The following example shows that Cesàro boundedness does not imply power
boundedness.

######  Remark 3.22.

The Assani matrix $T=\begin{pmatrix}-1&2\\\ 0&-1\end{pmatrix}$ is Cesàro
bounded, but not power bounded.

The following is an analogue of Lemma 3.3

######  Proposition 3.23.

Let $H$ be an ultraspherical hypergroup associated to a locally compact group
$G$. If $u\in B(H)$ is Cesàro bounded, then
$\dot{F}_{u}\in{\mathscr{R}}_{c}(H)$.

###### Proof.

Since $u\in B(H)$ is Cesàro bounded, the corresponding $\tilde{u}\in B(G)$ is
Cesàro bounded. Thus, by [Mus19, Proposition 2.4],
$p^{-1}(\dot{F}_{u})=F_{\tilde{u}}\in{\mathscr{R}}_{c}(G),$ which is
equivalent to saying that $\dot{F}_{u}\in{\mathscr{R}}_{c}(H).$ ∎

The following is the analogue of Lemma 3.10.

######  Lemma 3.24.

Let $H$ be an ultraspherical hypergroup associated to a locally compact group
$G$. Let $u\in B(H)$ and suppose that

$\theta=\text{weak${}^{*}$-$\lim_{n}$}\frac{1}{n}\sum_{k=0}^{n-1}u^{k}$

exists. Then

$\theta$ is an idempotent and satisfies $\theta u=\theta$. More precisely,
$\theta={\bf 1}_{\dot{F}_{u}^{\circ}}.$

The set $F_{u}^{\circ}$ is closed in $H$ and $\lambda(F_{u}\setminus
F_{u}^{\circ})=0$.

###### Proof.

First note that

$\displaystyle\theta u$
$\displaystyle=\text{weak${}^{*}$-$\lim_{n}$}\frac{1}{n}\sum_{k=1}^{n}u^{k}$
$\displaystyle=\text{weak${}^{*}$-$\lim_{n}$}\left[\frac{1}{n}\sum_{k=0}^{n}u^{k}-\frac{1}{n}\cdot
1\right]$
$\displaystyle=\text{weak${}^{*}$-$\lim_{n}$}\left[\frac{n+1}{n}(\frac{1}{n+1}\sum_{k=0}^{n}u^{k})-\frac{1}{n}\cdot
1\right]$
$\displaystyle=\text{weak${}^{*}$-$\lim_{n}$}\frac{1}{n+1}\sum_{k=0}^{n}u^{k}=\theta.$

By Lemma 2.4 multiplication in $B(H)$ is separately weak* continuous and
therefore $\theta=\theta^{2},$ i.e., $\theta$ is an idempotent in $B(H).$ So,
by Lemma 3.6 and Remark 3.7 there is clopen subset $\dot{E}$ of $H$ in
${\mathscr{R}}_{c}(H)$ such that $\theta={\bf 1}_{\dot{E}}.$ Now, the
remaining can be proved by using the same arguments as those employed in the
proof of Lemma 3.10. ∎

Our next result is an analogue of Corollary 3.14.

######  Proposition 3.25.

Let $H$ be an ultraspherical hypergroup associated to a locally compact group
$G$. Let $u$ be a Cesàro bounded element of $B(H)$. Then

${\bf
1}_{\dot{F}^{\circ}_{u}}=\text{weak${}^{*}$-$\lim_{n}$}\frac{1}{n}\sum_{k=0}^{n-1}u^{k}.$

###### Proof.

Let $u$ be a Cesàro bounded element of $B(H)$. Then for each $f\in L^{1}(H)$,
the sequence $(\frac{1}{n}\sum_{k=0}^{n-1}u^{k})_{n}$ converges pointwise to
${\bf 1}_{\dot{F}_{u}}f$ as $n\rightarrow\infty$ and is also dominated $|f|.$
Hence by the dominated convergence theorem

$\displaystyle\langle\psi_{n},\lambda(f)\rangle$
$\displaystyle=\int_{H}\psi_{n}(\dot{x})f(\dot{x})d\dot{x}\rightarrow\int_{H}{\bf
1}_{\dot{F}_{u}}(\dot{x})f(\dot{x})d\dot{x}$ $\displaystyle=\int_{H}{\bf
1}_{\dot{F}^{\circ}_{u}}(\dot{x})f(\dot{x})d\dot{x}+\int_{H}{\bf
1}_{\partial\dot{F}_{u}}(\dot{x})f(\dot{x})d\dot{x}$
$\displaystyle=\int_{H}{\bf
1}_{\dot{F}^{\circ}_{u}}(\dot{x})f(\dot{x})d\dot{x}\qquad(\text{since}~{}\lambda(F_{u}\setminus
F_{u}^{\circ})=0)$ $\displaystyle=\langle{\bf
1}_{\dot{F}^{\circ}_{u}},\lambda(f)\rangle.$

Now, using the density of $\lambda(L^{1}(H))$ in $C^{*}(H)$ and the
boundedness of the sequence $(\frac{1}{n}\sum_{k=0}^{n-1}u^{k})_{n},$ the
conclusion follows. ∎

######  Proposition 3.26.

Let $H$ be an ultraspherical hypergroup and let $u$ be a Cesàro bounded
element of $B(H)$ such that $\lim_{n}\frac{1}{n}\|u^{n}v\|_{A(H)}=0$ for all
$v\in A(H)$. Consider the following statements:

$\theta_{v}=\lim_{n}\frac{1}{n}\sum_{k=0}^{n-1}u^{k}v$ exists in $A(H)$-norm
topology for all $v\in A(H).$

$\dot{F}_{u}$ is an open subset of $H.$

$\theta_{v}={\bf 1}_{\dot{F}_{u}}v$ and $\|{\bf
1}_{\dot{F}_{u}}v\|_{A(H)}=dist(\overline{(1-u)A(H)},v)$ for all $v\in A(H).$

Then the following assertions hold:
$(i)\Longleftrightarrow(ii)\Longrightarrow(iii).$ Furthermore, if u is Cesàro
contractive, then (iii) implies the following:

$\|{\bf 1}_{\dot{F}_{u}}v\|_{A(H)}=dist(\overline{(1-u)A(H)},v)$ for all $v\in
A(H)$.

###### Proof.

The proof of this is exactly the same as in [Mus19, Theorem 2.8] ∎

######  Proposition 3.27.

Let $H$ be an ultraspherical hypergroup such that the underlying locally
compact group $G$ is amenable. Then for any Cesàro bounded element $u\in
B(H)$, we have the following:

The sets $\overline{(1-u)A(H)}$ and $I(\dot{F}_{u})$ are equal.

The ideal $\overline{(1-u)A(H)}$ has a bounded approximate identity.

Let $(e_{\alpha})$ be a bounded approximate identity of
$\overline{(1-u)A(H)}$. Then, after passing to a subnet if necessary,
$e_{\alpha}\rightarrow{\bf 1}_{H\setminus\dot{F}_{u}^{\circ}}$ in the
weak∗-topology $\sigma(B_{\lambda}(H),C^{*}_{\lambda}(H))$.

There exists an idempotent $\theta\in B(H)$ such that
$\overline{(1-u)A(H)}^{weak^{*}}=\theta B(H)$. More precisely, $\theta={\bf
1}_{H\setminus\dot{F}_{u}^{\circ}}$.

Moreover, If $\dot{F}_{u}$ is an open subset of $H$, then

$\overline{(1-u)A(H)}=I(\dot{F}_{u})={\bf 1}_{H\setminus\dot{F}_{u}}A(H)$.

###### Proof.

(i). Suppose that $u\in B(H)$ is Ces̀aro bounded. Then, by Proposition 3.23
and Lemma 4.11, $\dot{F}_{u}$ is a set of synthesis for $A(H)$. Now, since
$Z(\overline{(1-u)A(H)})=\dot{F}_{u}$, the equality of $\overline{(1-u)A(H)}$
and $I(\dot{F}_{u})$ holds.

(ii). This follows from (i) and Lemma 4.11.

(iii). Let $(e_{\alpha})$ be a bounded approximate identity of
$\overline{(1-u)A(H)}$. Passing to a subnet if necessary, we can assume that
$e_{\alpha}\rightarrow\theta$ in the weak∗-topology
$\sigma(B_{\lambda}(H),C^{*}_{\lambda}(H))$ for some $\theta\in B(H)$. Let
$f\in C_{c}(H)$. Then

$\displaystyle\langle e_{\alpha},\lambda(f)\rangle$
$\displaystyle=\int_{H}e_{\alpha}(\dot{x})f(\dot{x})d\dot{x}$
$\displaystyle=\int_{H\setminus\dot{F}_{u}}e_{\alpha}(\dot{x})f(\dot{x})d\dot{x}+\int_{\dot{F}_{u}}e_{\alpha}(\dot{x})f(\dot{x})d\dot{x}$
$\displaystyle=\int_{\dot{F}\setminus\dot{F}_{u}^{\circ}}e_{\alpha}(\dot{x})f(\dot{x})d\dot{x}+\int_{H\setminus\dot{F}_{u}^{\circ}}e_{\alpha}(\dot{x})f(\dot{x})d\dot{x}\qquad(\text{since}~{}e_{\alpha}|_{\dot{F}_{u}}=0)$
$\displaystyle=\int_{H\setminus\dot{F}_{u}^{\circ}}e_{\alpha}(\dot{x})f(\dot{x})d\dot{x}\qquad(\text{since
by Lemma \ref{lem10}},~{}\lambda(\dot{F}_{u}\setminus\dot{F}_{u}^{\circ})=0)$
$\displaystyle=\int_{H\setminus\dot{F}_{u}^{\circ}}e_{\alpha}(\dot{x})v(\dot{x})f(\dot{x})d\dot{x}\qquad(v\in
I(\dot{F}_{u})~{}\text{with}~{}v\mid_{\text{supp}(f)\cap(H\setminus\dot{F}_{u})}=1)$
$\displaystyle=\int_{H}e_{\alpha}(\dot{x})v(\dot{x}){\bf
1}_{H\setminus\dot{F}_{u}^{\circ}}(\dot{x})f(\dot{x})d\dot{x}$
$\displaystyle=\langle e_{\alpha}v,\lambda({\bf
1}_{H\setminus\dot{F}_{u}^{\circ}}f)\rangle\rightarrow\langle v,\lambda({\bf
1}_{H\setminus\dot{F}_{u}^{\circ}}f)\rangle$ $\displaystyle=\langle{\bf
1}_{H\setminus\dot{F}_{u}^{\circ}},\lambda(f)\rangle,$

where the last equality holds because

$H\setminus\dot{F}_{u}^{\circ}=(H\setminus\dot{F}_{u})\cup(\dot{F}_{u}\setminus\dot{F}_{u}^{\circ})\quad\text{and
}\quad\lambda(\dot{F}_{u}\setminus\dot{F}_{u}^{\circ})=0.$

Now, using the density of $\lambda(C_{c}(H))$ in $C^{*}(H)$ and boundedness of
the net $(e_{\alpha})$, we have

$\text{weak${}^{*}$-$\lim_{\alpha}$}e_{\alpha}={\bf
1}_{H\setminus\dot{F}_{u}^{\circ}}.$

(iv). The statement follows from (iii) and the fact that ${\bf
1}_{H\setminus\dot{F}_{u}^{\circ}}$ is an identity for
$\overline{(1-u)A(H)}^{weak^{*}}$ in $B(H)$.

(v). If $\dot{F}_{u}$ is open, then, by (iv), ${\bf
1}_{H\setminus\dot{F}_{u}}$ is an idempotent in $B(H)$. Thus ${\bf
1}_{H\setminus\dot{F}_{u}}A(H)$ is closed ideal of $A(H)$. Let $(e_{\alpha})$
be a bounded approximate identity of $A(H)$. Then $({\bf
1}_{H\setminus\dot{F}_{u}}e_{\alpha})$ is a bounded approximate identity for
$I(\dot{F}_{u})$. By (i), $\overline{(1-u)A(H)}=I(\dot{F}_{u})$, thus for each
$v\in\overline{(1-u)A(H)}$, we have

$v=\lim_{\alpha}{\bf 1}_{H\setminus\dot{F}_{u}}e_{\alpha}v.$

Hence, $v\in\overline{{\bf 1}_{H\setminus\dot{F}_{u}}A(H)}={\bf
1}_{H\setminus\dot{F}_{u}}A(H)$. The reverse inclusion is clear. ∎

The following proposition gives a description the set $F_{u},$ where $u$ is a
Cesàro bounded.

######  Proposition 3.28.

Let $H$ be an ultraspherical hypergroup. Let $u$ be a Cesàro bounded element
of $B(H)$ satisfying $\lim_{n}\frac{1}{n}\|u^{n}v\|_{A(H)}=0$ for all $v\in
A(H)$ and $\dot{F}_{u}$ is an open subset of $H$, then

$\overline{(1-u)A(H)}={\bf 1}_{H\setminus\dot{F}_{u}}A(H).$

###### Proof.

The proof of this is exactly same as [Mus19, Corollary 2.10]. ∎

######  Corollary 3.29.

Let $H$ be an ultraspherical hypergroup associated to an amenable locally
compact group $G$. Then for any power bounded element $u\in B(H)$, we have the
following:

The sets $\overline{(1-u)A(H)}$ and $I(\dot{F}_{u})$ are equal.

Let $A_{0}(u)=\\{v\in A(H):\lim_{n}\|u^{n}v\|=0\\}$. Then $A_{0}(u)$ and
$I(\dot{E}_{u})$ are equal.

The ideal $\overline{(1-u)B(H)}$ has a bounded approximate identity.

There exists an idempotent $\theta\in B(H)$ such that
$\overline{(1-u)B(H)}^{weak^{*}}=\theta B(H)$.

###### Proof.

As powerboundedness implies Cesàro boundedness, this corollary is an immediate
consequence of Proposition 3.27. ∎

######  Definition 3.30.

Let $H$ be a discrete ultraspherical hypergroup. For any subset $\dot{E}$ of
$C^{*}_{\lambda}(H)$, we write

$C^{*}_{\lambda}(E)=\overline{\text{span}\\{\lambda(\dot{x}):\dot{x}\in\dot{E}\\}}^{\|.\|_{C^{*}_{\lambda}(H)}}$

The following lemma generalises Lemma 4.3 of [KLU14] to the context of
ultraspherical hypergroups. As the proof of this is same as the proof of
[KLU14, Lemma 4.3] we omit the proof.

######  Lemma 3.31.

Let $H$ be a discrete ultraspherical hypergroup. If $E\in{\mathscr{R}}(H)$,
then ${\bf 1}_{\dot{E}}\cdot C^{*}_{\lambda}(H)=C^{*}_{\lambda}(E)$.

######  Corollary 3.32.

Let $H$ be a discrete ultraspherical hypergroup,
$\dot{E}\in{\mathscr{R}}_{c}(H)$ and $u\in B_{\lambda}(H)$. Then the following
are equivalent.

$u^{n}{\bf 1}_{\dot{E}}\rightarrow 0$ in weak∗ topology of $B_{\lambda}(H)$.

For each $T\in C^{*}_{\lambda}(\dot{E}),\lim_{n\rightarrow\infty}\langle
u^{n},T\rangle=0$.

##  4\. Power boundedness of $A(H)$ and $B(H)$

In this section, we study the power boundedness of the Fourier algebra $A(H)$
and Fourier-Stieltjes algebra $B(H).$

###  4.1. Power bounded property of $A(H)$

We shall begin with a simple lemma.

######  Lemma 4.1.

Let $U$ be an open subset of $H.$ Then there exists $u\in A(H)$ and an open
set $V$ of $H$ such that the following holds:

$0\leq u\leq 1$.

$\|u\|_{A(H)}=1=u(\dot{e})$.

$\text{supp}(u)\subset U$.

$u(\dot{x})>0$ for all $\dot{x}\in V$.

###### Proof.

There exists a symmetric, relatively compact neighbourhood $V_{1}$ of
$\dot{e}$ in $H$ such that $V_{1}^{2}\subset U.$ Let
$u=\dfrac{1}{\lambda(V_{1})}{\bf 1}_{V_{1}}\ast{\bf 1}_{V_{1}}$ and
$V=\\{\dot{x}\in H:~{}u(\dot{x})>0\\}.$ Then $u$ and $V$ satisfy the
requirements of the lemma. ∎

######  Lemma 4.2.

Let $H$ be a second countable ultraspherical hypergroup and let $K$ a closed
subset of $H.$ Then there exists $u\in A(H)$ such that

$0\leq u\leq 1$ and

$K=u^{-1}(0)$.

###### Proof.

Let $\mathcal{V}$ be a neighbourhood basis of $\dot{e}$ in $H$ such that for
each $V\in\mathcal{V},$ there exists $u_{V}\in A(H)$ with the property that
$0\leq u_{V}\leq 1,$ $\|u_{V}\|_{A(H)}=1=u(\dot{e}),$
$\text{supp}(u_{V})\subset V$ and $u_{V}(\dot{x})>0$ for all $\dot{x}\in V$
(such neighbourhoods exists by the previous lemma). Let $K$ be any closed set
in $H$ and $W=H\setminus K.$ Since $H$ is second countable, there are
sequences $\\{\dot{a}_{n}\\}_{n\in\mathbb{N}}\subset W$ and
$\\{V_{n}\\}_{n\in\mathbb{N}}\subset\mathcal{V}$ such that
$W=\bigcup_{n\in\mathbb{N}}\dot{a}_{n}\ast V_{n}.$ For each $n\in\mathbb{N},$
let $u_{n}=u_{V_{n}}$ and let $u=\sum_{1}^{\infty}2^{-n}L_{\dot{a}_{n}}u_{n}$
where $L_{\dot{a}}f(\dot{x}):=f(\check{\dot{a}}\ast\dot{x})$. Then $u\in
A(H),$ since $\|L_{\dot{a}_{n}}u_{n}\|\leq\|u_{n}\|=1.$ By [Jew75, Lemma 4.1B]
it is obvious that $u(\dot{x})>0$ for all $\dot{x}\in W$ and $u(\dot{x})=0$
for all $\dot{x}\in K.$ ∎

######  Theorem 4.3.

Let $H$ be an ultraspherical hypergroup. The Banach algebra $A(H)$ has the
power boundedness property if and only if $H$ is discrete.

###### Proof.

If $H$ is discrete, then the power boundedness of $A(H)$ is a consequence of
[Sch70, Corollary 2.3]. Thus, we are left with the proof of the forward part.
Towards a contradiction, assume that $H$ is nondiscrete.

Suppose that $H$ is second countable. In view of Lemma 2.2 and Lemma 4.2 the
proof now follows exactly as in [Gra76, Proposition 1] .

Now let $H$ be an ultraspherical hypergroup which is not second countable. Let
$U$ be a symmetric relatively compact neighbourhood of $\dot{e}$ and let
$H^{\prime}=\underset{n\in\mathbb{N}}{\bigcup}{U^{n}},$ where
$U^{n}=U*U*\ldots*U\ (n$-times$).$ It is clear that $H^{\prime}$ is an open
subhypergroup of $H$ and hence closed. Since $A(H)$ has the power bounded
property, it follows from [SK14, Lemma 5.1] that $A(H^{\prime})$ also has the
power bounded property. Since $H$ is non-discrete, so is $H^{\prime}.$ Choose
a sequence $\\{U_{n}\\}_{n\in\mathbb{N}}$ of relatively compact neighbourhoods
of $\dot{e}$ such that $\lambda(U_{n})\rightarrow 0.$ By [SK17, Theorem 1.1],
there exists a compact subhypergroup
$H^{\prime\prime}\subseteq\underset{n\in\mathbb{N}}{\cap}U_{n}$ such that
$H^{\prime}//H^{\prime\prime}$ is a second countable ultraspherical
hypergroup. Since $H^{\prime}//H^{\prime\prime}$ is also an ultraspherical
hypergroup based on the subgroup $p^{-1}(H^{\prime})$ and the spherical
projector $\pi|_{C_{c}(p^{-1}(H^{\prime}))},$ it follows that
$A(H^{\prime}//H^{\prime\prime})$ can be identified isometrically inside
$A(H^{\prime}).$ As $A(H^{\prime})$ has the power bounded property, it follows
that $A(H^{\prime}//H^{\prime\prime})$ also has the power bounded property.
Therefore, by the preceding paragraph it follows that
$H^{\prime}//H^{\prime\prime}$ is discrete, which forces us to conclude that
$H^{\prime\prime}$ is open and hence $\lambda(H^{\prime\prime})>0,$ a
contradiction. ∎

###  4.2. Power bounded property for $B(H)$

Our main aim of this part is to show that $B(H)$ is power bounded if and only
if $H$ is finite. Our idea is to make use of weakly almost periodic functions
on $H$ and weakly almost periodic functionals on $L^{1}(H).$ We shall begin
this section by defining the same.

######  Definition 4.4.

  1. (i)

Let $H$ be an arbitrary locally compact hypergroup. A function $f\in C_{b}(H)$
is called weakly almost periodic if the left orbit $O_{L}(f)=\\{\ell_{x}f:x\in
H\\},$ where $\ell_{x}f(t)=f(x\ast t)~{}(t\in H),$ is relatively weakly
compact in $C_{b}(H)$. We denote the set of all weakly almost periodic
functions on $H$ by $wap(H).$

  2. (ii)

Let $H$ be a locally compact hypergroup with a Haar measure $\lambda$. Let
$f\in L^{\infty}(H)$ and $\mathcal{O}_{L}(f)=\\{a\cdot f:a\in
L^{1}(H),\|a\|\leq 1\\},$ where $a\cdot f=\frac{1}{\Delta}\check{a}\ast
f~{}(a\in L^{1}(H))$. The functional $f$ is called weakly almost periodic on
$L^{1}(H)$ if $\mathcal{O}_{L}(f)$ is
$\sigma(L^{\infty}(H),L^{\infty}(H)^{*})$-relatively compact. We denote the
set of all weakly almost periodic functionals on $L^{1}(H)$ by
$WAP(L^{1}(H))$.

######  Lemma 4.5.

Let $H$ be a locally compact hypergroup with a Haar measure $\lambda$. Let
$f\in L^{\infty}(H)$ which is in $WAP(L^{1}(H))$. Then $f$ is $\lambda$-a.e
continuous on $H$.

###### Proof.

Let $f\in WAP(L^{1}(H))$ and $(e_{\alpha})_{\alpha}$ be an approximate
identity for $L^{1}(H)$ with $\|e_{\alpha}\|\leq 1$ for all $\alpha$. As $f\in
WAP(L^{1}(H))$ and also as the net $(e_{\alpha})$ converges to a right
identity of $L^{\infty}(H)^{*},$ after passing to a subnet if necessary, we
can assume that $\text{weak-$\lim_{\alpha}$}e_{\alpha}\cdot f=f.$ Now, since
$e_{\alpha}\cdot f\in C_{b}(H)$ for all $\alpha$, we conclude that
$f\in\overline{C(H)}^{weak}=C(H).$ ∎

######  Lemma 4.6.

Let $H$ be a locally compact hypergroup with a Haar measure $\lambda$ and let
$f\in C_{b}(H)$. Then $f\in wap(H)$ if and only if the set
$\mathbb{O}_{L}(f)=\\{f\cdot\mu:\mu\in M(H),\|\mu\|\leq 1\\}$ is relatively
weakly compact in $C_{b}(H)$.

###### Proof.

Assume that $\mathbb{O}_{L}(f)$ is relatively weakly compact. Since
$\mathcal{O}_{L}(f)\subset\mathbb{O}_{L}(f)$, we have $f\in wap(H)$. For the
converse, let $\mathcal{O}_{L}(f)$ be relatively weakly compact in $C_{b}(H)$.
Then, by Krein-Šmulian theorem [Meg12, Theorem 2.8.14], the closure of
$co(O_{L}(f))$ in $C_{b}(H)$ is weakly compact. Let $\mu\in M(H)$ with
$\|\mu\|\leq 1$ such that $C=supp(\mu)$ is compact. By [Bou65, p.71, Corollary
2], there exists a net $(\mu_{\alpha})_{\alpha}$ in $co\\{\delta_{x}:x\in
C\\}$ such that

$\mu=\text{weak${}^{*}$-$\lim_{\alpha}$}\ \mu_{\alpha}.$

Now, since the net $(f\cdot\mu_{\alpha})_{\alpha}$ is contained in
$\overline{co}(O_{L}(f))$ and as $\overline{co}(O_{L}(f))$ is weakly compact,
by passing to a subnet if necessary, we can assume that
$f\cdot\mu_{\alpha}\rightarrow h$ for some $h\in C_{B}(H)$ in the weak
topology. Thus for each $t\in H$, we have

$\displaystyle h(t)=\langle\delta_{t},h\rangle$
$\displaystyle=\lim_{\alpha}\langle\delta_{t},f\cdot\mu_{\alpha}\rangle$
$\displaystyle=\lim_{\alpha}\langle\mu_{\alpha}\ast\delta_{t},f\rangle$
$\displaystyle=\langle\mu\ast\delta_{t},f\rangle$
$\displaystyle=\langle\delta_{t},f\cdot\mu\rangle.$

Hence, $f\cdot\mu=h\in\overline{co}(O_{L}(f))$. Now, let $\mu$ be an arbitrary
measure in the unit ball of $M(H)$. Since the set of compactly supported
measures is norm-dense in $M(H)$, there is a sequence
$\\{\mu_{n}\\}_{n\in\mathbb{N}}$ of compactly supported measures in $M(H)$
such that $\lim_{n}\|\mu_{n}-\mu\|=0$. Thus

$\|f\cdot\mu_{n}-f\cdot\mu\|_{\infty}\leq\|f\|_{\infty}\|\mu_{n}-\mu\|\rightarrow
0,$

Hence, $f\cdot\mu\in\overline{co}(O_{L}(f))$ and therefore,
$\mathbb{O}_{L}(f)\subseteq\overline{co}(O_{L}(f)),$ which implies that it is
relatively weakly compact. ∎

######  Proposition 4.7.

Let $H$ be a locally compact hypergroup with a Haar measure $\lambda$. Then
$WAP(L^{1}(H))=wap(H)$.

###### Proof.

Assume that $f\in WAP(L^{1}(H))$ and $(e_{\alpha})_{\alpha}$ be an approximate
identity of $L^{1}(H)$ of bound 1. Let $\mu$ be an arbitrary measure in the
unit ball of $M(H)$. Then, since $f\in WAP(L^{1}(H))$, After passing to a
subnet if necessary, we can assume that $f\cdot\mu\ast e_{\alpha}\rightarrow
g$ in the $\sigma(L^{\infty}(H),L^{\infty}(H)^{*})$-topology for some $g\in
C_{b}(H)$. Now, since $\sigma(L^{\infty}(H),L^{\infty}(H)^{*})$-topology on
$C_{b}(H)$ is stronger than $\sigma(C_{b}(H),M(H))$-topology, we have

$\lim_{\alpha}\langle f\cdot\mu\ast e_{\alpha},\nu\rangle=\lim_{\alpha}\langle
f,\mu\ast e_{\alpha}\ast\nu\rangle=\langle f\cdot\mu,\nu\rangle$

for all $\nu\in M(H)$, so $g=f\cdot\mu$. Thus $f\cdot\mu\ast
e_{\alpha}\rightarrow f\cdot\mu$ in the
$\sigma(L^{\infty}(H),L^{\infty}(H)^{*})$-topology. Hence,
$f\cdot\mu\in\mathcal{O}_{L}(f)$ and consequently
$\mathbb{O}_{L}(f)\subseteq\mathcal{O}_{L}(f)$ which implies that
$\mathbb{O}_{L}(f)$ is relatively weakly compact in $C_{b}(H)$. Therefore, by
Lemma 4.6, $f\in wap(H).$ Thus, $WAP(L^{1}(H))\subseteq wap(H).$ Now the other
way inclusion is a consequence of Lemma 4.6. ∎

As an immediate consequence of [Ska89, Proposition 4.2.8] we obtain the
following lemma.

######  Lemma 4.8.

Let $H$ be a locally compact hypergroup with a Haar measure. Then $B(H)\subset
wap(H)$.

Here is the main result of this section. It has been known since the work of
Arens [Are51] that there are two Banach algebra products on the bidual
$L^{1}(H)^{**}$, each extending the convolution on $L^{1}(H)$, and $L^{1}(H)$
is called Arens regular if those products coincide. According to [DL05,
Theorem 3.14], the Banach algebra $L^{1}(H)$ is Arens regular if and only if
$WAP(L^{1}(H))=L^{\infty}(H)$.

######  Theorem 4.9.

Let $H$ be an ultraspherical hypergroup. Then the following statements are
equivalent:

Every subset of $H$ belongs to ${\mathscr{R}}(H)$.

$L^{1}(H)$ is Arens regular.

$H$ is finite.

The Fourier-Stieltjes algebra $B(H)$ has power boundedness property.

###### Proof.

(i) $\Longrightarrow$ (ii). First note that, by hypothesis and Lemma 3.6,
${\bf 1}_{\\{\dot{e}\\}}\in B(H).$ This implies that $H$ is discrete and hence
${\mathscr{R}}(H)={\mathscr{R}}_{c}(H)$. Now, let $\dot{E}$ be any subset of
$H$. Applying hypothesis and Lemma 3.6 again, we get ${\bf 1}_{\dot{E}}\in
B(H)\subseteq WAP(L^{1}(H))$. Now density of simple functions in
$L^{\infty}(H)$ implies that $WAP(L^{1}(H))=L^{\infty}(H),$ which in turn
implies that $L^{1}(H)$ is Arens regular.

(ii) $\Longrightarrow$ (iii). This follows from [Ska89, Theorem 5.2.3].

(iii) $\Longrightarrow$ (i) is obvious.

(iii) $\Longrightarrow$ (iv). If $H$ is finite, then clearly $B(H)=A(H)$ and
hence, by Theorem 4.3, $B(H)$ has the power boundedness property.

(iv) $\Longrightarrow$ (iii). Suppose that $B(H)$ is power bounded. Then
$A(H)$ is also power bounded. Thus, by Theorem 4.3, $H$ is discrete. Towards a
contradiction, assume that $H$ has an infinite countable subhypergroup $K.$ As
$B(H)$ is power bounded, by Lemma 3.4, $B(K)$ also possesses the power
boundedness property. Note that, by Lemma 3.5, for any subset $\dot{E}$ of $K$
there exists $u\in B(K)$ with $\|u\|_{\infty}=1$ such that $F_{u}=\dot{E}$.
Now since $u$ is power bounded, by Corollary 3.8,
$\dot{E}\in{\mathscr{R}}_{c}(K)$. Thus $K$ is finite, a contradiction. Hence,
$H$ is finite. ∎

###  4.3. Power boundedness of $I(\dot{E})$

Our next result deals with the power boundedness of the ideal $k(\dot{B}).$
This is an analogue of [KLU11, Proposition 2.5].

######  Proposition 4.10.

Let $H$ be an ultraspherical hypergroup and $\dot{E}$ a closed subset of $H$.
If the ideal $k(\dot{E})$ has the power boundedness property, then every
compact subset of $H\setminus\dot{E}$ belongs to ${\mathscr{R}}_{c}(H)$.

###### Proof.

Let $\dot{C}$ be a compact subset of $H\setminus\dot{E}$. Then, by [Jew75,
Lemma 10.1 C], there exists an open $\sigma$-compact subhpergroup $K$ of $H$
which contains $\dot{C}$. Since $K\setminus\dot{C}$ is $\sigma$-compact, by
Lemma 3.5, there exists $u\in B(K)$ with $\|u\|_{\infty}=1$ such that
$\dot{F}_{u}=\dot{C}$. Now, use Lemma 3.4, to get $u_{0}\in B(H)$ that is $u$
on $K$ and vanishes outside $K$. Also, using regularity of $A(H)$, one can
find $v\in A(H)$ such that $0\leq v\leq 1,$ $v\equiv 1$ on $\dot{C}$ and
$\text{supp}(v)\cap\dot{E}=\emptyset.$ Let $u_{1}=vu_{0}$. Then $u_{1}\in
k(\dot{E}),\|u_{1}\|_{\infty}=1$ and $\dot{F}_{u_{1}}=\dot{C}$. By hypothesis,
$u_{1}$ is power bounded and hence, by Lemma 3.3,
$\dot{C}\in{\mathscr{R}}_{c}(H).$ ∎

We shall end this section with a result which is of independent interest. This
result is about the existence of bounded approximate identity in the ideals
$I(\dot{E}).$

######  Lemma 4.11.

Let $H$ be an ultraspherical hypergroup associated to an amenble locally
compact group $G$. If $\dot{E}\in{\mathscr{R}}_{c}(H)$, then

$\dot{E}$ is a set of synthesis for $A(H).$

$I(\dot{E})$ has a bounded approximate identity.

###### Proof.

(i). This follows from [DSKL14, Theorem 3.1].
(ii). Since $p^{-1}(\dot{E})\in{\mathscr{R}}_{c}(G)$, by [FKLS03, Lemma 2.2]
the ideal $I(p^{-1}(\dot{E}))$ has a bounded approximate identity. Let $v\in
I(p^{-1}(\dot{E}))$ and let $v_{0}$ be the element of $A(H)$ associated with
$\pi(v)$. Then for each $\dot{x}\in E$, we have

$v_{0}(\dot{x})=\pi(v)(x)=\int_{\mathcal{O}_{x}}v(z)d\pi^{*}(\delta_{x})(z),$

since $\mathcal{O}_{x}\subseteq p^{-1}(\dot{E})$. Thus, we can identify
$\pi(I(p^{-1}(\dot{E})))$ with $I(\dot{E})$. Suppose that
$(e_{\alpha})_{\alpha}$ is a bounded approximate identity of
$I(p^{-1}(\dot{E}))$ and let $(\dot{e}_{\alpha})_{\alpha}$ be its associated
net in $A(H),$ i.e., $\tilde{\dot{e}}_{\alpha}=\pi(e_{\alpha}).$ Then, for
each $w\in I(\dot{E})$, we have,

$\displaystyle\|w\dot{e}_{\alpha}-w\|_{A(H)}$ $\displaystyle=$
$\displaystyle\|\tilde{w}\tilde{\dot{e}}_{\alpha}-\tilde{w}\|_{A(G)}$
$\displaystyle=$ $\displaystyle\|\pi(\tilde{w}e_{\alpha}-\tilde{w})\|_{A(G)}$
$\displaystyle\leq$
$\displaystyle\|\tilde{w}e_{\alpha}-\tilde{w}\|_{A(G)}\rightarrow 0.$

This completes the proof. ∎

## Acknowledgement

The second author would like to thank the Science and Engineering Board,
India, for the MATRICS project fund with the Project No:MTR/2018/000849.

## References

  * [Are51] R. Arens, _The adjoint of a bilinear operation_ , Proc. Amer. Math. Soc. 2 (1951), 839–848.
  * [BH11] W. R Bloom and H. Heyer, _Harmonic analysis of probability measures on hypergroups_ , vol. 20, DeGruyter, 2011.
  * [Bou65] N. Bourbaki, _Éléments de mathématique. Fasc. XIII. Livre VI: Intégration. Chapitres 1, 2, 3 et 4: Inégalités de convexité, Espaces de Riesz, Mesures sur les espaces localement compacts, Prolongement d’une mesure, Espaces $L^{p}$_, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1175, Hermann, Paris, 1965, Deuxième édition revue et augmentée.
  * [DL05] H. G. Dales and A. T.-M Lau, _The second duals of beurling algebras_ , vol. 836, Mem. Amer. Math. Soc., 2005.
  * [DR92] E. Damek and F. Ricci, _Harmonic analysis on solvable extensions of h-type groups_ , J. Geoe. Anal. 2 (1992), 213–248.
  * [DSKL14] S. Degenfeld-Schonburg, E. Kaniuth, and R. Lasser, _Spectral synthesis in Fourier algebras of ultrapherical hypergroups_ , J. Fourier Anal. Appl. 20 (2014), no. 2, 258–281.
  * [Eym64] P. Eymard, _L’algèbre de Fourier d’un groupe localement compact_ , Bull. Soc. Math. France 9 (1964), 181–236.
  * [FKLS03] B. Forrest, E. Kaniuth, A. T.-M. Lau, and N. Spronk, _Ideals with bounded approximate identities in Fourier algebras_ , J. Funct. Anal. 203 (2003), 286–304.
  * [Gra76] E. E. Granirer, _A characterisation of discreteness for locally compact groups in terms of the banach algebras $a_{p}(g)$_, Proc. Amer. Math. Soc. 54 (1976), 189–192.
  * [Hos86] B. Host, _Le théorème des idempotents dans $B(G)$_, Bull. Soc. Math. France 114 (1986), 215–223.
  * [Jew75] R. I. Jewett, _Spaces with an abstract convolution of measures_ , Adv. Math. 18 (1975), 1–101.
  * [Kan09] E. Kaniuth, _A course in commutative Banach algebras_ , Graduate Texts in Mathematics, vol. 246, Springer, New York, 2009.
  * [KLU10] E. Kaniuth, A. T.-M. Lau, and A. Ülger, _Multipliers of commutative Banach algebras, power boundedness and Fourier–Stieltjes algebras_ , J. London Math. Soc. 81 (2010), 255–275.
  * [KLU11] by same author, _Power boundedness in Fourier and Fourier–Stieltjes algebras and other commutative Banach algebras_ , J. Funct. Anal. 260 (2011), 2366–2386.
  * [KLU14] by same author, _Power boundedness in Banach algebras associated with locally compact groups_ , Studia Math. 222 (2014), 165–189.
  * [KU13] E. Kaniuth and A. Ülger, _The structure of power bounded elements in Fourier–Stieltjes algebras of locally compact groups_ , Bull. Sci. Math. 137 (2013), 45–62.
  * [Meg12] R. E. Megginson, _An introduction to Banach space theory_ , vol. 183, Springer, 2012.
  * [Mur07] V. Muruganandam, _Fourier algebra of a hypergroup. I_ , J. Aust. Math. Soc. 82 (2007), 59–83.
  * [Mur08] by same author, _Fourier algebra of a hypergroup. II. Spherical hypergroups_ , Math. Nachr. 281 (2008), 1590–1603.
  * [Mus19] H. Mustafayev, _Mean ergodic theorems for multipliers on Banach algebras_ , J. Fourier. Anal. Appl. 25 (2019), 393–426.
  * [Sch70] B. Schreiber, _Measures with bounded convolution powers_ , Trans. Amer. Math. Soc. 151 (1970), 405–405.
  * [SK14] N. Shravan Kumar, _Invariant mean on a class of von Neumann algebras related to ultraspherical hypergroups_ , Studia. Math. 225 (2014), 235–247.
  * [SK17] by same author, _Invariant means on a class of von Neumann algebras related to ultraspherical hypergroups II_ , Canad. Math. Bull. 60 (2017), 402–410.
  * [Ska89] M. Skantharajah, _Amenable hypergroups_ , ProQuest LLC, Ann Arbor, MI, 1989, Thesis (Ph.D.)–University of Alberta (Canada).