#### Abstract

We study the concept of -module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. Also, we compare the notions of -amenability and -module amenability of Banach algebras. As a consequence, we show that, if is an inverse semigroup with finite set of idempotents and is a commutative Banach -module, then is -module amenable if and only if is finite, when is an epimorphism. Indeed, we have generalized a well-known result due to Ghahramani et al. (1996).

#### 1. Introduction

The concept of amenability for Banach algebras was first introduced by Johnson in [1]. For a locally compact group , Ghahramani et al. showed that is amenable if and only if is finite [2]. The notion of module amenability for a Banach algebra, which is a Banach module over another Banach algebra with compatible actions, was introduced and studied by the third author in [3]. The notion of -module amenability was introduced by Bodaghi in [4]; he obtained some results for a specific compatible action (i.e., trivial left action). In [5, 6], the authors investigated the module amenability of the second dual of the semigroup Banach algebra , for an inverse semigroup with the set of idempotents . They showed that is module amenable, if and only if an appropriate group homomorphic image of is finite, when acts on by the compatible actions and , for and . Indeed, for the very specific compatible actions, they presented a generalization of the result due to Ghahramani et al. (in the discrete case).

The aim of this paper is to investigate the structure of -module amenable Banach algebras (we do not restrict ourselves to some specific compatible actions). In particular, we give the generalization of the result of Ghahramani et al. for arbitrary commutative compatible actions. The paper is organized as follows. In Section 1 we give the definitions which are needed throughout the paper. In Section 2 we introduce the notions of -module virtual diagonal and -module approximate diagonal and study the structure of -module amenable Banach algebras. We also find relations between -module amenability and -amenability (that generalize the concepts of module amenability and amenability, respectively) without the extra assumption that the compatible action is trivial from one direction, or the assumption that has a bounded approximate identity for . We assume that is either idempotent or surjective. The former is used to ensure that fixes points of its range. The latter is used in particular in Proposition 10 (and then in Theorem 13) to ensure that a -module approximate diagonal is also a module approximate diagonal.

In Section 3 we apply main results of Section 2 to semigroup Banach algebras.

#### 2. Preliminaries

Let be a Banach algebra and let be a endomorphism on . Suppose that is a Banach -bimodule. A bounded linear map is called a -derivation if For each , we define the -derivation by These are called -inner derivations. The Banach algebra is called -amenable if, for any Banach -bimodule , every -derivation from to is -inner.

Throughout this paper, and are Banach algebras such that is a Banach -bimodule with compatible actions; that is Let be a Banach -bimodule and a Banach -bimodule with compatible actions; that is for , , , and similarly for the right or two-side actions. Then is called a Banach --module. If, moreover, then is called a commutative Banach --module.

It is obvious that, if is a (commutative) Banach --module, then so is under the following compatible actions: and similarly for the right actions.

Note that, when acts on itself by algebra multiplication, it need not be a Banach --module, as we have not assumed the compatibility condition for and . But when is a commutative -module and acts on itself by multiplication from both sides, then it is a commutative Banach --module.

Let and be -modules. A continuous mapping is called an -module morphism if also, We denote the space of all such -module morphisms by and denote by .

Let , and be as above and let . A bounded map is called a *-module derivation* if

also, Note that is bounded if there exist such that (), although is not necessarily linear, but still its boundedness implies its norm continuity. Let and if we define as in (2); then is a -module derivation that is called a -module inner derivation.

The Banach algebra is called *-module amenable* if, for any commutative Banach --module , each -module derivation form to is -module inner.

We note that, if is the identity map on , then -module amenability is the same as module amenability. Also, when , everything reduces to the classical case.

#### 3. -Amenability and -Module Amenability

Throughout this section is a Banach algebra, is a Banach -module with compatible actions, and , unless otherwise specified. We start this section by the following lemma, which is proved similar to Proposition in [7].

Lemma 1. *Let be a commutative Banach --module. Then every -module derivation from to is -module inner, when one of the following is satisfied. *(i)* has a bounded right approximate identity and .*(ii)* has a bounded left approximate identity and .*

*Definition 2. *Let be a Banach algebra. A Banach -bimodule is called -pseudo-unital if

The proof of the following proposition is routine, but we give it for the sake of completeness.

Proposition 3. *Let be idempotent or surjective and let have a bounded approximate identity. Suppose that, for any commutative Banach --module which is -pseudo-unital, each -module derivation form to is -module inner. Then is -module amenable.*

*Proof. *Let be a commutative Banach --module and let be a -module derivation. Let , , and . Let be the restriction map . In the case where is idempotent, then we turn into an another commutative Banach --module, by letting the same actions of and the following actions of :
Also, in the above actions, is again a -module derivation. By Cohen’s factorization theorem, and are closed --submodules of (with respect to the module actions).

Let be a -module derivation; then so is . Since is -pseudo-unital, there is such that . Choose such that and consider . Then is a -module derivation. Therefore, there is such that , by Lemma 1. Thus, . Hence, any -module derivation from into is -module inner.

Now, by the above assertion, let such that . From Hahn-Banach theorem, we obtain an extension of , so that is a -module derivation. Since , there is such that . Let . In the case where is an idempotent, we have
Therefore , where is idempotent or surjective (similarly). Consequently, is -module inner.

Let be a Banach algebra with a bounded approximate identity and let be idempotent or surjective. Consider ; then is automatically a commutative Banach -module. Also, -derivations and -module derivations are the same; hence -module amenability is the same as -amenability for . Consequently, is -amenable if and only if, for any Banach -bimodule which is -pseudo-unital, each -derivation form to is -inner, by Proposition 3.

Proposition 4. *Let be a commutative Banach -module. If is -module amenable, then has a bounded approximate identity for .*

*Proof. *Consider , then is a commutative Banach --module, with the same actions of and the following actions of :
Let be the canonical embedding of into its second dual. Then is a -module derivation. Thus, there is in such that for all . Now, as the proof of Proposition in [7], we can obtain a bounded net such that it is an approximate identity for .

Lemma 5. *Let be linear and idempotent or surjective, let be a commutative Banach -module, and let be a -derivation for some -pseudo-unital Banach -bimodule . If has a bounded approximate identity such that and are convergent to , then there is a commutative Banach --module such that is a -module derivation.*

*Proof. *Let be the -closed linear span of the following set:
In the case where is idempotent, we turn into an another Banach -bimodule via , as follows. Since is -pseudo-unital, we conclude that is a Banach -submodule of such that . Let be a Banach -bimodule such that , which exists by Exercise 2.1.2 of [7]. For , let , and be such that . For , define
We claim that is well defined; that is, it is independent of the choices of , and . Let , and such that . Then, for each , we have
similarly, is well defined. Clearly, by the above actions of and the given actions of , is a Banach --module. For and , we have
similarly, . Thus is a Banach -submodule of . So is a Banach --module. For all and in , is an element of , so for each and , we have
similarly, . Also, -commutativity of , implies that
Thus, by linearity and the -continuity of the compatible actions, for and
thus . Therefore, is a commutative Banach --module. Also,
Consequently, is a -module derivation.

Proposition 6. *Let be a commutative Banach -module with a bounded approximate identity . Suppose that is -module amenable and has a -amenable, closed subalgebra such that . Then is -amenable, when is linear and idempotent or surjective.*

*Proof. *Let be a Banach -bimodule and be a -derivation, without loss of generality; we may suppose that is -pseudo-unital. By -amenability of , there is such that
Let . Then is a -derivation such that , for and . Let ; then both and are bounded approximate identities for .

In the case where is idempotent, for , we have
thus, . Moreover, since is -pseudo-unital, we obtain that . Similarly , for all .

Therefore, if is idempotent or surjective, then and are convergent to , for all and . Thus, by Lemma 5, there is a commutative Banach --module such that is a -module derivation. Hence, there is such that . Since, by in the proof of Lemma 5, is an -submodule of (via , in the case is idempotent), we obtain that .

Theorem 7. *Let be an epimorphism or an idempotent homomorphism. Suppose that is a unital, commutative Banach -module and is amenable. Then -module amenability of implies its -amenability.*

*Proof. *Suppose that is an identity for . Let be the closed linear span of . Since is an identity for , is a closed subalgebra of under the following multiplication:
Let be defined by , for . Then is a continuous homomorphism and is dense in . Hence is amenable, by Proposition of [7]. By definition of , we have that is an endomorphism on . Therefore, is -amenable (by Corollary 2.2 in [8]) and satisfies conditions of Proposition 6.

Let be the projective tensor product of by itself. Then is a Banach --module with the canonical actions [5]. Consider the closed ideal of generated by elements of the form , for and . Let be the closed ideal of generated by elements of the form , for and . It is clear that and are both -submodules and -submodules of and , respectively. Hence, the module projective tensor product [9] and the quotient Banach algebra are both Banach -modules and Banach -modules. Define by and by , extended by linearity and continuity. Clearly, is an -module homomorphism and an -module homomorphism.

Suppose that and is a closed ideal of such that . Then we may define by . In particular, for all and that is, . Therefore, we can define .

In the remainder of this section, we use to denote the coset of in .

Lemma 8. * is -module amenable if and only if is -module amenable.*

*Proof. *Let be -module amenable. Suppose that is a commutative Banach --module and is a -module derivation. Clearly , so is a commutative Banach --module, by the same actions of and and (). For and , we have . Hence, vanishes on and induces a map from into which is clearly a -module derivation. Hence , for some in . Thus,
Consequently, .

The converse follows from Proposition 2.5 in [4].

Now, we define the concepts of -module virtual diagonal and -module approximate diagonal as a generalization of the earlier notions of virtual diagonal and approximate diagonal

*Definition 9. *Let . (i)An element is called a -module virtual diagonal for if
(ii)A bounded net in is called a -module approximate diagonal for if is a bounded approximate identity for and

We note that, if is the identity map, then -module virtual (or approximate) diagonal is the same as module virtual (or approximate) diagonal [6]. Moreover, in the case where , -module virtual (or approximate) diagonal and virtual (or approximate) diagonal coincide.

The next proposition follows from Corollary 2.3 of [4] and Theorem 2.1 of [3].

Proposition 10. *If has a -module approximate diagonal such that is surjective, then is -module amenable.*

Proposition 11. *Let be a commutative Banach --module. If is -module amenable and has a bounded approximate identity, then has a -module virtual diagonal.*

*Proof. *Let be a bounded approximate identity for and let in be a -accumulation point of . Hence,
Thus, is a -module derivation into . Since is a commutative Banach --module, so is . By -module amenability of , there is such that . Consequently, it is clear that is a -module virtual diagonal for .

Lemma 12. * has a -module virtual diagonal if and only if it has a -module approximate diagonal.*

*Proof. *This is essentially the same as the proof of Lemma of [10].

In Proposition 2.1 of [3] and Proposition 3.3 of [6], the authors proved that module amenability of follows from amenability of and , respectively, under the strong condition that has a bounded approximate identity for . According to Lemma 8, we present the generalization of Proposition 2.1 of [3] and Proposition 3.3 of [6] without the extra assumption that has a bounded approximate identity for . Indeed, (as an application of the following theorem) we show that the class of amenable Banach algebras is contained in the class of module amenable Banach algebras.

Theorem 13. *Let be a Banach -module and let be linear. Then -module amenability of follows from its -amenability, when one of the following is satisfied:*(i)* is surjective.*(ii)* is an idempotent and is unital.*

*Proof. *(i) Since is linear, -amenability of implies that is -module amenable as a commutative Banach -module. Also, automatically is a commutative Banach --module. Therefore from Proposition 4 and 2.11, and Lemma 12, there is a bounded net in such that is a bounded approximate identity for and , for all . Now define in by . Then it is clear that is a -module approximate diagonal for . Consequently is -module amenable, by Proposition 10.

(ii) Let be a commutative Banach --module which is -pseudo-unital and let be a -module derivation. Let be an identity for . Clearly is zero and for , additivity of implies that . Thus, . Hence, by continuity of , we have . Moreover,
Thus, and therefore . Now, it is routinely checked that is linear. Consequently is -module inner, by -amenability of .

Lemma 14. *Let be a closed ideal and an -submodule of such that . If is -module amenable and is -module amenable, then is -module amenable.*

*Proof. *Let be a commutative Banach --module. Suppose that is the space of all elements such that and is the subspace of generated by . Since , the following module actions are well defined
and similar for the right actions. Therefore, is a commutative Banach --module and so is .

Now, let be a -module derivation. Consider such that and let . Since vanishes on , so it induces a -module derivation from to , which we denote likewise by . Also, for all and we obtain that . Hence, . Therefore, -module amenability of implies that for some . Consequently, .

Now we are ready to prove the main result of this section. In Theorem 13 we obtained sufficient conditions that -amenability of implies -module amenability of . The next corollary together with Theorem 7 may be considered as the converse of Theorem 13.

Theorem 15. *Let be a commutative Banach -module and let be an epimorphism. Then -module amenability of implies its -amenability, when is commutative and amenable.*

*Proof. *First we suppose that has an identity for itself and . Consider with the following multiplication:
It is straightforward that is a unital Banach algebra with the norm algebra and is a closed ideal of . Also, is a commutative Banach -module with the following compatible actions:
Define
then such that . Since is amenable, it is -amenable and -module amenability of follows from Theorem 13. Therefore is -module amenable, by Lemma 14. Hence, Theorem 7 implies that is -amenable. Now, by Proposition 3.1 of [11] and Proposition 4, we obtain that is -amenable.

In the case is not unital we consider as the unitization of . We also define the compatible actions of on that extend the compatible actions of on , by letting
Then is a commutative Banach -module and is an identity for the actions on . Also, . Since , any -module derivation on where is a Banach -module is a -module derivation on where is a Banach -module. Therefore, if is -module amenable as a Banach -module, then it is -module amenable as a Banach -module. Consequently is -amenable, by the first case.

Let be the inverse semigroup of positive integers with maximum operation. Then is not amenable, by Theorem 2 of [12]. On the other hand, as in the proof of the last example of [13], we obtain that is module amenable on itself by the multiplication algebra. Consequently, Theorem 7 and Theorem 15 are not valid, when is not amenable.

#### 4. Semigroup Algebras

Recall that a discrete semigroup is called an* inverse semigroup* if for each there is a unique element such that and . Elements of the form are called idempotents of and form a commutative subsemigroup . An inverse semigroup whose idempotents are in the center is called a* Clifford semigroup*.

The Banach algebra could be regarded as a subalgebra of (see [14]) and thereby is a Banach algebra and a Banach -module with proper compatible actions. It is possible to consider arbitrary actions of on and prove certain module amenability results. Here we do not restrict ourselves to any particular action.

In the following theorem, we generalize the well-known result of Ghahramani et al. (in the case discrete), which assert that is finite if and only if is amenable, when is a locally compact group.

Theorem 16. *Let be an inverse semigroup with set of idempotents , let be a commutative Banach -module, and let be an epimorphism. Assume that is amenable as a Banach algebra. Then is -module amenable if and only if is finite.*

*Proof. *Since is -dense in and the compatible actions are -continuous, is a commutative Banach -module and is an epimorphism. If is -module amenable, then -amenability of follows from Theorem 15 and its amenability follows from Proposition 2.3 of [8]. Therefore is finite, by Theorem 11.8 of [15].

Conversely, if is finite, then is amenable and so it is -amenable. Consequently, -module amenability of follows from Theorem 13.

In the main results of [5, 6] (see Theorem 3.4 and Theorem 2.11, resp.), the authors studied the module amenability of , when is a Banach -module with very specific compatible actions. Also, in [13] we studied the super module amenability of , when is a commutative Banach -module with some commutative compatible actions that is pseudo-unital (see Corollary 3.5 of [13]). Now, in the following corollary, we investigate the module amenability of , when is a commutative Banach -module with arbitrary commutative compatible actions.

Corollary 17. *Let be an inverse semigroup with finitely many idempotents. If is a commutative Banach -module, then is module amenable if and only if is finite.*

*Proof. *This is immediate from Theorem 16, when is the identity map on .

Let be a Clifford semigroup. Given , consider the following commutative compatible actions: or Consequently, there are large extra numbers of commutative compatible actions that turn into a commutative Banach -module (note that, with the above second actions, is not necessary pseudo-unital. For instance, if we let be a discrete group, then the second actions above are zero).

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

Massoud Amini was partly supported by a Grant from IPM (no. 90430215).