# Super module amenability of inverse semigroup algebras

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DOI: 10.1007/s00233-012-9432-0

- Cite this article as:
- Lashkarizadeh Bami, M., Valaei, M. & Amini, M. Semigroup Forum (2013) 86: 279. doi:10.1007/s00233-012-9432-0

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## Abstract

In this paper we compare the notions of super amenability and super module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. We find conditions for the two notions to be equivalent. In particular, we study arbitrary module actions of *l*^{1}(*E*_{S}) on *l*^{1}(*S*) for an inverse semigroup *S* with the set of idempotents *E*_{S} and show that under certain conditions, *l*^{1}(*S*) is super module amenable if and only if *S* is finite. We also study the super module amenability of *l*^{1}(*S*)^{∗∗} and module biprojectivity of *l*^{1}(*S*), for arbitrary actions.

### Keywords

Module Arense regularModule biprojectiveModule derivationSuper module amenable## 1 Introduction

The notion of module amenability for a Banach algebra \(\mathcal{A}\) which is a Banach module over another Banach algebra \(\mathfrak{A}\) was introduced and studied by the third author in [1]. The super module amenability of \(\mathcal{A}\) as a \(\mathfrak{A}\)-module was studied by Poormahmood Aghababa in [8]. The main example in [8] is an inverse semigroup *S* with the set of idempotents *E* where \(\mathfrak{A}=l^{1}(E_{S})\) acts on \(\mathcal{A}=l^{1}(S)\) by *δ*_{e}.*δ*_{s}=*δ*_{s} and *δ*_{s}.*δ*_{e}=*δ*_{se}, for *s*∈*S* and *e*∈*E*_{S}, where *δ*_{x} denotes the point mass at *x*. Then it is shown that *l*^{1}(*S*) is super module amenable if and only if *l*^{1}(*S*)^{∗∗} is module amenable, and this holds exactly when the maximal group homomorphic image \(\frac{S}{\approx}\) of *S* is finite. In [3] the authors have introduced the concept of weak module amenability for a Banach algebra and proved that *l*^{1}(*S*) is weak module amenable, when *S* is a commutative inverse semigroup and a *l*^{1}(*E*_{S}) acts on *l*^{1}(*S*) by the compatible actions *δ*_{e}.*δ*_{s}=*δ*_{s}.*δ*_{e}=*δ*_{se}, for *s*∈*S*, *e*∈*E*_{S}.

It is natural to ask that if these results remain valid for an arbitrary action. This is the main theme of the present paper which distinguishes it from all the previous work [1–3, 8, 10]. We investigate super module amenability and the module biprojectivity of *l*^{1}(*S*) for arbitrary compatible actions of *l*^{1}(*E*_{S}), when *E*_{S} is finite. The paper is organized as follows. Section 1 is devoted to the notation and definitions which are needed throughout the paper. In Sect. 2 we find sufficient conditions for super amenability of a quotient \(\mathcal{A}/J\) of \(\mathcal{A}\) to be equivalent to super module amenability of \(\mathcal{A}\). This is done without the extra assumption that the action is trivial from one direction. In Sect. 3 we consider the main example of an inverse semigroup *S* with finite set of idempotents *E*_{S}. We show that if *l*^{1}(*S*) is a pseudo-unital Banach *l*^{1}(*E*_{S})-module and a commutative Banach *l*^{1}(*E*_{S})-module under an arbitrary module action, then *l*^{1}(*S*) is super module amenable if and only if *l*^{1}(*S*)^{∗∗} is super-module amenable, and this holds exactly when *S* is finite. This is a generalization of a result due to Selivanov, which asserts that for a locally compact group *G*, the Banach algebra *L*^{1}(*G*) is super-amenable if and only if *G* is finite. We also prove that if *l*^{1}(*S*) is unital and a pseudo-unital Banach *l*^{1}(*E*_{S})-module with arbitrary commutative compatible actions, then *l*^{1}(*S*) is module biprojective if and only if *S* is finite. This extends a previously known result of Helemskii in [6], which asserts that for a discrete group *G*, the Banach algebra *l*^{1}(*G*) is biprojective if and only if *G* is finite.

## 2 Preliminaries

*X*be a Banach \(\mathcal{A}\)-bimodule and a Banach \(\mathfrak{A}\)-bimodule with compatible actions

*x*∈

*X*and similarly for the right or two-side actions. We call

*X*a Banach \(\mathcal{A}\)-\(\mathfrak{A}\)-module. If moreover,

*X*is called a commutative Banach \(\mathcal{A}\)-\(\mathfrak{A}\)-module.

*X*is a commutative Banach \(\mathcal{A}\)-\(\mathfrak{A}\)-module, then so is

*X*

^{∗}under the actions

*a*.

*α*).

*b*=

*a*.(

*α*.

*b*) for \(\alpha \in \mathfrak{A}\) and \(a,b\in \mathcal{A}\). But when \(\mathcal{A}\) is a commutative \(\mathfrak{A}\)-module and acts on itself by multiplication from both sides, then it is a commutative Banach \(\mathcal{A}\)-\(\mathfrak{A}\)-module.

*a*.

*α*⊗

*b*−

*a*⊗

*α*.

*b*for \(\alpha\in \mathfrak{A}\) and \(a,b\in \mathcal{A}\). Let

*J*be the closed ideal of \(\mathcal{A}\) generated by elements of the form (

*a*.

*α*)

*b*−

*a*(

*α*.

*b*) for \(\alpha\in \mathfrak{A}\) and \(a,b\in \mathcal{A}\), then the module projective tensor product \({\mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\, {\mathcal{A}}\cong (\mathcal{A}\,\widehat{\otimes}\, \mathcal{A})/\mathcal{I}\) and the quotient Banach algebra \(\mathcal{A}/J\) are Banach \(\mathfrak{A}\)-modules with the following compatible actions

Define \(\omega\in {\mathcal{L}}({\mathcal{A}}\,\widehat{\otimes}\, \mathcal{A}, \mathcal{A})\) by *ω*(*a*⊗*b*)=*ab* and \(\widetilde{\omega}\in {\mathcal{L}}({\mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\,{\mathcal{A}}, \mathcal{A}/J)\) by \(\widetilde{\omega}(a\otimes b+{\mathcal{I}})=ab+{J}\), then extending by linearity and continuity, \(\widetilde{\omega}\) is an \(\mathfrak{A}\)-module homomorphism. Moreover if \(\mathcal{A}/J\) and \({\mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\, {\mathcal{A}}\) are commutative \(\mathfrak{A}\)-modules, then \(\mathcal{A}/J\) and \({\mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\, {\mathcal{A}}\) are \(\mathcal{A}/J\)-\({\mathfrak{A}}\)-module and \(\widetilde{\omega}\) is an \(\mathcal{A}/J\)-\({\mathfrak{A}}\)-module homomorphism.

*X*be as above. A bounded map \(D:{\mathcal{A}} \longrightarrow X\) is called a

*module derivation*if

*M*>0 such that ∥

*D*(

*a*)∥≤

*M*∥

*a*∥ (\(a\in{\mathcal{A}}\)), although

*D*is not necessarily linear, but still its boundedness implies its norm continuity. For every

*x*∈

*X*we define the inner module derivation ad

_{x}by

*super module amenable*(or more explicitly, super \(\mathfrak{A}\)-module amenable) if for any commutative Banach \(\mathcal{A}\)-\(\mathfrak{A}\)-module

*X*, each module derivation \(D : {\mathcal{A}} \longrightarrow X\) is inner.

*module Arens regular*if the module topological center \(Z_{\mathfrak{A}}({\mathcal{A}}^{**})\) of \({\mathcal{A}}^{**}\) is equal to \(\mathcal{A}^{**}\) (see [10]), where

*E*has the

*approximation property*if there is a net (

*S*

_{λ})

_{λ∈Λ}in the space of the finite-rank operators on

*E*such that

*S*

_{λ}⟶id

_{E}, uniformly on compact subsets of

*E*(see Sect. 3.1 in [11]).

Throughout the rest of this paper, we fix \(\mathfrak{A}\), \(\mathcal{A}\), \(\mathcal{I}\) and *J* as above, unless otherwise specified.

## 3 Relation between super amenability and super module amenability

We start this section by the following proposition which the proof of it is the same as proof of Corollary 2.3 in [8].

### Proposition 3.1

*Let*\(\mathcal{A}\)*be a Banach*\(\mathfrak{A}\)-*module with commutative compatible actions*. *If*\(\mathcal{A}^{**}\)*is super module amenable then so is*\(\mathcal{A}\).

### Lemma 3.2

*Let*\(\mathcal{A}\)*be a Banach*\(\mathfrak{A}\)-*module with compatible actions such that any module derivation from*\(\mathcal{A}\)*to the dual of each commutative Banach*\(\mathcal{A}\)-\(\mathfrak{A}\)-*module is*\(\Bbb{C}\)-*linear*. *If*\(\mathcal{A}\)*is super amenable*, *then it is super module amenable*.

### Proof

*φ*∈

*X*

^{∗}, we have So \(\widetilde{D}(\alpha.a)=\alpha.\widetilde{D}(a)\). Similarly \(\widetilde{D}(a.\alpha)=\widetilde{D}(a).\alpha\). Thus, \(\widetilde{D}\) is a module derivation, therefore it is \(\Bbb{C}\)-linear. It follows that

*D*is \(\Bbb{C}\)-linear. From the super amenability of \(\mathcal{A}\) we conclude that

*D*=ad

_{x}, for some

*x*∈

*X*. □

### Theorem 3.3

*Let*\(\mathcal{A}\)*be a Banach*\(\mathfrak{A}\)-*module with compatible actions such that*\(\mathfrak{A}\)*has a bounded approximate identity for*\(\mathcal{A}\). *If*\(\mathcal{A}/J\)*is super amenable*, *then*\(\mathcal{A}\)*is super module amenable*.

### Proof

Suppose that *X* is a commutative Banach \(\mathcal{A}/J\)-\(\mathfrak{A}\)-module and \(D:{\mathcal{A}/J}\longrightarrow X^{*}\) is a module derivation. Clearly if \((e_{\lambda})_{\lambda\in\varLambda}\subseteq \mathfrak{A}\) is a bounded approximate identity for \(\mathcal{A}\), then it is also a bounded approximate identity for \(\mathcal{A}/J\). By the proof of Proposition 2.1 in [1], we conclude that *D* is \(\Bbb{C}\)-linear. Since \(\mathcal{A}\) is super amenable, then so is \(\mathcal{A}/J\). From Lemma 3.2, it follows that \(\mathcal{A}/J\) is super module amenable. Now Lemma 3.6 in [8] implies that \(\mathcal{A}\) is super module amenable. □

In Theorem 3.3 we obtained a sufficient condition that super amenability of \(\mathcal{A}/J\) implies super module amenability of \(\mathcal{A}\). The next theorem may be considered as a converse of Theorem 3.3.

### Theorem 3.4

*Let*\({\mathcal{A}}\)*be super module amenable and*\(\mathfrak{A}\)*be super amenable*. *If*\({\mathcal{A}/J}\)*is a commutative Banach*\(\mathfrak{A}\)-*module*, *then*\(\mathcal{A}/J\)*is super amenable*.

### Proof

*e*+

*J*is an identity for \(\mathcal{A}/J\), which exists by Proposition 3.2 in [8]. Let \(\mathcal{L}\) be the closed linear span of \(\{\alpha.(e+{J})~:~\alpha\in \mathfrak{A} \}\), then \(\mathcal{L}\) is a closed subalgebra of \(\mathcal{A}/J\) with the following multiplication:

*θ*is a continuous algebra homomorphism with dense range such that ∥

*θ*∥≤∥

*e*+

*J*∥. Now the super amenability of \(\mathcal{L}\) follows from the super amenability of \(\mathfrak{A}\), by Exercise 4.1.4(i) of [11].

*X*and let \(D:{\mathcal{A}/J}\longrightarrow X\) be a derivation. Since \(D|_{\mathcal{L}}:{\mathcal{L}}\longrightarrow X\) is a derivation and \(\mathcal{L}\) is super amenable, there is

*x*

_{1}∈

*X*such that

*Y*be the closed linear span of the set

*X*is pseudo-unital, we conclude that

*Y*is a Banach \({\mathcal{A}/J}\)-bimodule such that \(\widetilde{D}({\mathcal{A}/J})\subseteq Y\subseteq X\). We prove that \(\widetilde{D}\) is an inner derivation. To see this, define compatible actions

*e*+

*J*is an identity for \(\mathcal{A}/J\), we have Thus for all \(\alpha \in \mathfrak{A}\) and \(a,b,c\in \mathcal{A}\), from \(\mathfrak{A}\)-commutativity of \(\mathcal{A}/J\) it follows that By continuity and linearity, we have

*Y*is a commutative Banach \({\mathcal{A}}/J\)-\(\mathfrak{A}\)-module. Also, Hence \(\widetilde{D}:{\mathcal{A}/J}\longrightarrow Y\) is a module derivation. Since \(\mathcal{A}\) is super module amenable, so is \(\mathcal{A}/J\), by Lemma 3.6 of [8]. Hence there is

*x*

_{2}∈

*Y*such that \(\widetilde{D}=\mathrm{ad}_{x_{2}}\). Consequently, \(D=\mathrm{ad}_{x_{1}+x_{2}}\) for

*x*

_{1}+

*x*

_{2}∈

*X*. □

By the above theorem and Theorem 3.3 we get the following result.

### Corollary 3.5

*Let*\(\mathfrak{A}\)*be super amenable and*\(\mathcal{A}\)*be a pseudo*-*unital Banach*\(\mathfrak{A}\)-*module with commutative compatible actions*. *Then*\(\mathcal{A}\)*is super module amenable if and only if*\(\mathcal{A}\)*is super amenable*.

The following corollary follows from the above theorem and Theorem 4.1.5 of [11].

### Corollary 3.6

*Let*\({\mathcal{A}}\)

*be super module amenable and*\({\mathcal{A}/J}\)

*be a commutative Banach*\(\mathfrak{A}\)-

*module with the approximation property*.

*If*\({\mathfrak{A}}\)

*is super amenable then*

*for some*\(n_{1},\ldots,n_{k}\in \Bbb{N}\),

*where*\(\Bbb{M}_{n}(\Bbb{C})\)

*is an*

*n*×

*n*

*matrix over*\(\Bbb{C}\).

We give a direct simple proof for the following lemma, for an alternative proof one may combine I.3.68 and VII.1.74 of [6].

### Lemma 3.7

*Every finite*-*dimensional amenable Banach algebra is super amenable*.

### Proof

*E*is a pseudo-unital Banach \(\mathcal{B}\)-bimodule. Let \(D:{\mathcal{B}}\longrightarrow E\) be a derivation and

*F*be the closed linear span of \(\{a.D(b).c~|~a,b,c \in {\mathcal{B}} \}\). Then

*F*is a finite-dimensional Banach \(\mathcal{B}\)-bimodule, so it is reflexive. Since \(\mathcal{B}\) is amenable, it has a bounded approximate identity, thus \(D({\mathcal{B}})\subseteq F\). It is clear that if \(\widetilde{D}(a):=\widehat{D(a)}\) for each \(a\in \mathcal{B}\), then \(\widetilde{D}\in {\mathcal{Z}}^{1}({\mathcal{B}},F^{**})\). Therefore there is

*x*∈

*F*such that for all \(a\in {\mathcal{B}}\), it follows that

*D*is inner. □

### Definition 3.8

An element \(\textbf{m}\in {\mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\,{\mathcal{A}}\) is called a module diagonal (or \(\mathfrak{A}\)-module diagonal) if \(\widetilde{\omega}(\textbf{m})\) is an identity for \({\mathcal{A}}/J \) and *a*.**m**=**m**.*a*, for all \(a\in {\mathcal{A}}\).

*Y*,

*Z*and

*W*be commutative Banach \(\mathcal{A}/J\)-\(\mathfrak{A}\)-modules, and

*f*:

*Y*→

*Z*and

*g*:

*Z*→

*W*be \(\mathcal{A}/J\)-\(\mathfrak{A}\)-module homomorphisms, then the short, exact sequence

- (i)
is admissible if there is an \(\mathfrak{A}\)-module homomorphism

*h*:*W*→*Z*such that*gh*=id_{W}. - (ii)
splits if there is an \(\mathcal{A}/J\)-\(\mathfrak{A}\)-module homomorphism

*h*:*W*→*Z*such that*gh*=id_{W}.

The proof of the following proposition is routine but we state it briefly.

### Proposition 3.9

*Let*\(\mathcal{A}/J\)

*and*\({\mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\,{\mathcal{A}}\)

*be commutative Banach*\(\mathfrak{A}\)-

*modules*.

*Consider the following short exact sequence*

- (i)
*If*\(\mathcal{A}/J\)*has an identity*,*then**Π**is admissible*. - (ii)
\(\mathcal{A}\)

*is super module amenable if and only if**Π**splits and*\(\mathcal{A}/J\)*is unital*.

### Proof

(i) Suppose *e*+*J* is an identity for \(\mathcal{A}/J\), then \(\theta(a+{J})=a\otimes e+{\mathcal{I}}\) (\(a\in {\mathcal{A}}\)) defines an \(\mathfrak{A}\)-module homomorphism and is a right inverse for \(\widetilde{\omega}\).

**m**. If we define

*θ*(

*a*+

*J*)=

*a*.

**m**\((a\in \mathcal{A})\), then

*θ*is an \(\mathcal{A}/J\)-\(\mathfrak{A}\)-module homomorphism such that \(\widetilde{\omega}\theta=\mathrm{id}_{\mathcal{A}/J}\).

Conversely, if *e*+*J* is an identity for \(\mathcal{A}/J\) and *θ* is an \(\mathcal{A}/J\)-\(\mathfrak{A}\)-module homomorphism with \(\widetilde{\omega}\theta=\mathrm{id}_{\mathcal{A}/J}\), then **m**=*θ*(*e*+*J*) is a module diagonal for \(\mathcal{A}\). Therefore \(\mathcal{A}\) is super module amenable by Theorem 3.5 in [8]. □

## 4 Super module amenability and module biprojectivity of semigroup algebras

*S*is an

*inverse semigroup*if for each

*s*∈

*S*there is a unique element

*s*

^{∗}∈

*S*such that

*s*

^{∗}

*ss*

^{∗}=

*s*

^{∗}and

*ss*

^{∗}

*s*=

*s*. If we denote the set of idempotents of

*S*by

*E*

_{S}, then

*E*

_{S}is a semilattice with the following order

*Clifford semigroup*. It is easy to see that

*E*

_{S}is a commutative subsemigroup of

*S*. In particular

*l*

^{1}(

*E*

_{S}) could be regarded as a subalgebra of

*l*

^{1}(

*S*) [7]. Consequently,

*l*

^{1}(

*S*) is a Banach algebra and a Banach

*l*

^{1}(

*E*

_{S})-module with compatible actions. It is possible to consider arbitrary actions of

*l*

^{1}(

*E*

_{S}) on

*l*

^{1}(

*S*) and prove certain module amenability results. In this section we do not restrict ourself to any particular action. This distinguishes our results from that of [8].

### Remark 4.1

Throughout this section we suppose that *S* is an inverse semigroup for which *E*_{S} is finite.

Selivanov showed that for any locally compact group *G*, the Banach algebra *L*^{1}(*G*) is super amenable if and only if *G* is finite [12]. The following theorem is the module version of Selivanov’s result for inverse semigroups. This is quite different from Theorem 3.7 in [8], as the latter is proved for a specific action of *E*_{S} on *S* which makes a homomorphic image of *S* into a discrete group. Here we prove the same result for an arbitrary action under some conditions.

### Theorem 4.2

*Let**l*^{1}(*S*) *be a pseudo*-*unital Banach**l*^{1}(*E*_{S})-*module*. *If**l*^{1}(*S*)/*J**is a commutative Banach**l*^{1}(*E*_{S})-*module and**J**is finite*-*dimensional*, *then**l*^{1}(*S*) *is super module amenable if and only if**S**is finite*.

### Proof

Since *E*_{S} is a finite semilattice, *l*^{1}(*E*_{S}) is a finite dimensional amenable Banach algebra, by Theorem 8 of [4]. From Lemma 3.7, it follows that *l*^{1}(*E*_{S}) is super amenable. Since each semigroup algebra has the approximation property [11] and *J* is complemented in *l*^{1}(*S*), so *l*^{1}(*S*)/*J* has the approximation property. Suppose that *l*^{1}(*S*) is super module amenable, from Corollary 3.6 it follows that *l*^{1}(*S*)/*J* is finite-dimensional and so is *l*^{1}(*S*). Thus, *S* is finite.

Conversely, if *S* is finite, then all maximal subgroups of *S* are amenable. From Theorem 8 of [4], we conclude that *l*^{1}(*S*) is amenable and so it is super amenable. Also super amenability of *l*^{1}(*E*_{S}) implies that *l*^{1}(*E*_{S}) has an identity, hence super module amenability of *l*^{1}(*S*) follows from Theorem 3.3. □

Ghahremani, Loy and Willis in [5] showed that if *G* is a locally compact group, then *L*^{1}(*G*)^{∗∗} is amenable if and only if *G* is finite.

Now by Lemma 3.7, we conclude that *L*^{1}(*G*)^{∗∗} is super amenable if and only if *G* is finite.

The following theorem can be regarded as a module version of this result for inverse semigroups. Again in contrast to Corollary 2.13 in [8] we do not assume any specific action.

### Theorem 4.3

*Let the hypothesis of preceding theorem holds*, *then**l*^{1}(*S*)^{∗∗}*is super module amenable if and only if**S**is finite*.

### Proof

If *l*^{1}(*S*)^{∗∗} is super module amenable, then so is (*l*^{1}(*S*)/*J*)^{∗∗}≅*l*^{1}(*S*)^{∗∗}/*J*^{⊥⊥}. Since *l*^{1}(*S*)/*J* is a commutative *l*^{1}(*E*_{S})-module, from Proposition 3.1 it follows that *l*^{1}(*S*)/*J* is super module amenable. Therefore *l*^{1}(*S*)/*J* is finite-dimensional by Corollary 3.6, hence *S* is finite.

The converse follows from Theorem 4.2. □

### Definition 4.4

A Banach algebra \(\mathcal{A}\) is called module biprojective if \(\widetilde{\omega}\) has a bounded right inverse which is also \(\mathcal{A}/J\)-\(\mathfrak{A}\)-module homomorphism.

Now we give the main result of this section for discrete inverse semigroup *S*, that present the module version of known result of Helemskii in [6].

### Corollary 4.5

*Let*

*l*

^{1}(

*S*)

*be a pseudo*-

*unital Banach*

*l*

^{1}(

*E*

_{S})-

*module with commutative compatible actions*.

*Then the following statements are equivalent*:

- (i)
*S**is finite*. - (ii)
*l*^{1}(*S*)*is super module amenable*. - (iii)
*l*^{1}(*S*)^{∗∗}*is super module amenable*. - (iv)
*l*^{1}(*S*)*is module Arens regular*.

*In the case where*

*l*

^{1}(

*S*)

*is unital*,

*then the above statements are equivalent to the following*:

- (v)
*l*^{1}(*S*)*is module biprojective*.

### Proof

By commutativity of compatible actions we have *J*=0 and \({\mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\,{\mathcal{A}}\) is a commutative Banach \(\mathfrak{A}\)-module. Therefore (i), (ii) and (iii) are equivalent by Theorems 4.2 and 4.3. Since *E*_{S} is finite, we conclude that *l*^{1}(*S*) is Arens regular if and only if *S* is finite [9]. Also *J*=0 implies that *J*^{⊥}=*l*^{1}(*S*)^{∗}, it follows that *l*^{1}(*S*) is Arens regular if and only if it is module Arens regular. This shows that (i) and (iv) are equivalent. Now let *l*^{1}(*S*) be unital, then equivalently of (ii) and (v) follows from Proposition 3.9. □

*l*

^{1}(

*S*) were only known for the very specific compatible actions

*δ*

_{e}.

*δ*

_{s}=

*δ*

_{s},

*δ*

_{s}.

*δ*

_{e}=

*δ*

_{se}(see Theorems 3.7 in [8] and 3.3 in [10]). The above corollary shows that these results could be applied to arbitrary commutative actions as well. For instance, let

*S*be a discrete Clifford semigroup such that

*E*

_{S}is finite. If

*l*

^{1}(

*E*

_{S}) acts on

*l*

^{1}(

*S*) by the following compatible actions

*l*

^{1}(

*S*) is a pseudo-unital Banach

*l*

^{1}(

*E*

_{S})-module and the above four conditions are equivalent.

*S*is infinite and each element of

*S*is an idempotent. Since

*l*

^{1}(

*S*) is commutative, then it is a commutative Banach

*l*

^{1}(

*E*

_{S})-module with the above compatible actions. Let

*X*be a commutative Banach

*l*

^{1}(

*S*)-

*l*

^{1}(

*E*

_{S})-module and

*D*be a module derivation from

*l*

^{1}(

*S*) into

*X*. Then as in the classical case, without loss of generality we may assume that

*X*is a pseudo-unital Banach ł

^{1}(

*S*)-bimodule. Since

*δ*

_{1}is an identity for

*l*

^{1}(

*S*), we have

*D*is zero. Therefore \(\mathcal{A}\) is super module amenable and module biprojective. This example shows that Corollary 4.5 is not valid, when the set of idempotents of

*S*is infinite.

## Acknowledgements

The authors would like to thank the referee for careful reading. We are grateful to the office of graduate studies of the University of Isfahan for their support.