The Structure of 𝜑 -Module Amenable Banach Algebras

We study the concept of 𝜑 -module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. Also, we compare the notions of 𝜑 -amenability and 𝜑 -module amenability of Banach algebras. As a consequence, we show that, if 𝑆 is an inverse semigroup with finite set 𝐸 of idempotents and 𝑙 1 (𝑆) is a commutative Banach 𝑙 1 (𝐸) module, then 𝑙 1 (𝑆) ∗∗ is 𝜑 ∗∗ -module amenable if and only if 𝑆 is finite, when 𝜑 ∈ Hom 𝑙 1 (𝐸) (𝑙 1 (𝑆)) is an epimorphism. Indeed, we have generalized a well-known result due to Ghahramani et al. (1996).


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 1 ( ) * * is amenable if and only if is finite [2]. The notion of module amenability for a Banach algebra, A which is a Banach module over another Banach algebra A 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 1 ( ) * * of the semigroup Banach algebra 1 ( ), for an inverse semigroup with the set of idempotents . They showed that 1 ( ) * * is module amenable, if and only if an appropriate group homomorphic image / ≈ of is finite, when A := 1 ( ) acts on A := 1 ( ) 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 ofmodule 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 A has a bounded approximate identity for A. 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 amodule approximate diagonal is also a module approximate diagonal.
In Section 3 we apply main results of Section 2 to semigroup Banach algebras.
For each ∈ , we define the -derivation by These are called -inner derivations. The Banach algebra A is called -amenable if, for any Banach A-bimodule , every -derivation from A to * is -inner. Throughout this paper, A and A are Banach algebras such that A is a Banach A-bimodule with compatible actions; that is Let be a Banach A-bimodule and a Banach A-bimodule with compatible actions; that is for ∈ A, ∈ A, ∈ , and similarly for the right or two-side actions. Then is called a Banach A-A-module. If, moreover, then is called a commutative Banach A-A-module.
It is obvious that, if is a (commutative) Banach A-A-module, then so is * under the following compatible actions: and similarly for the right actions. Note that, when A acts on itself by algebra multiplication, it need not be a Banach A-A-module, as we have not assumed the compatibility condition ( ⋅ ) ⋅ = ⋅ ( ⋅ ) for ∈ A and , ∈ A. But when A is a commutative A-module and acts on itself by multiplication from both sides, then it is a commutative Banach A-A-module.
Let A and B be A-modules. A continuous mapping : also, We denote the space of all such A-module morphisms by Hom A (A, B) and denote Hom A (A, A) by Hom A (A). Let A, A and be as above and let ∈ Hom A (A). A bounded map : A → is called a -module derivation if ( ± ) = ( ) ± ( ) , also, Note that : A → is bounded if there exist > 0 such that ‖ ( )‖ ≤ ‖ ‖ ( ∈ A), although is not necessarily linear, but still its boundedness implies its norm continuity. Let ∈ Hom A (A) and ∈ if we define as in (2); then is a -module derivation that is called a -module inner derivation.
The Banach algebra A is called -module amenable if, for any commutative Banach A-A-module , each -module derivation form A to * is -module inner.
We note that, if is the identity map on A, then Amodule amenability is the same as module amenability. Also, when A := C, everything reduces to the classical case.

-Amenability and -Module Amenability
Throughout this section A is a Banach algebra, A is a Banach A-module with compatible actions, and ∈ Hom A (A), unless otherwise specified. We start this section by the following lemma, which is proved similar to Proposition 2.1.3 in [7].

Lemma 1.
Let be a commutative Banach A-A-module. Then every -module derivation from A to * is -module inner, when one of the following is satisfied. (ii) A has a bounded left approximate identity and ⋅ (A) = 0.
The proof of the following proposition is routine, but we give it for the sake of completeness. In the case where is idempotent, then we turn into an another commutative Banach A-Amodule, by letting the same actions of A and the following actions of A: Also, in the above actions, is again a -module derivation. By Cohen's factorization theorem, 1 and 2 are closed A-A-submodules of (with respect to the module actions). Let : A → * 2 be a -module derivation; then so is 1 ∘ : Now, by the above assertion, let 2 ∈ * 2 such that 2 ∘ = 2 . From Hahn-Banach theorem, we obtain an extension In the case where is an idempotent, we have Therefore = ℎ , where is idempotent or surjective (similarly). Consequently, is -module inner.
Let B be a Banach algebra with a bounded approximate identity and let ∈ Hom C (B) be idempotent or surjective. Consider A := C; then B is automatically a commutative Banach C-module. Also, -derivations and -module derivations are the same; hence -module amenability is the same as -amenability for B. Consequently, B is -amenable if and only if, for any Banach B-bimodule which ispseudo-unital, each -derivation form B to * is -inner, by Proposition 3.

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

Proof. Consider
:= A, then is a commutative Banach A-A-module, with the same actions of A and the following actions of A: Let : A → A * * be the canonical embedding of A into its second dual. Then ∘ is a -module derivation. Thus, there is in * * such that ( ) = ( ) ⋅ for all ∈ A. Now, as the proof of Proposition 2.2.1 in [7], we can obtain a bounded net ( ) ⊆ A such that it is an approximate identity for (A).

Lemma 5. Let be linear and idempotent or surjective, let A be a commutative Banach A-module, and let : A → * be a -derivation for some -pseudo-unital Banach A-bimodule
. If (A) has a bounded approximate identity ( ( )) such that ( ( ) ⋅ ( ⋅ )) and ( ( ⋅ ) ⋅ ( )) are convergent to Proof. Let be the * -closed linear span of the following set: In the case where is idempotent, we turn into an another Banach A-bimodule via , as follows. Since is -pseudounital, we conclude that is a Banach A-submodule of * such that (A) ⊆ . Let be a Banach A-bimodule such that * = , which exists by Exercise 2.1.2 of [7]. For ∈ , let , ∈ A, and ∈ be such that = ( ) ⋅ ⋅ ( ). For ∈ A, define We claim that • is well defined; that is, it is independent of the choices of , , and . Let , ∈ A, and ∈ such that = ( ) ⋅ ⋅ ( ). Then, for each ∈ A, we have similarly, • is well defined. Clearly, by the above actions of A and the given actions of A, is a Banach A-A-module.
For ∈ A and ∈ , we have thus • = • . Therefore, is a commutative Banach A-A-module. Also, Consequently, : A → * is a -module derivation.

Proposition 6. Let A be a commutative Banach A-module with a bounded approximate identity ( ) . Suppose that
A is -module amenable and has a | -amenable, closed subalgebra such that ( ⋅ ) ∈ ( ∈ A). Then A isamenable, when is linear and idempotent or surjective.
In the case where is idempotent, for ∈ A, we havẽ Therefore, if is idempotent or surjective, then (̃( ) ⋅ ( ⋅ )) and ( ( ⋅ )⋅̃( )) are convergent tõ( ⋅ ), for all ∈ A and ∈ A. Thus, by Lemma 5, there is a commutative Banach A-A-module such that̃: A → * is a -module derivation. Hence, there is ∈ * such that̃= . Since, by in the proof of Lemma 5, * is an A-submodule of * (via , in the case is idempotent), we obtain that = + .

Theorem 7.
Let ∈ Hom A (A) be an epimorphism or an idempotent homomorphism. Suppose that A is a unital, commutative Banach A-module and A is amenable. Thenmodule amenability of A implies its -amenability.
Proof. Suppose that is an identity for A. Let be the closed linear span of { ⋅ ( ) : ∈ A}. Since ( ) is an identity for (A), is a closed subalgebra of A under the following multiplication:
Let A⊗A be the projective tensor product of A by itself. Then A⊗A is a Banach A-A-module with the canonical actions [5]. Consider the closed ideal I of A⊗A generated by elements of the form ⋅ ⊗ − ⊗ ⋅ , for , ∈ A and ∈ A. Let J be the closed ideal of A generated by elements of the form ( ⋅ ) − ( ⋅ ), for , ∈ A and ∈ A. It is clear that J and I are both A-submodules and A-submodules of A and A⊗A, respectively. Hence, the module projective tensor product A⊗ A A ≅ (A⊗A)/I [9] and the quotient Banach algebra A/J are both Banach A-modules and Banach Amodules. Define : A⊗A → A by ( ⊗ ) = and : A⊗ A A → A/J bỹ( ⊗ + I) = ( ⊗ ) + J, extended by linearity and continuity. Clearly,̃is an Amodule homomorphism and an A-module homomorphism.
Suppose that ∈ Hom A (A) and is a closed ideal of A such that ( ) ⊆ . Then we may define : A/ → A/ by ( + ) = ( ) + . In particular, for all , ∈ A and ∈ A that is, (J) ⊆ J. Therefore, we can define J : A/J → A/J. In the remainder of this section, we use to denote the coset of ∈ A in A/J.

Lemma 8. A is -module amenable if and only if
Proof. Let A/J be J -module amenable. Suppose that is a commutative Banach A-A-module and : A → * is a -module derivation. Clearly J ⋅ = ⋅ J = 0, so is a commutative Banach A/J-A-module, by the same actions of A and • = ⋅ and • = ⋅ ( ∈ A). For , ∈ A and ∈ A, we have (( ⋅ ) ) = ( ( ⋅ )). Hence, vanishes on J and induces a map̃from A/J into * which is clearly a J -module derivation. Hencẽ= J , 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 We note that, if is the identity map, then id A -module virtual (or approximate) diagonal is the same as module virtual (or approximate) diagonal [6]. Moreover, in the case where A := C, id A -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 11. Let A⊗ A A be a commutative Banach A-Amodule. If A is -module amenable and
Thus, +I ⊥⊥ : A → (A⊗ A A) * * is a -module derivation into ker̃ * * ≅ (ker̃) * * . Since A⊗ A A is a commutative Banach A-A-module, so is ker̃. By -module amenability of A, there is + I ⊥⊥ ∈ (ker̃) * * such that +I ⊥⊥ = +I ⊥⊥ . Consequently, it is clear that ( − ) + I ⊥⊥ is a -module virtual diagonal for A.

Lemma 12. A 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 2.9.64 of [10].
In Proposition 2.1 of [3] and Proposition 3.3 of [6], the authors proved that module amenability of A follows from amenability of A and A/J, respectively, under the strong condition that A has a bounded approximate identity for A. 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 A has a bounded approximate identity for A. 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 A be a Banach A-module and let
∈ Hom A (A) be linear. Then -module amenability of A follows from its -amenability, when one of the following is satisfied: (ii) is an idempotent and A is unital.
Proof. (i) Since is linear, -amenability of A implies that A is -module amenable as a commutative Banach C-module. Also, automatically A⊗ C A = A⊗A is a commutative Banach A-C-module. Therefore from Propositions 4 and 11, and Lemma 12, there is a bounded net (m ) in A⊗A such that ( m ) is a bounded approximate identity for (A) and ( )⋅ m − m ⋅ ( ) → 0, for all ∈ A. Now define (m ) in A⊗ A A by m = m + I. Then it is clear that (m ) is amodule approximate diagonal for A. Consequently A ismodule amenable, by Proposition 10.

Lemma 14. Let be a closed ideal and an
and similar for the right actions. Therefore, / is a commutative Banach A/ -A-module and so is ≅ ( / ) * . Now, let : A → * be a -module derivation.
Now we are ready to prove the main result of this section. In Theorem 13 we obtained sufficient conditions thatamenability of A implies -module amenability of A. The next corollary together with Theorem 7 may be considered as the converse of Theorem 13.

Theorem 15. Let A be a commutative Banach A-module and let
∈ Hom A (A) be an epimorphism. Thenmodule amenability of A implies its -amenability, when A is commutative and amenable.
Proof. First we suppose that A has an identity for itself and A. Consider B = A ⊕ A with the following multiplication: It is straightforward that B is a unital Banach algebra with the norm algebra ‖ + ‖ := ‖ ‖ + ‖ ‖ and A is a closed ideal of B. Also, B is a commutative Banach A-module with the following compatible actions: Define then ∈ Hom A (B) such that | A = . Since B/A ≅ A is amenable, it is A -amenable and A -module amenability of B/A follows from Theorem 13. Therefore B ismodule amenable, by Lemma 14. Hence, Theorem 7 implies that B is -amenable. Now, by Proposition 3.1 of [11] and Proposition 4, we obtain that A is -amenable.
In the case A is not unital we consider A ♯ as the unitization of A. We also define the compatible actions of A ♯ on A that extend the compatible actions of A on A, by letting Then A is a commutative Banach A ♯ -module and is an identity for the actions on A. Also, ∈ Hom A ♯ (A). Since A ⊆ A ♯ , any -module derivation on A where A is a Banach A ♯module is a -module derivation on A where A is a Banach A-module. Therefore, if A is -module amenable as a Banach A-module, then it is -module amenable as a Banach A ♯module. Consequently A is -amenable, by the first case.
Let = (N, ∨) be the inverse semigroup of positive integers with maximum operation. Then A := 1 ( ) 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 A is module amenable on itself by the multiplication algebra. Consequently, Theorem 7 and Theorem 15 are not valid, when A is not amenable.

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 1 ( ) could be regarded as a subalgebra of 1 ( ) (see [14]) and thereby 1 ( ) is a Banach algebra and a Banach 1 ( )-module with proper compatible actions. It is possible to consider arbitrary actions of 1 ( ) on 1 ( ) 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 1 ( ) * * is amenable, when is a locally compact group.