Amenability and Orlicz Figa-Talamanca Herz Algebras

TL;DR

This paper explores the relationship between the amenability of locally compact groups and the properties of their associated Orlicz Figa-Talamanca Herz algebras, providing new characterizations.

Contribution

It offers a novel characterization of group amenability using the properties of Orlicz Figa-Talamanca Herz algebras, linking harmonic analysis and algebraic properties.

Findings

Characterization of amenability via algebra properties

New criteria for group amenability

Connections between harmonic analysis and algebraic structures

Abstract

In this paper, we characterize the amenablity of locally compact groups in terms of the properties of the Orlicz Figa-Talamanca Herz algebras.

Equations

$$u=\underset{n=1}{\sum ^{\infty }}{f}_n\ast \check{g_n},$$

$$\underset{n=1}{\sum ^{\infty }}{N}_{\Phi }(f_n)\Vert g_n{\Vert }_{\psi }<\infty .$$

$$\Psi (y):=\sup \{x|y|-\Phi (x):x\ge 0\},y\in \mathbb{R}.$$

$$L^{\Phi }(G)=\left\{f:G\to \mathbb{C}:\text{mbox}fismeasurableand{\int }_G\Phi (\beta |f|)dx<\infty \text{ for some}\beta >0\right\}$$

$$N_{\Phi }(f)=\inf \left\{k>0:{\int }_G\Phi \left(\frac{|f|}{k}\right)dx\le 1\right\}.$$

$$\|f\|_{\Phi}=\sup\Bigg{\{}\int_{G}|fg|dx:\int_{G}\Psi(|g|)dx\leq 1\Bigg{\}}.$$

$${\Phi }^{\prime }((1+ϵ)x)\ge c_{ϵ}{\Phi }^{\prime }(x),x\ge x_1(ϵ).$$

$$A_{\Phi }(G)=\left\{u=\underset{n=1}{\sum ^{\infty }}{f}_n\ast \check{g_n}:\{f_n\}\subset L^{\Phi }(G),\{g_n\}\in L^{\Psi }(G)\text{mbox}and\underset{n=1}{\sum ^{\infty }}{N}_{\Phi }(f_n)\Vert g_n{\Vert }_{\Psi }<\infty \right\}.$$

$$\Vert u{\Vert }_{A_{\Phi }}:=\inf \left\{\underset{n=1}{\sum ^{\infty }}{N}_{\Phi }(f_n)\Vert g_n{\Vert }_{\Psi }:u=\underset{n=1}{\sum ^{\infty }}{f}_n\ast \check{g_n}\right\}.$$

$$Z(I)=\{x\in E:\hat{a}(x)=0\forall a\in I\}.$$

$$j_{\mathcal{A}}(E)$$

$$J_{\mathcal{A}}(E)$$

$$I_{\mathcal{A}}(E)$$

$$\Vert u_{K,ϵ}{\Vert }_{A_{\Phi }}\le$$

$$\le$$

$$<$$

$$\Vert u_{K,ϵ}f-f{\Vert }_{A_{\Phi }}=\frac{ϵ}{1+ϵ}\Vert f{\Vert }_{A_{\Phi }}\le ϵ\Vert f{\Vert }_{A_{\Phi }}.$$

$$N_{\Phi }(f_0\ast f)\le \Vert f_0{\Vert }_1{N}_{\Phi }(f)\le 1.$$

$$⟨f^{\prime }\ast (f_0\ast f\check{)},\mu ⟩=⟨f^{\prime }\ast \check{f},\mu \ast f_0⟩.$$

$$\Vert L_{\mu }{\Vert }_{C{V}_{\Psi }(G)}$$

$$\Vert u_{\alpha }u{\Vert }_{A_{\Phi }(G)}\le \Vert u_{\alpha }{\Vert }_{A_{\Phi }(G)}\Vert u{\Vert }_{B_{\Phi }(G)}\le 2\Vert u{\Vert }_{B_{\Phi }(G)}.$$

$$\mathcal{S}(A,B)$$

$$s(A,B)$$

$$\mathcal{F}(A)$$

$$s_{\Phi }(A)$$

$$v\mapsto \int v(x)\varphi (x)dx$$

$$u.z:$$

$$z.u:$$

$$⟨T,u.m⟩=⟨u.T,m⟩=u(e)⟨T,m⟩\forall T\in P{M}_{\Psi }(G),\forall u\in A_{\Phi }(G).$$

Full text

Keywords - Orlicz Space, Orlicz Figa-Talamanca Herz algebra, Amenable group, Bounded approximate identity, Derivation, Splittings

2000 Mathematics Subject Classification 2000 - Primary 43A07, 43A15; Secondary 46J10

1 Introduction

Let $G$ be a locally compact group and let $A_{p}(G)$ the Figa-Talamanca Herz algebra introduced by Herz [10]. The following theorem on the characterization of amenability in terms of the $A_{p}(G)$ algebras is well-known.

Theorem 1.1**.**

Let $G$ be a locally compact group. Then the following are equivalent:

1. a)

The group $G$ is amenable. 2. b)

The Banach algebra $A_{p}(G)$ possesses a bounded approximate identity. 3. c)

Every closed cofinite ideal is of the form $I(E),$ where $E$ is a finite subset of $G.$ 4. d)

The Banach algebra $A_{p}(G)$ factorizes weakly. 5. e)

Each homomorphism from $A_{p}(G)$ with finite dimensional range is continuous. 6. f)

Every derivation of $A_{p}(G)$ into a Banach $A_{p}(G)$-bimodule is continuous.

The equivalence of the statements a) and b) was due to Herz [11]. The equivalence of the statements a), c), e) and f) were due to Forrest [8]. The equivalence of the statements a) and d) was due to Losert [15].

In [16], we have introduced and studied the $L^{\Phi}$-versions of the Figa-Talamanca Herz algebras. Here $L^{\Phi}$ denotes the Orlicz space corresponding to the Young function $\Phi.$ The space $A_{\Phi}(G)$ is defined as the space of all continuous functions $u,$ where $u$ is of the form

[TABLE]

where $f_{n}\in L^{\Phi}(G),$ $g_{n}\in L^{\Psi}(G),$ $(\Phi,\Psi)$ is a pair of complementary Young functions satisfying the $\Delta_{2}$-condition and

[TABLE]

This paper has the modest aim of proving the above said equivalent statements in the context of $A_{\Phi}(G)$ algebras. We shall begin with some preliminaries that are needed in the sequel.

2 Preliminaries

Let $\Phi:\mathbb{R}\rightarrow[0,\infty]$ be a convex function. Then $\Phi$ is called a Young function if it is symmetric and satisfies $\Phi(0)=0$ and $\underset{x\rightarrow\infty}{\lim}\Phi(x)=+\infty$. If $\Phi$ is any Young function, then define $\Psi$ as

[TABLE]

Then $\Psi$ is also a Young function and is termed as the complementary function to $\Phi.$ Further, the pair $(\Phi,\Psi)$ is called a complementary pair of Young functions.

Let $G$ be a locally compact group with a left Haar measure $dx.$ We say that a Young function $\Phi$ satisfies the $\Delta_{2}$-condition, denoted $\Phi\in\Delta_{2},$ if there exists a constant $K>0$ and $x_{0}>0$ such that $\Phi(2x)\leq K\Phi(x)$ whenever $x\geq x_{0}$ if $G$ is compact and the same inequality holds with $x_{0}=0$ if $G$ is non compact.

The Orlicz space, denoted $L^{\Phi}(G),$ is a vector space consisting of measurable functions, defined as

[TABLE]

The Orlicz space $L^{\Phi}(G)$ is a Banach space when equipped with the norm

[TABLE]

The above norm is called as the Luxemburg norm or Gauge norm. If $(\Phi,\Psi)$ is a complementary Young pair, then there is a norm on $L^{\Phi}(G),$ equivalent to the Luxemberg norm, given by,

[TABLE]

This norm is called as the Orlicz norm.

Let $C_{c}(G)$ denote the space of all continuous functions on $G$ with compact support. If a Young function $\Phi$ satisfies the $\Delta_{2}$ -condition, then $C_{c}(G)$ is dense in $L^{\Phi}(G).$ Further, if the complementary function $\Psi$ is such that $\Psi$ is continuous and $\Psi(x)=0$ iff $x=0,$ then the dual of $(L^{\Phi}(G),N_{\Phi}(\cdot))$ is isometrically isomorphic to $(L^{\Psi}(G),\|\cdot\|_{\Psi}).$ In particular, if both $\Phi$ and $\Psi$ satisfies the $\Delta_{2}$-condition, then $L^{\Phi}(G)$ is reflexive.

We say that an Young function $\Phi$ satisfies the Milnes-Akimovi$\check{\mbox{c}}$ condition (in short MA-condition) if for each $\epsilon>0$ there exists $c_{\epsilon}>1$ and an $x_{1}(\epsilon)\geq 0$ such that

[TABLE]

This condition will be used again and again in many of the theorems because of the following result due to M. M. Rao [19, Theorem 8].

Theorem 2.1**.**

Let $G$ be a locally compact group. Then $G$ is amenable if and only if for each $N$-function $\Phi$ satisfying the MA-condition and for each $\nu\in M_{1}^{+}(G),$ the operator $T_{\nu}:L^{\Phi}\rightarrow L^{\Phi}$ has norm 1, where $T_{\nu}(f)=\nu*f.$

For more details on Orlicz spaces, we refer the readers to [20].

Let $\Phi$ and $\Psi$ be a pair of complementary Young functions satisfying the $\Delta_{2}$ condition. Let

[TABLE]

Note that if $u\in A_{\Phi}(G)$ then $u\in C_{0}(G).$ If $u\in A_{\Phi}(G),$ define $\|u\|_{A_{\Phi}}$ as

[TABLE]

The space $A_{\Phi}(G)$ equipped with the above norm and with the pointwise addition and multiplication becomes a commutative Banach algebra [16, Theorem 3.4]. In fact, $A_{\Phi}(G)$ is a commutative, regular and semisimple banach algebra with spectrum homeomorphic to $G$ [16, Corollary 3.8]. This Banach algebra $A_{\Phi}(G)$ is called as the Orlicz Figà-Talamanca Herz algebra.

Let $\mathcal{B}(L^{\Phi}(G))$ be the linear space of all bounded linear operators on $L^{\Phi}(G)$ equipped with the operator norm. For a bounded complex Radon measure $\mu$ on $G$ and $f\in L^{\Phi}(G),$ define $T_{\mu}:L^{\Phi}(G)\rightarrow L^{\Phi}(G)$ by $T_{\mu}(f)=\mu*f.$ It is clear that $T_{\mu}\in\mathcal{B}(L^{\Phi}(G)).$ Let $PM_{\Phi}(G)$ denote the closure of $\{T_{\mu}:\mu\mbox{ is a bounded complex Radon measure}\}$ in $\mathcal{B}(L^{\Phi}(G))$ with respect to the ultraweak topology. It is proved in [16, Theorem 3.5], that for a locally compact group $G,$ the dual of $A_{\Phi}(G)$ is isometrically isomorphic to $PM_{\Psi}(G).$

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 by $Z(I),$ is a closed subset of $\Delta(\mathcal{A})$ defined as

[TABLE]

For a closed subset $E\subset\Delta(\mathcal{A}),$ we define the following ideals in $\mathcal{A}:$

[TABLE]

Note that $J_{\mathcal{A}}(E)$ and $I_{\mathcal{A}}(E)$ are closed ideals in $\mathcal{A}$ with the zero set equal to $E$ and $j_{\mathcal{A}}(E)\subseteq I\subseteq I_{\mathcal{A}}(E)$ for any ideal $I$ with zero set $E.$ $E$ is said to be a set of spectral synthesis (or a spectral set) for $\mathcal{A}$ if $I_{\mathcal{A}}(E)=J_{\mathcal{A}}(E).$ Let $I^{c}_{\mathcal{A}}(E)$ denote the elements in $I_{\mathcal{A}}(E)$ with compactly supported Gelfand transforms. We say that $E$ is a set of local spectral synthesis if $I^{c}_{\mathcal{A}}(E)\subseteq J_{\mathcal{A}}(E).$ By [16, Theorem 3.6] singletons are sets of spectral synthesis for $A_{\Phi}(G).$ Further, every closed subgroup is a set of local synthesis for $A_{\Phi}(G).$

The closed set $E$ is a Ditkin set if for every $u\in I_{\mathcal{A}}(E),$ there exists a sequence $\{u_{n}\}\subset j_{\mathcal{A}}(E)$ such that $u.u_{n}$ converges in norm to $u;$ if the condition holds for every compactly supported $u\in I_{\mathcal{A}}(E)$ then $E$ is called a local Ditkin set. If the sequence can be chosen in such a way that it is bounded and is the same for all $u\in I_{\mathcal{A}}(E),$ then we say that $E$ is a strong Ditkin set. Note that every Ditkin set is a set of spectral synthesis. The Banach algebra $\mathcal{A}$ is called a strong Ditkin algebra if all the singletons and the empty set are strong Ditkin sets.

For more on spectral synthesis see [14, 21].

Throughout this paper, $G$ will denote a locally compact group and $(\Phi,\Psi)$ will denote a complementary pair of Young functions satisfying the $\Delta_{2}$-condition.

3 Amenability and bounded approximate identities

We begin this section with the main result of this paper on the characterization of amenable groups in terms of the existence of bounded approximate identities in $A_{\Phi}(G).$

Theorem 3.1**.**

Let $G$ be a locally compact group and let $\Phi$ satisfy the MA condition. Then $G$ is amenable if and only if $A_{\Phi}(G)$ posseses a bounded approximate identity.

Proof.

Suppose that $G$ is amenable. Let $K$ be a compact subset of $G$ and let $\epsilon>0.$ It follows from Leptin’s condition [18, Definition 7.1] that there exists a compact set $C$ in $G$ of non-zero measure such that $|KC|<(1+\epsilon)|C|.$ Let $u_{K,\epsilon}=\frac{1}{(1+\epsilon)|C|}\chi_{KC}*\check{\chi}_{C}.$ Then $u_{K,\epsilon}\in A_{\Phi}(G)$ and

[TABLE]

Consider the set $\Lambda=\{(K,\epsilon):K\mbox{ is a compact subset of }G\mbox{ and }\epsilon>0\}$ directed as follows: $(K_{1},\epsilon_{1})\prec(K_{2},\epsilon_{2})$ if $K_{1}\subset K_{2}$ and $\epsilon_{2}<\epsilon_{1}.$ Now consider the net $\{u_{K,\epsilon}\}_{(K,\epsilon)\in\Lambda}$ in $A_{\Phi}(G).$ We now claim that $\{u_{K,\epsilon}\}_{(K,\epsilon)\in\Lambda}$ is an approximate identity for $A_{\Phi}(G).$ Let $f\in A_{\Phi}(G)\cap C_{c}(G)$ be such that $suppf=K$ and let $\epsilon>0.$ Then $(u_{K,\epsilon}f)(x)=\frac{f(x)}{1+\epsilon}$ if $x\in K$ and [math] otherwise. Therefore

[TABLE]

We now proceed further to prove the converse. Suppose that $A_{\Phi}(G)$ posseses an approximate identity $\{u_{\alpha}\}_{\alpha\in\Lambda}$ bounded by $c,$ for some $c>0.$ For a positive function $\psi\in C_{c}(G),$ using [16, Theorem 3.5], it can be shown as in [18, Theorem 10.4], that $\|\psi\|_{1}=\|L_{\psi}\|_{CV_{\Psi}(G)}.$ We now show that a similar equality holds if we replace $\psi$ by a positive measure having compact support. Let $\mu\in M(G)$ be a positive measure having compact support. Choose $f_{0}\in C_{c}(G)$ such that $f_{0}$ is positive, $\check{f_{0}}=f_{0}$ and $\|f_{0}\|_{1}=1.$ Note that, for every $f\in C_{c}(G)$ with $N_{\Phi}(f)\leq 1,$ we have $f_{0}*f$ is also positive and has compact support. Further,

[TABLE]

Also, if $f^{\prime}\in C_{c}^{+}(G),$ then

[TABLE]

Thus

[TABLE]

As $\|L_{\mu}\|_{CV_{\Psi}(G)}\leq\|\mu\|_{M(G)}$ for all $\mu\in M(G),$ we have $\|L_{\mu}\|_{CV_{\Psi}(G)}=\|\mu\|_{M(G)}$ for all positive $\mu$ having compact support. Thus $G$ is amenable, thanks to Theorem 2.1. ∎

Remark 3.2**.**

Note that the condition that $\Phi$ satisfies the MA condition in the above theorem is needed only while proving the converse, i.e., while invoking Theorem 2.1. As mentioned in [19], the assumption that $\Phi$ satisfies the MA condition is needed only to avoid the Riesz-convexity theorem. Note that the proof of the above theorem for the $A_{p}(G)$-algebras uses the Riesz-convexity theorem. Although an extended Riesz-convexity theorem for Orlicz spaces is available, it cannot be used here.

We now begin to prove some corollaries. In the first corollary, we characterize amenability in terms of certain weak*-closed $A_{\Phi}(G)$-submodules of $PM_{\Psi}(G).$

Corollary 3.3**.**

Let $G$ be a locally compact group, $\Phi$ satisfy the MA-condition and let $X$ be a weak-closed $A_{\Phi}(G)$-submodule of $PM_{\Psi}(G).$ Then $G$ is amenable if and only if the following statements about $X$ are equivalent:*

1. a)

The space $X$ is invariantly complemented 2. b)

The space ${}^{\perp}X$ has a bounded approximate identity.

Proof.

The proof of the if part follows from Theorem 3.1 and [6, Proposition 6.4]. The only if part follows again from Theorem 3.1 by choosing $X=\{0\}.$ ∎

Let $B_{\Phi}(G)=\left\{u\in C(G):uv\in A_{\Phi}(G)~{}\forall~{}v\in A_{\Phi}(G)\right\}.$ Then the space $B_{\Phi}(G)$ when equipped with the operator norm becomes a commutative banach algebra.

Corollary 3.4**.**

Let $G$ be an amenable group and let $\Phi$ satisfy the MA-condition. Then the two norms $\|\cdot\|_{A_{\Phi}(G)}$ and $\|\cdot\|_{B_{\Phi}(G)}$ are equivalent.

Proof.

By definition of $B_{\Phi}(G),$ it is clear that, for any $u\in A_{\Phi}(G),$ $\|u\|_{B_{\Phi}(G)}\leq\|u\|_{A_{\Phi}(G)}.$ For this inequality, the assumption on the group to be amenable is not needed.

For the other inequality, note that, since $G$ is amenable, by Theorem 3.1, $A_{\Phi}(G)$ possesses a bounded approximate identity $\{u_{\alpha}\}$ such that $\|u_{\alpha}\|_{A_{\Phi}(G)}\leq 2\ \forall\ \alpha.$ Thus, for any $u\in A_{\Phi}(G),$ we have,

[TABLE]

Hence the proof. ∎

One of the classical results of Reiter states that every closed subgroup of a locally compact abelian group is a set of spectral synthesis for the Fourier algebra $A(G).$ This result is known as the subgroup lemma [21]. This result was generalized to locally compact groups by Takesaki and Tatsuuma [22]. For $1<p<\infty,$ Herz generalized the subgroup lemma to $A_{p}(G)$ algebras under the assumption that $G$ is amenable. For other generalisations see [2]. Our next corollary is the subgroup lemma for spectral synthesis. The proof of this is an immediate consequence of Theorem 3.1 and [16, Theorem 3.6].

Corollary 3.5**.**

Let $G$ be an amenable group and let $\Phi$ satisfy the MA-condition. Then every closed subgroup is a set of spectral synthesis for $A_{\Phi}(G).$

4 Ideals with bounded approximate identities

In this section, our aim is to characterize amenable groups in terms of Ditkin sets.

We shall begin this section by introducing some notations. Let $A,~{}B\subset G$ be closed set of $G.$ Let

[TABLE]

Our first result is an analogue of [8, Proposition 3.4]. This theorem proves the existence of bounded approximate identities with some properties, in certain closed ideals.

Theorem 4.1**.**

Let $G$ be a amenable locally compact group and let $E$ be a closed subset of $G.$ If $E$ is a set of synthesis for $A_{\Phi}(G)$ and $s_{\Phi}(E)<\infty,$ then the ideal $I(E)$ has a bounded approximate identity $\{u_{\alpha}\}_{\alpha\in\Lambda}$ such that the following holds:

1. a)

$\|u_{\alpha}\|_{A_{\Phi}(G}\leq 8+4s_{\Phi}(E)\ \forall\ \alpha\in\Lambda,$** 2. b)

$u_{\alpha}\in A_{\Phi}(G)\cap C_{c}(G)\ \forall\ \alpha\in\Lambda,$** 3. c)

for every compact subset $K$ of $G$ with $K\cap E=\emptyset,$ there exists a sequence $\{u_{n}\}$ from $\{u_{\alpha}\}$ such that for every $u\in A_{\Phi}(G)$ with $supp(u)\subset K,$ we have $\|uu_{n}-u\|_{A_{\Phi}(G)}\leq\frac{1}{n}.$

Proof.

Since $G$ is amenable, it follows from the proof of Theorem 3.1, that $A_{\Phi}(G)$ possesses an approximate identity $\{u_{K,\epsilon}\}_{\mathcal{F}(E)\times\mathbb{R}^{+}}$ such that

1. i)

$\|u_{K,\epsilon}\|\leq 2$ 2. ii)

$supp(u_{K},\epsilon)$ is compact and 3. iii)

if $v\in A_{\Phi}(G)$ such that $supp(v)\subseteq K$ then $u_{K,\epsilon}v=\frac{v}{1+\epsilon}.$

Since $s_{\Phi}(E)$ is finite, there exist $u_{K}\in\mathscr{S}(E,K)$ such that $\|u_{K}\|_{B_{\Phi}(G)}\leq s_{\phi}(E)+1.$ Let $v_{K,\epsilon}=u_{K,\epsilon}-u_{K,\epsilon}u_{K}.$ It is clear that $v_{K,\epsilon}\in I(E).$ This $\{v_{K,\epsilon}\}_{(K,\epsilon)\in\mathcal{F}(E)\times\mathbb{R}^{+}}$ will satisfy the requirements of the theorem. ∎

Lemma 4.2**.**

Let $K$ be a compact subgroup of a locally compact group $G.$ Then $s_{\Phi}(K)$ is finite.

Proof.

Let $C$ be a compact subset of $G$ such that $C\cap K=\emptyset.$ Choose an open neighbourhood $U$ of $e$ such that $U$ is symmetric, relatively compact and $C^{-1}K\cap KU^{2}=\emptyset.$ Let $u(x)=\frac{1}{|KU|}\chi_{KU}*\check{\chi}_{KU}(x).$ Now, it is clear that $u(e)=1,\|u\|_{B_{\Phi}(G)}\leq 1.$ Further, note that $u$ is 1 on $K$ and 0 on $C,$ i,e., $u\in\mathscr{S}(K,C).$ Hence the proof. ∎

As an immediate consequence we have the following corollary.

Corollary 4.3**.**

Let $G$ be a locally compact amenable group. Then for each $x\in G,$ $I(\{x\})$ contains a bounded approximate identity.

Proof.

The proof of this follows from [16, Theorem 3.6], Theorem 4.1 and Lemma 4.2. ∎

Here is the characterization of amenable groups in terms of the Ditkin sets.

Corollary 4.4**.**

Let $G$ be a locally compact group and let $\Phi$ satisfy the MA-condition. Then $G$ is amenable if and only if $A_{\Phi}(G)$ is a strong Ditkin algebra.

Proof.

The proof of this follows from Theorem 3.1 and Corollary 4.3. ∎

5 Weak factorization and cofinite ideals

In this section, we characterize amenable groups in terms of weak factorization and cofinite ideals.

For $A_{p}(G)$ algebras, the following theorem was proved by Losert [15].

Theorem 5.1**.**

Let $G$ be a locally compact group $G$ and let $\Phi$ satisfy the MA condition. Then $G$ is amenable if and only if $A_{\Phi}(G)$ factorizes weakly.

Proof.

Let $G$ be amenable. Then the if part follows from the Cohen’s factorization theorem. We shall now prove the converse. Suppose that $A_{\Phi}(G)$ weakly factorizes. Note that $A_{\Phi}(G)$ is a self-adjoint Banach algebra. Thus, by [9, Theorem 1.3], there exists $c>0$ such that for each compact subset $C$ of $G$ there exists a positive function $u\in A_{\Phi}(G)$ such that $u\geq 1$ on $C$ and $\|u\|_{A_{\Phi}(G)}\leq c.$ Observe that, for $\phi\in C_{c}^{+}(G),$ the norm of the convolution operator $\|L_{\phi}\|_{CV_{\Psi}(G)}$ is equal to the norm of the linear functional

[TABLE]

on $A_{\Phi}(G).$ Thus, $\left|\int\phi(x)\ dx\right|\leq c\|L_{\phi}\|_{CV_{\Psi}(G)},$ which implies that $\|\phi\|_{1}^{n}\leq c\|L_{\phi}\|^{n}_{CV_{\Psi}(G)}$ and hence it follows that $\|\phi\|_{1}\leq\|L_{\phi}\|_{CV_{\Psi}(G)}.$ Now proceeding as in the proof of the converse of Theorem 3.1, one can show that $G$ is amenable. ∎

Before we proceed to our next characterization, here are some preparatory lemmas.

Lemma 5.2**.**

Let $G$ be a amenable group and let $\Phi$ satisfy the MA-condition. Then every finite subset is a set of spectral synthesis.

Proof.

The proof of this is an immediate consequence of [12, Theorem 39.24] and Corollary 4.4. ∎

Lemma 5.3**.**

Let $G$ be a non-amenable locally compact group and let $I=I(\{e\}).$ Then $I^{2}$ is not closed in $A_{\Phi}(G),$ where $\Phi$ satisfies the MA-condition.

Proof.

Using [16, Theorem 3.6] and Theorem 5.1, the proof of this follows similar lines as in [6, Lemma 5.7]. ∎

Our next result is the characterization of amenable groups in terms of cofinite ideals.

Theorem 5.4**.**

Let $G$ be a locally compact group and let $\Phi$ satisfy the MA condition. Then the following are equivalent:

1. a)

$G$* is amenable.* 2. b)

Every cofinite ideal in $A_{\Phi}(G)$ is of the form $I(E)$ for some finite subset $E$ of $G.$ 3. c)

Each homomorphism from $A_{\Phi}(G)$ with finite dimensional range is continuous.

Proof.

a) $\Rightarrow$ b). Let $G$ be amenable and let $I$ be a cofinite ideal in $A_{\Phi}(G).$ By [5, Theorem 2.3], it is enough to show that every closed cofinite ideal is idempotent. So, let us assume that $I$ is a closed cofinite ideal in $A_{\Phi}(G).$ Since $I$ is cofinite, the zero set $Z(I)$ is finite and hence, by Lemma 5.2, $Z(I)$ is a set of spectral synthesis. Thus, it follows that $I=I(Z(I)).$ Further, by Theorem 4.1, $I(Z(I))$ has a bounded approximate identity and hence it follows from Cohen’s factorization theorem that $I$ is idempotent.

b) $\Rightarrow$ a) follows from [5, Theorem 2.3] and Lemma 5.3. The equivalence of b) and c) follows again from [5, Theorem 2.3]. ∎

6 Derivations and splittings

In this section, we characterize amenable groups in terms of continuous derivations. Next we study algebraic splittings and strong splittings of the extensions of the algebra $A_{\Phi}(G)$ in the spirit of [17].

We begin this section by showing the existence of a discontinuous derivation. The proof of this Lemma follows from Lemma 5.3 and [5, Pg. 402].

Lemma 6.1**.**

Let $G$ be a nonamenable group and let $\Phi$ satisfy the MA-condition. Then there exists a discontinuous derivation of $A_{\Phi}(G)$ into a finite dimensional commutative Banach $A_{\Phi}(G)$-bimodule.

Lemma 6.2**.**

Let $G$ be a amenable group and let $I$ be a closed ideal in $A_{\Phi}(G)$ of infinite codimension. Then there exists sequences $\{u_{n}\},\{v_{n}\}$ in $A_{\Phi}(G)$ such that $u_{n}v_{1}\ldots v_{n-1}\notin I$ but $u_{n}v_{1}\ldots v_{n-1}u_{1}v_{n}\in I$ for all $n\geq 2.$

Proof.

As in the proof of Theorem 5.4, one can show that the zero set $Z(I),$ of an ideal of infinite codimension, is infinite. Now the remaining proof follows exactly as in the proof of [7, Lemma 2]. ∎

Here is the characterization of amenable groups in terms of continuous derivations.

Theorem 6.3**.**

Let $G$ be a locally compact group and let $\Phi$ satisfy the MA-condition. Then the following are equivalent:

1. a)

Every derivation of $A_{\Phi}(G)$ into a Banach $A_{\Phi}(G)$-bimodule is continuous. 2. b)

Every derivation of $A_{\Phi}(G)$ into a finite dimensional commutative Banach $A_{\Phi}(G)$-bimodule is continuous. 3. c)

$G$* is amenable.*

Proof.

a) $\Rightarrow$ b) is trivial and b) $\Rightarrow c)$ follows from Lemma 6.1. We shall now prove c) $\Rightarrow$ a). In order to prove this, it is enough to verify the conditions of [13, Theorem 2] for a closed cofinite ideal of $A_{\Phi}(G).$ But this follows from Theorem 5.4 and Lemma 6.2. ∎

Before we proceed to study algebraic and strong splittings, here are some notations. Let $x,y\in G\cup\{0\}.$ For $z\in\mathbb{C}$ and $u\in A_{\Phi}(G),$ define $u.z$ and $z.u$ as follows:

[TABLE]

Note that the above left and right action turns $\mathbb{C}$ into a $A_{\Phi}(G)$-bimodule. In order to emphasize the role of $x$ and $y,$ we shall denote this bimodule as $\mathbb{C}_{x,y}.$

A linear functional m on $PM_{\Psi}(G)$ is called a mean if $\|m\|=m(I)=1.$ A mean $m$ on $PM_{\Psi}(G)$ is said to be topologically invariant if $u.m=u(e)m~{}~{}~{}\forall~{}u\in A_{\Phi}(G)$, that is,

[TABLE]

It is shown in [16, Corollary 6.2] that the set of all topological invariant means on $PM_{\Psi}(G)$ is non-empty. As a result, by following the arguments given for [17, Lemma 3.1], we have the following Lemma for $A_{\Phi}(G).$

Lemma 6.4**.**

If $x\in G,$ then there exists an $A_{\Phi}(G)$-bimodule homomorphisms $\Theta_{x}:A_{\Phi}(G)^{\prime}\rightarrow\mathbb{C}_{x,x}$ such that $\|\Theta_{x}\|=\Theta_{x}(\delta_{x})=1.$

For algebraic and strong splittings of extensions of Banach algebras, we shall refer to [4].

As a consequence of the above Lemma along with [16, Theorem 3.6] and [16, Corollary 3.8], we have the following Lemma, whose proof is similar to [17, Lemma 3.4].

Lemma 6.5**.**

Let $X$ be a finite-dimensional Banach $A_{\Phi}(G)$-bimodule. Suppose that $X$ is also essential as a left module. Then every singular extension of $A_{\Phi}(G)$ by $X$ splits strongly.

Corollary 6.6**.**

Let $X$ be a finite-dimensional $A_{\Phi}(G)$-bimodule. Then $X$ is isomorphic to $\underset{i=1}{\overset{n}{\oplus}}\mathbb{C}_{x_{i},y_{i}},$ for some $n\in\mathbb{N}$ and $x_{i},y_{i}\in G\cup\{0\}.$

Theorem 6.7**.**

Let $\Phi$ satisfy the MA-condition. If $G$ is amenable, then all finite-dimensional extensions of $A_{\Phi}(G)$ split strongly.

Proof.

Let $X$ be a finite-dimensional $A_{\Phi}(G)$-module. As $G$ is amenable, by Theorem 3.1, it follows that $X$ is an essential $A_{\Phi}(G)$-module. Thus, by Lemma 6.5, every singular extension of $A_{\Phi}(G)$ by $X$ splits strongly. Now the conclusion follows from [4, Corollary 1.9.8].

Another proof of this follows from [1, Theorem 4.18], Theorem 4.1 and Theorem 5.4. ∎

Proposition 6.8**.**

Suppose that $A_{\Phi}(G)$ posseses an approximate identity. Then all the singular finite-dimensional extensions of $A_{\Phi}(G)$ split algebraically.

Proof.

Let $X$ be a finite-dimensional $A_{\Phi}(G)$-bimodule. By Corollary 6.6, it follows that $\widetilde{H}^{2}(A_{\Phi}(G),X)=\underset{i=1}{\overset{n}{\oplus}}\widetilde{H}^{2}(A_{\Phi}(G),\mathbb{C}_{x_{i},y_{i}}),$ where $n$ is the dimension of $X.$ Note that if $x_{i}$ or $y_{i}$ are non-zero, then by [1, Pg. 21-22], it follows that $\widetilde{H}^{2}(A_{\Phi}(G),\mathbb{C}_{x_{i},y_{i}})=\{0\}.$ Further, by [4, Proposition 2.9.34], it follows that $\widetilde{H}^{2}(A_{\Phi}(G),\mathbb{C}_{0,0})=\{0\}.$ Thus the proof follows from [4, Corollary 2.8.13]. ∎

Acknowledgement

The first author would like to thank the University Grants Commission, India, for research grant.