Compact and weakly compact multipliers on Fourier algebras of ultraspherical hypergroups

TL;DR

This paper explores the properties of Fourier algebras on ultraspherical hypergroups, linking algebraic features like compact multipliers to the hypergroup's discreteness and analyzing Arens regularity of ideals.

Contribution

It extends known results about Fourier algebras from groups to ultraspherical hypergroups, providing new characterizations of hypergroup discreteness and studying ideal regularity.

Findings

Discreteness of hypergroups characterized by compact multipliers

Several algebraic conditions equivalent to hypergroup discreteness

Analysis of Arens regularity of closed ideals

Abstract

A locally compact group $ G $ is discrete if and only if the Fourier algebra $ A(G) $ has a non-zero (weakly) compact multiplier. We partially extend this result to the setting of ultraspherical hypergroups. Let $H$ be an ultraspherical hypergroup and let $A(H)$ denote the corresponding Fourier algebra. We will give several characterizations of discreteness of $ H $ in the terms of the algebraic properties of $A(H)$. We also study Arens regularity of closed ideals of $ A(H)$.

Equations

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

$$⟨\Phi \cdot u,v⟩$$

$$⟨m\cdot \Phi ,u⟩$$

$$⟨m\square n,\Phi ⟩$$

$$\mathcal{Z}({\mathcal{A}}^{\ast \ast },\square )=\{$$

$${\text{is weak}}^{\ast }{\text{-weak}}^{\ast }\text{continuous on }{\mathcal{A}}^{\ast \ast }\}.$$

$$G(H):=\{\dot{x}\in H;{\delta }_{\dot{x}}\ast {\delta }_{\check{\dot{x}}}={\delta }_{\dot{e}}={\delta }_{\check{\dot{x}}}\ast {\delta }_{\dot{x}}\}.$$

$$\Phi (vw)u=(vw)u=(vu)(wu)=\Phi (v)\Phi (w)u,$$

$$v(\dot{x})u(\dot{t})=vu(\dot{t})=v(\dot{t})u(\dot{t}),$$

$$Soc(A(H{)}^{\ast \ast })=Soc(A(H))=A(H)=A(H{)}^{\ast \ast }.$$

$$⟨u\cdot {\Phi }_{\alpha },m⟩=⟨u\cdot {\Phi }_{\alpha },\phi ⟩$$

$$⟶⟨\Phi ,u\phi ⟩$$

$$⟨{\rho }_u^{\ast }(\lambda (\dot{a})),v⟩=⟨\lambda (\dot{a}),{\rho }_u(v)⟩=u(\dot{a})⟨\lambda (\dot{a}),v⟩.$$

$${\rho }_{T(m)}(v)=vT(m)=v\square T(m)=T(v\square m)=T\circ {\rho }_m(v).$$

$$⟨u\cdot \Phi ,v⟩=⟨\Phi ,uv⟩=0.$$

$$⟨m,\Phi ⟩=u(\dot{e})⟨m,\Phi ⟩=⟨m,u\cdot \Phi ⟩=0.$$

$$⟨m_2\square \Phi ,v⟩=v(\dot{e})⟨m_2,\Phi ⟩=⟨m_2,\Phi ⟩⟨\lambda (\dot{e}),v⟩(v\in A(H)).$$

$$⟨m_1\square m_2,\Phi ⟩=⟨m_1,m_2\square \Phi ⟩=⟨m_1,\lambda (\dot{e})⟩⟨m_2\square \Phi ⟩=⟨m_2,\Phi ⟩.$$

$$m_2=m_1\square m_2=m_2\square m_1=m_1.$$

$$\iota (I)\circ I^{\ast \ast }=\iota (I\cdot I)\circ I^{\perp \perp }\subseteq \iota (I)\circ (\iota (I)\circ A(H{)}^{\ast \ast })\subseteq I\cdot A(H)\subseteq I.$$

Full text

1. Introduction

Let $G$ be a locally compact group. We let $A(G)$ and $VN(G)$ denote the Fourier and group von Neumann algebras of $G$ which is introduced by Eymard [6]. It is known from [11] by Lau that the Fourier algebra $A(G)$ has a non-zero (weakly) compact left multiplier if and only if $G$ is discrete. He also proved that for a discrete and amenable group $G$, $A(G)$ is precisely the algebra of all weakly compact multipliers on $A(G)$. In addition, Ghahramani and Lau in [9] showed that $G$ is discrete if and only if the second dual of $A(G)$ equipped with the first Arens product has a (weakly) compact left multiplier $T$ such that $\langle T(n),\lambda(e)\rangle\neq 0$ for some $n\in A(G)^{**}$.

In recent years, some classes of hypergroups whose Fourier space forms a Banach algebra under pointwise multiplication have been discovered [1, 14, 15, 18]. Most notably, this concerns the class of ultraspherical hypergroups, which includes in particular all double coset hypergroups and hence all orbit hypergroups. Since ultraspherical hypergroups are generalized versions of locally compact groups, it is natural to ask whether the above results hold in the ultraspherical hypergroup setting. One of the purpose of this paper is to provide an affirmative answer to this question.

In Section 2, after recalling some background notations and definitions, we discuss some basic properties of the Fourier algebra of ultraspherical hypergroups that we shall need in establishing our main results.

In section 3, we will give some characterizations for discreteness of the ultraspherical hypergroup $H$ in terms of the existence of minimal idempotents in $A(H)$.

In section 4, we study (weakly) compact multipliers of $A(H)$. As an application of this result, we show that $A(H)$ is an ideal in $VN(H)^{*}$ if an only if $H$ is discrete. We also prove that $H$ is discrete if and only if there is a weakly compact right (equivalently, left) multiplier $T$ of $VN(H)^{*}$ and $m\in VN(H)^{*}$ such that $T(m)\in A(H)$ and $\langle T(m),\lambda(\dot{e})\rangle\neq 0$.

In the final section we study Arens regularity of closed ideals of $A(H)$. Furthermore, we give a characterization of $H$ to be finite in terms of weakly compact multipliers of $A(H)$.

2. Preliminaries

Let $H$ be an ultraspherical hypergroup associated to a locally compact group $G$ and a spherical projector $\pi:C_{c}(G)\rightarrow C_{c}(G)$ which was introduced and studied in [15]. Let $A(H)$ denote the Fourier algebra corresponding to the hypergroup $H$. A left Haar measure on $H$ is given by $\int_{H}f(\dot{x})d\dot{x}=\int_{G}f(p(x))dx$, $f\in C_{c}(H)$, where $p:G\rightarrow H$ is the quotient map. Recall that the Fourier algebra $A(H)$ is semisimple, regular and Tauberian[15, Theorem 3.13]. As in the group case, let $\lambda$ also denote the left regular representation of $H$ on $L^{2}(H)$ given by

[TABLE]

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 $C_{\lambda}^{*}(H)$.

A bounded linear operator on Banach algebra ${\mathcal{A}}$ is called a right (resp. left) multiplier if it satisfies $T(ab)=aT(b)$ (resp. $T(ab)=T(a)b$) for all $a,b\in{\mathcal{A}}$. We denote by $RM({\mathcal{A}})$ (resp. $LM({\mathcal{A}})$) the space of all right (resp. left) multipliers for ${\mathcal{A}}$. Clearly $RM({\mathcal{A}})$ and $LM({\mathcal{A}})$ are Banach algebras as subalgebras of $B({\mathcal{A}})$, the space of all bounded linear operator on ${\mathcal{A}}$. For any $a\in\mathcal{A}$, let $\rho_{a}:\mathcal{A}\longrightarrow\mathcal{A}$ (resp. $\ell_{a}:\mathcal{A}\longrightarrow\mathcal{A}$) be the multiplication map defined by $\rho_{a}(b)=ba$ (resp. $\ell_{a}(b)=ab$) for all $b\in\mathcal{A}$. Then $\rho_{a}\in RM(\mathcal{A})$ and $\ell_{a}\in LM(\mathcal{A})$. If $\mathcal{A}$ is commutative, then $RM({\mathcal{A}})=LM({\mathcal{A}})$ and we denote it by $M({\mathcal{A}})$. An element $a\in\mathcal{A}$ is called (weakly) completely continuous if $\rho_{a}$ is a (weakly) compact operator on $\mathcal{A}$. For the general theory of multipliers we refer to Larsen [10].

Let $\mathcal{A}$ be a commutative Banach algebra. The Arens products on $\mathcal{A}^{**}$ is defined as following three steps. For $u,v$ in $\mathcal{A}$, $\Phi$ in $\mathcal{A}^{*}$ and $m,n\in\mathcal{A}^{**},$ we define $\Phi\cdot u,u\cdot\Phi$, $m\cdot\Phi,\Phi\cdot m\in\mathcal{A}^{*}$ and $m\square n,m\diamondsuit n\in\mathcal{A}^{**}$ as follows:

[TABLE]

$\mathcal{A}$ is said to be Arens regular if $\square$ and $\diamondsuit$ coincide on $\mathcal{A}^{**}$. For any $m\in\mathcal{A}^{**}$ the mapping $n\mapsto m\square n$ is weak∗-weak∗ continuous on $\mathcal{A}^{**}$. However, the mapping $n\mapsto n\square m$ need not to be weak∗-weak∗-continuous. The left topological center of $\mathcal{A}^{**}$ is defined as

[TABLE]

A linear functional $m\in VN(H)^{*}$ is called a topologically invariant mean on $VN(H)$ if $\lVert m\rVert=\langle m,\lambda({\dot{e}})\rangle=1$ and $\langle m,u\cdot\Phi\rangle=u(\dot{e})\langle m,\Phi\rangle$ for every $\Phi\in VN(H)$, $u\in A(H).$ We denote by $TIM(\widehat{H})$ the set of all topologically invariant means on $VN(H)$. It has been shown by Kumar [16] that $VN(H)$ always admits a topologically invariant mean.

3. $Soc(A(H))$ and discreteness of $H$

In this section, we will give some characterizations of discreteness of an ultraspherical hypergroup $H$ in the terms of the algebraic properties of $A(H)$. Recall that in a commutative Banach algebra $\mathcal{A}$ a non-zero element $e$ satisfying $e^{2}=e$ and $e\mathcal{A}=\mathbb{C}e$ is called minimal idempotent. If $\mathcal{A}$ has minimal ideals, the smallest ideal containing all of them is called Socle of $\mathcal{A}$ and is denoted by Soc$(\mathcal{A})$. If $\mathcal{A}$ does not have minimal ideals, we define Soc$(\mathcal{A})=\{0\}$.

Let $H$ be a hypergroup. Then the set

[TABLE]

is a locally compact group and is called maximum subgroup of $H$. In what follows, $H$ will always be an ultraspherical hypergroup associated to a locally compact group $G$ and a spherical projector $\pi:C_{c}(G)\rightarrow C_{c}(G)$.

Proposition 3.1**.**

$H$* is discrete if and only if there exists a minimal idempotent $u\in A(H)$ such that $u(\dot{a})\neq 0$ for some $\dot{a}\in G(H)$. In this case $u={\bf 1}_{\dot{a}}$, where ${\bf 1}_{\dot{a}}$ denote the characteristic function at $\{\dot{a}\}$ on $H$.*

Proof.

If $H$ is discrete, then it suffices to take $u={\bf 1}_{\dot{e}}.$ Conversely, let $u$ be a minimal idempotent in $A(H)$ $u\in A(H)$ such that $u(\dot{a})\neq 0$ for some $\dot{a}\in G(H)$. Then by [3, Proposition IV.31.3] and using the commutativity of $A(H)$, there is $\Phi\in A(H)^{*}$ such that $vu=\Phi(v)u$ for all $v\in A(H).$ Now, we show that $\Phi=\Phi_{\dot{x}}$ for some $\dot{x}\in H$, where $\Phi_{\dot{x}}(v)=v(\dot{x})$ for all $v\in A(H)$. To prove this, note that for each $v,w\in A(H)$, we have

[TABLE]

which implies that $\Phi$ is a multiplicative functional. Hence by [15, Theorem 3.13] there is $\dot{x}\in H$ such that $\Phi=\Phi_{\dot{x}}$. It follows that $vu=v(\dot{x})u$ for all $v\in A(H)$ and in particular $u(\dot{x})=1$. Now, we show that $u={\bf 1}_{\dot{x}}.$ Let $\dot{t}\in H-\{\dot{x}\}$ and choose $v\in A(H)$ such that $v(\dot{x})\neq v(\dot{t}).$ Then we have

[TABLE]

which implies that $u(\dot{t})=0.$ Since $u(\dot{a})\neq 0,$ it follows that $\dot{x}=\dot{a}$. Therefore, $u={\bf 1}_{\dot{a}}$ and $H$ must be discrete. ∎

For an ideal $I$ of $A(H)$, we denote by $Z(I)$ the set of all $\dot{x}\in H$ such that $v(\dot{x})=0$ for all $v\in I$.

Corollary 3.2**.**

There is a minimal ideal $\mathfrak{m}$ in $A(H)$ with $G(H)\nsubseteq Z(\mathfrak{m})$ if and only if $H$ is discrete and $\mathfrak{m}=\mathbb{C}{\bf 1}_{\dot{a}}$ for some $\dot{a}\in G(H).$

Proof.

Let $\mathfrak{m}$ be a minimal ideal in $A(H)$ such that $G(H)\nsubseteq Z(\mathfrak{m})$. Then by [3, Proposition IV.30.6] there is a minimal idempotent $u$ in $A(H)$ such that $\mathfrak{m}=uA(H)$. Since $G(H)\nsubseteq Z(\mathfrak{m})$, it follows from Proposition 3.1 that $H$ is discrete and $u={\bf 1}_{\dot{a}}$ for some $\dot{a}\in G(H)$. Thus, $\mathfrak{m}=\mathbb{C}{\bf 1}_{\dot{a}}$. Conversely, if $H$ is discrete, then $\mathfrak{m}=\mathbb{C}{\bf 1}_{\dot{e}}$ is the desired minimal ideal in $A(H).$ ∎

Proposition 3.3**.**

The following conditions are equivalent.

(i)* $H$ is discrete.*

(ii)* There is $u\in Soc(A(H))$ such that $u(G(H))\neq\{0\}$.*

(iii)* There is $u\in Soc(B_{\lambda}(H))$ such that $u(G(H))\neq\{0\}$.*

Proof.

(i)$\Leftrightarrow$(ii). By Corollary 3.2, discreteness of $H$ is equivalent to the existence of a minimal ideal $\mathfrak{m}$ in $A(H)$ with $G(H)\nsubseteq Z(\mathfrak{m})$, which is equivalent to the existence of $u\in Soc(A(H))$ with $u(G(H))\neq\{0\}$.

(ii)$\Leftrightarrow$(iii). Since $A(H)$ is an ideal in $B_{\lambda}(H)$ and separates the points of $H$, it follows from [8, Proposition 1] that $Soc(A(H))=Soc(B_{\lambda}(H))$. ∎

Proposition 3.4**.**

Suppose that $A(H)$ admits a bounded approximate identity. Then $H$ is finite if and only if $Soc(A(H)^{**})=A(H)^{**}$.

Proof.

If $H$ is finite, then $A(H)$ is finite dimensional. Therefore,

[TABLE]

The converse is an immediate consequence of [8, Theorem 1]. ∎

4. Weakly compact multipliers of $A(H)$

In the group setting Lau in [11] proved that a locally compact group $G$ is discrete if and only if $A(G)$ admits (weakly) completely continuous elements. Among other things, we extend this result to the ultraspherical hypergroup setting. The proof of the following lemma is an adaptation of the proof given in [12, Lemma 4.7].

Lemma 4.1**.**

Let $H$ be a non-discrete ultraspherical hypergroup. Then $\lambda(\dot{x})\notin C^{*}_{\lambda}(H)$ for all $\dot{x}\in G(H).$

Proof.

Suppose, in contrary, that $\lambda(\dot{x})\in C^{*}_{\lambda}(H)$ for some $\dot{x}\in G(H)$. Then $\lambda({\check{\dot{x}}})\notin C^{*}_{\lambda}(H)$ by [5, Corollary 4.4]. Therefore, by Hahn-Banach theorem there is $m\in VN(H)^{*}$ such that $\langle m,\Psi\rangle=0$ for all $\Psi\in C^{*}_{\lambda}(H)$ and $\langle m,\lambda(\check{\dot{x}})\rangle\neq 0.$ Let $n\in VN(H)^{*}$ be defined by $\langle n,\Phi\rangle=\overline{\langle m,\Phi^{*}\rangle}$ for all $\Phi\in VN(H)$. Then $\langle n,\lambda(\dot{x})\rangle=\overline{\langle m,\lambda(\check{\dot{x}})\rangle}\neq 0$ and $\langle n,\Psi\rangle=0$ for all $\Psi\in C^{*}_{\lambda}(H)$, which is impossible. ∎

Proposition 4.2**.**

Let $u\in A(H).$ Then $\rho_{u}$ is weakly compact if and only if $\rho_{u}^{*}(VN(H))\subseteq C^{*}_{\lambda}(H).$

Proof.

If $\rho_{u}$ is weakly compact, then $\rho_{u}^{*}$ is weak∗-weak continuous. Suppose that $\Phi\in\rho_{u}^{*}(VN(H))$. Since $C^{*}_{\lambda}(H)$ is weak∗ dense in $VN(H),$ there is a net $(\Psi_{\alpha})$ in $C^{*}_{\lambda}(H)$ such that weak-$\lim\limits_{\alpha}\rho_{u}^{*}(\Psi_{\alpha})=\rho_{u}^{*}(\Phi).$ Hence, $\rho_{u}^{*}(VN(H))\subseteq\overline{\rho_{u}^{*}(C^{*}_{\lambda}(H))}^{w}\subseteq C^{*}_{\lambda}(H)$.

Conversely, let $\rho_{u}^{*}(VN(H))\subseteq C^{*}_{\lambda}(H)$. Suppose that $(\Phi_{\alpha})$ is a net in $VN(H)$ such that weak∗-$\lim\limits_{\alpha}\Phi_{\alpha}=\Phi$ for some $\Phi\in VN(H).$ Now, let $m\in VN(H)^{*}$ and $\varphi\in B_{\lambda}(H)$ be the restriction of $m$ on $C^{*}_{\lambda}(H)$. Using the fact that $A(H)$ is an ideal in $B_{\lambda}(H)$, we have

[TABLE]

Hence, $\rho_{u}^{*}$ is weak∗-weak continuous. It follows from [13, Theorem 3.5.14] that $\rho_{u}$ is weakly compact. ∎

Theorem 4.3**.**

The following conditions are equivalent.

$(\rm{i})$* $H$ is discrete.*

$(\rm{ii})$* $\rho_{u}$ is compact for every $u\in A(H).$*

$(\rm{iii})$* There exists $u\in A(H)$ such that $\rho_{u}$ is weakly compact and $u(\dot{a})\neq 0$ for*

some $\dot{a}\in G(H)$.

Proof.

$(i)\Rightarrow(ii)$. If $H$ is discrete and $u={\bf{1}}_{\dot{a}}$ for some ${\dot{a}}\in H$, then $\rho_{u}(A(H))=\{\lambda{\bf{1}}_{\dot{a}}:\lambda\in\mathbb{C}\}.$ This implies that $\rho_{u}$ is compact. Therefore, $\rho_{u}$ is compact for every $u\in A(H)$ with finite support. Now, since the set of all $u\in A(H)$ such that $u$ has finite support, is dense in $A(H)$, a simple approximation argument gives that $\rho_{u}$ is compact for all $u\in A(H).$

$(ii)\Rightarrow(iii)$. This is obvious.

$(iii)\Rightarrow(i)$. Let $u\in A(H)$ be such that $u(\dot{a})\neq 0$ for some $\dot{a}\in G(H)$ and $\rho_{u}$ is weakly compact. Then for each $v\in A(H),$ we have

[TABLE]

Since $u(\dot{a})\neq 0$, it follows that $\lambda(\dot{a})\in\rho_{u}^{*}(VN(H))\subseteq C^{*}_{\lambda}(H).$ Hence, $H$ is discrete by Lemma 4.1. ∎

It is known that a Banach algebra $\mathcal{A}$ is an ideal in $\mathcal{A}^{**}$ if and only if multiplication operators in ${\mathcal{A}}$ are weakly compact; see [4, page 248]. The next result generalizes [12, Theorem 3.7].

Corollary 4.4**.**

$H$* is discrete if and only if $A(H)$ is an ideal in $VN(H)^{*}$.*

Let $H$ be a locally compact group. Since $G(H)=H$, the following result is an immediate consequence of Theorem 4.3.

Corollary 4.5**.**

A locally compact group $G$ is discrete if and only if there is a non-zero $u\in A(G)$ such that $\rho_{u}$ is weakly compact.

Since $G(H)$ always possesses $\dot{e}$, we have the following corollary.

Corollary 4.6**.**

$H$* is discrete if and only if there is $u\in A(H)$ with $u(\dot{e})\neq 0$ such that $\rho_{u}$ is weakly compact.*

Corollary 4.7**.**

The following conditions are equivalent.

(i)* $H$ is discrete.*

(ii)* $A(H)$ has a 1-dimensional ideal $I$ such that $G(H)\nsubseteq Z(I)$.*

(iii)* $A(H)$ has a finite dimensional ideal $I$ such that $G(H)\nsubseteq Z(I)$.*

Proof.

(i)$\Rightarrow$(ii). Suppose that $H$ is discrete. Then for each $\dot{a}\in G(H)$ the set $\mathbb{C}{\bf{1}}_{\dot{a}}$ is a non-zero 1-dimensional ideal in $A(H).$

(ii)$\Rightarrow$(iii). This is clear.

(iii)$\Rightarrow$(i). Let $I$ be a finite dimensional ideal in $A(H)$ with $G(H)\nsubseteq Z(I)$. Then there is $u\in I$ such that $u(G(H))\neq\{0\}$ and $\rho_{u}$ has finite rank. Therefore, Theorem 4.3 implies that $H$ is discrete. ∎

Corollary 4.8**.**

$H$* is discrete if and only if there exists $u\in B_{\lambda}(H)$ such that $\rho_{u}$ is weakly compact on $B_{\lambda}(H)$ and $u(\dot{a})\neq 0$ for some $\dot{a}\in G(H)$.*

Proof.

Let $v\in A(H)$ be such that $v(\dot{a})\neq 0$. Then $\rho_{vu}$ is weakly compact on $A(H)$. Hence, $H$ is discrete by Theorem 4.3. For the converse, choose $u={\bf 1}_{\dot{e}}$. ∎

Proposition 4.9**.**

Let $H$ be an ultraspherical hypergroup on amenable locally compact group $G$ and let $u\in B_{\lambda}(H)$. If $\rho_{u}$ is weakly compact, then $u\in A(H)$.

Proof.

Define the map $\widetilde{\rho}_{u}:C^{*}_{\lambda}(H)\longrightarrow C^{*}_{\lambda}(H)$ by $\widetilde{\rho}_{u}(\Psi)=u\cdot\Psi$ for all $\Psi\in C^{*}_{\lambda}(H)$. It is now easily verified that $\widetilde{\rho}_{u}^{*}=\rho_{u}$. Hence, $\rho_{u}$ is weak∗-weak continuous. Using arguments similar to those in the proof of Proposition 4.2, replacing $C^{*}_{\lambda}(H)$ by $A(H)$ and $VN(H)$ by $B_{\lambda}(H)$, one can show that the range of $\rho_{u}$ is contained in $A(H)$. Now, since $G$ is amenable, it follows from [5, Theorem 3.4] that $1\in B_{\lambda}(H)$. Therefore, $u=\rho_{u}(1)\in A(H)$. ∎

Proposition 4.10**.**

$H$* is discrete if and only if there is a weakly compact right multiplier $T:VN(H)^{*}\longrightarrow VN(H)^{*}$ and $m\in VN(H)^{*}$ such that $T(m)\in A(H)$ and $\langle T(m),\lambda(\dot{e})\rangle\neq 0$.*

Proof.

Suppose that $H$ is discrete and consider $u={\bf 1}_{\dot{e}}\in A(H)$. Then the map $\tilde{\rho}_{u}:VN(H)^{*}\longrightarrow VN(H)^{*}$ defined by $\tilde{\rho}_{u}(m)=m\square u,$ is a weakly compact right multiplier of $VN(H)^{*}$ with the desired properties.

Conversely, first note that for each $v\in A(H)$, we have

[TABLE]

Using this and the fact that the restriction of $T$ on $A(H)$ is weakly compact, we conclude that $\rho_{T(m)}=T\circ\rho_{m}$ is weakly compact on $A(H)$. Now, since $\langle T(m),\lambda(\dot{e})\rangle\neq 0$, $H$ must be discrete by Theorem 4.3. ∎

Remark 4.11*.*

We note that the condition $T(m)\in A(H)$ can not be removed in Proposition 4.10. In fact, let $m$ be a topologically invariant mean on $VN(H)$. Then, the map $T:VN(H)^{*}\longrightarrow VN(H)^{*}$ defined by $T(n)=n\square m=\langle n,\lambda(\dot{e})\rangle m$ is a rank one right multiplier of $VN(H)^{*}$, and hence is weakly compact.

Proposition 4.12**.**

Let $H$ be an ultraspherical hypergroup on amenable locally compact group $G$ and let $\Lambda:VN(H)\longrightarrow C^{*}_{\lambda}(H)$ be a $weak^{*}$-$weak^{*}$ continuous $A(H)$-module homomorphism. Then $\Lambda=\rho^{*}_{u}$ for some $u\in A(H)$.

Proof.

Since $\Lambda$ is weak∗-weak∗ continuous, by [13, Theorem 3.1.11], there is a bounded linear operator $S:A(H)\longrightarrow A(H)$ such that $S^{*}=\Lambda$. It is easy to see that $S$ is a multiplier of $A(H)$. As $G$ is amenable, we obtain from [5, Theorem 3.4] that $S=\rho_{u}$ for some $u\in B_{\lambda}(H)$. Now, since $\rho_{u}^{*}(VN(H))=\Lambda(VN(H))\subseteq C^{*}_{\lambda}(H)$, using arguments similar to those in the proof of Proposition 4.2, one can see that $\rho_{u}$ is weakly compact. Therefore, Proposition 4.9 implies that $u\in A(H)$. ∎

Proposition 4.13**.**

Let $H$ be a discrete ultraspherical hypergroup on amenable locally compact group $G$ and let $\Lambda:VN(H)\longrightarrow C^{*}_{\lambda}(H)$ be a bounded $A(H)$-module homomorphism. Then $\Lambda=\rho^{*}_{u}$ for some $u\in A(H)$.

Proof.

Suppose that $H$ is discrete. Then by Proposition 4.3, the multiplication in $A(H)$ is weakly compact. Since $A(H)$ is a weakly sequentially complete Banach algebra which admits a bounded approximate identity, it follows from [2, Theorem 3.1] that $\Lambda=\rho^{*}_{u}$ for some $u\in A(H)$. ∎

Next, we show that $\mathcal{Z}(A(H)^{**},\square)$ is invariant under the weak∗-weak∗ continuous elements of $LM(A(H)^{**},\square)$. For any $m\in A(H)^{**}$, let $L_{m}:A(H)^{**}\longrightarrow A(H)^{**}$ be the left multiplication operator defined by $L_{m}(n)=m\square n$.

Proposition 4.14**.**

Let $T$ be a weak∗-weak∗ continuous element of $LM(A(H)^{**},\square)$. Then $T(\mathcal{Z}(A(H)^{**},\square))\subseteq\mathcal{Z}(A(H)^{**},\square)$.

Proof.

Let $m\in\mathcal{Z}(A(H)^{**},\square)$. Then $L_{m}$ is weak∗-weak∗ continuous. It is easy to see that $L_{T(m)}=T\circ L_{m}$, hence the mapping $L_{T(m)}$ is weak∗-weak∗ continuous. Therefore, $T(m)\in\mathcal{Z}(A(H)^{**},\square)$. ∎

Corollary 4.15**.**

Let $T$ be a weak∗-weak∗-continuous weakly compact element of $LM(A(H)^{**},\square)$. Then $T(A(H)^{**},\square)\subseteq\mathcal{Z}(A(H)^{**},\square)$.

Proof.

Since $T$ is weak∗-weak∗ continuous and weakly compact, it follows from [13, Theorem 3.1.11, Theorem 3.5.14] that $T$ is weak∗-weak continuous. Using the weak∗ density of $\mathcal{Z}(A(H)^{**},\square)$ in $A(H)^{**}$, we conclude that $T(A(H)^{**})\subseteq T(\mathcal{Z}(A(H)^{**},\square)^{-w}$. Hence, by Proposition 4.14 and Mazur’s theorem, $T(A(H)^{**},\square)\subseteq\mathcal{Z}(A(H)^{**},\square)$. ∎

5. Arens regularity of closed ideals

In this section, we study Arens regularity of the closed ideals of $A(H)$. We obtain results similar to those of Forrest [7] obtained in the group setting.

Proposition 5.1**.**

Let $I$ be a closed ideal in $A(H)$ such that $\dot{e}\notin Z(I)$. If $I$ is Arens regular, then $VN(H)$ has a unique topologically invariant mean. In particular, $H$ is discrete.

Proof.

Let $m\in TIM(\widehat{H})$ and let $\Phi\in I^{\perp}.$ Take $u\in I$ with $u(\dot{e})=1.$ Then for each $v\in A(H)$, we have

[TABLE]

Therefore, $u\cdot\Phi=0,$ and then

[TABLE]

Hence, $m\in I^{\perp\perp}$. Identifying $I^{\perp\perp}$ with $I^{**}$, we get that $TIM(\widehat{H})\subseteq I^{**}$. Suppose now that $m_{1},m_{2}\in TIM(\widehat{H}).$ Then

[TABLE]

Hence, for each $\Phi\in VN(H),$ we have

[TABLE]

This shows that $m_{1}\square m_{2}=m_{2}.$ Finally, since $I$ is Arens regular, $I^{**}$ is commutative and hence,

[TABLE]

The last statement follows from [17, Theorem 1.7]. ∎

Proposition 5.2**.**

Let $I$ be a closed ideal in $A(H)$ with $\dot{e}\notin Z(I)$. If $I$ has a bounded approximate identity, then $I$ is Arens regular if and only if $I$ is reflexive.

Proof.

If $I$ is reflexive, then $I$ is obviously Arens regular.

For the converse, suppose that $I$ is Arens regular. Then, it follows from Proposition 5.1 that $H$ is discrete. Since $I$ has a bounded approximate identity, by Cohen’s factorization theorem, $I=I^{2}.$ Let $\iota:A(H)\longrightarrow A(H)^{**}$ be the natural embedding. Then, we have

[TABLE]

Therefore, $I$ is an ideal in $I^{**}$. On the other hand, since $A(H)$ is weakly sequentially complete, $I$ is also weakly sequentially complete. It follows from [4, Theorem 2.9.39] that $I$ is unital. The inclusion $I\subseteq C_{0}(H)$ and discreteness of $H$ implies that $I$ is finite dimensional and hence $I$ is reflexive. ∎

In the following result, we denote the set of all weakly compact multipliers of $A(H)$ by $M_{wc}(A(H))$.

Corollary 5.3**.**

Let $H$ be an ultraspherical hypergroup on amenable locally compact group $G$. Then $M_{wc}(A(H))=B_{\lambda}(H)$ if and only if $H$ is finite.

Proof.

Suppose that $M_{wc}(A(H))=B_{\lambda}(H)$. Since $A(H)\subseteq B_{\lambda}(H)$, we obtain from Theorem 4.3 that $H$ is discrete. On the other hand, $M(A(H))=B_{\lambda}(H)$ by [5, Theorem 3.4], which implies that $1\in B_{\lambda}(H)$ and hence the set $\{u\cdot 1:\lVert u\rVert_{A(H)}\leqslant 1\}$ in $A(H)$ is weakly compact. This means that $A(H)$ is reflexive. Therefore, $A(H)$ is Arens regular by Proposition 5.2. This and [4, Theorem 2.9.39] imply that $A(H)$ is unital. Finally, since $A(H)\subseteq C_{0}(H)$ separates the points of $H$, we conclude that $H$ is finite. The converse is trivial. ∎

We end this paper with the following questions.

1-If $T:VN(H)^{*}\longrightarrow VN(H)^{*}$ is a weakly compact left multiplier such that $\langle T(n),\lambda(\dot{e})\rangle\neq 0$ for some $n\in VN(H)^{*}$, must $H$ be discrete? In the group setting, Ghahramani and Lau in [9, Theorem 4.3] have given a positive answer to this question.

2-If there is a non-zero $u\in A(H)$ such that $\rho_{u}$ is (weakly) compact, must $H$ be discrete?