Semigroup Forum

, Volume 86, Issue 2, pp 279–288

Super module amenability of inverse semigroup algebras

Authors

  • M. Lashkarizadeh Bami
    • Department of MathematicsUniversity of Isfahan
    • Department of MathematicsUniversity of Isfahan
  • M. Amini
    • Department of Mathematics, Faculty of Mathematical SciencesTarbiat Modares University
    • School of MathematicsInstitute for Research in Fundamental Sciences (IPM)
RESEARCH ARTICLE

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

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 l1(ES) on l1(S) for an inverse semigroup S with the set of idempotents ES and show that under certain conditions, l1(S) is super module amenable if and only if S is finite. We also study the super module amenability of l1(S)∗∗ and module biprojectivity of l1(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 δe.δs=δs and δs.δe=δse, for sS and eES, where δx denotes the point mass at x. Then it is shown that l1(S) is super module amenable if and only if l1(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 l1(S) is weak module amenable, when S is a commutative inverse semigroup and a l1(ES) acts on l1(S) by the compatible actions δe.δs=δs.δe=δse, for sS, eES.

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 [13, 8, 10]. We investigate super module amenability and the module biprojectivity of l1(S) for arbitrary compatible actions of l1(ES), when ES 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 ES. We show that if l1(S) is a pseudo-unital Banach l1(ES)-module and a commutative Banach l1(ES)-module under an arbitrary module action, then l1(S) is super module amenable if and only if l1(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 L1(G) is super-amenable if and only if G is finite. We also prove that if l1(S) is unital and a pseudo-unital Banach l1(ES)-module with arbitrary commutative compatible actions, then l1(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 l1(G) is biprojective if and only if G is finite.

2 Preliminaries

Throughout this paper, \(\mathcal{A}\) and \(\mathfrak{A}\) are Banach algebras such that \(\mathcal{A}\) is a Banach \(\mathfrak{A}\)-bimodule with compatible actions
$$\alpha.(ab)=(\alpha.a)b,\qquad (ab).\alpha=a(b.\alpha)\quad(\alpha \in \mathfrak{A},\ a,b\in \mathcal{A}). $$
Let X be a Banach \(\mathcal{A}\)-bimodule and a Banach \(\mathfrak{A}\)-bimodule with compatible actions
$$\alpha.(a.x)=(\alpha.a)x,\qquad a.(\alpha.x)=(a.\alpha).x,\qquad (\alpha.x).a= \alpha.(x.a), $$
for \(\alpha \in \mathfrak{A}\), \(a\in {\mathcal{A}}\), xX and similarly for the right or two-side actions. We call X a Banach \(\mathcal{A}\)-\(\mathfrak{A}\)-module. If moreover,
$$\alpha.x=x.\alpha\quad (\alpha\in{\mathfrak{A}},\ x\in X), $$
then X is called a commutative Banach \(\mathcal{A}\)-\(\mathfrak{A}\)-module.
It is obvious that if X is a commutative Banach \(\mathcal{A}\)-\(\mathfrak{A}\)-module, then so is X under the actions
$$\langle \alpha.\varphi,x\rangle=\langle\varphi,x.\alpha\rangle,\qquad \langle a. \varphi,x\rangle=\langle \varphi,x.a\rangle\quad \bigl(\alpha\in{\mathfrak{A}},\ a \in {\mathcal{A}},\ \varphi\in X^*\bigr), $$
and similarly for the right actions. Note that when \(\mathcal{A}\) acts on itself by algebra multiplication, it need not be a Banach \(\mathcal{A}\)-\(\mathfrak{A}\)-module, as we have not assumed the compatibility condition (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.
Let \({\mathcal{A}}\,\widehat{\otimes}\, {\mathcal{A}}\) be the projective tensor product of \(\mathcal{A}\) by itself. Then \({\mathcal{A}}\,\widehat{\otimes}\, {\mathcal{A}}\) is a Banach \(\mathcal{A}\)-\({\mathfrak{A}}\)-module with canonical actions. Consider the closed ideal \(\mathcal{I}\) of \({\mathcal{A}}\,\widehat{\otimes}\, \mathcal{A}\) generated by elements of the form a.αbaα.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)ba(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
$$\alpha.(a\otimes b+{\mathcal{I}})=\alpha.a\otimes b+{\mathcal{I}}\quad \mbox{and}\quad \alpha.(a+{J})=\alpha.a+{J}, $$
for \(\alpha\in {\mathfrak{A}}\), \(a,b\in{\mathcal{A}}\), and similarly for the right actions.

Define \(\omega\in {\mathcal{L}}({\mathcal{A}}\,\widehat{\otimes}\, \mathcal{A}, \mathcal{A})\) by ω(ab)=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.

Let \(\mathfrak{A}\), \(\mathcal{A}\) and X be as above. A bounded map \(D:{\mathcal{A}} \longrightarrow X\) is called a module derivation if
$$D(a\pm b)=D(a)\pm D(b),\qquad D(ab)=D(a).b+a.D(b) \quad (a,b \in {\mathcal{A}}), $$
and
$$D(\alpha.a)=\alpha.D(a),\qquad D(a.\alpha)=D(a).\alpha \quad (a\in{\mathcal{A}}, \ \alpha \in{\mathfrak{A}}). $$
Note that \(D : {\mathcal{A}} \to X\) is bounded if there exist M>0 such that ∥D(a)∥≤Ma∥  (\(a\in{\mathcal{A}}\)), although D is not necessarily linear, but still its boundedness implies its norm continuity. For every xX we define the inner module derivation adx by
$$\mathrm{ad}_x(a)=a.x-x.a \quad (a\in {\mathcal{A}}). $$
The Banach algebra \({\mathcal{A}}\) is called 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.
Also \(\mathcal{A}\) is module Arens regular if the module topological center \(Z_{\mathfrak{A}}({\mathcal{A}}^{**})\) of \({\mathcal{A}}^{**}\) is equal to \(\mathcal{A}^{**}\) (see [10]), where
$$Z_{\mathfrak{A}}\bigl({\mathcal{A}}^{**}\bigr)= \bigl\{G\in { \mathcal{A}}^{**}\mid F\mapsto G\Box F \mbox{ is }\sigma\bigl({ \mathcal{A}}^{**},{J}^{\bot}\bigr)\mbox{-continuous} \bigr\}. $$
A Banach space E has the approximation property if there is a net (Sλ)λΛ in the space of the finite-rank operators on E such that Sλ⟶idE, 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

Suppose \(D:{\mathcal{A}}\longrightarrow X\) is a module derivation for some commutative Banach \(\mathcal{A}\)-\(\mathfrak{A}\)-module X. Let \(\widehat{\ \ \ }:X\to X^{**}\) be the canonical embedding map. Define \(\widetilde{D}:{\mathcal{A}}\longrightarrow X^{**}\) by \(\widetilde{D}(a)=\widehat{D(a)}\) (\(a\in \mathcal{A}\)). For \(\alpha \in{\mathfrak{A}}\), \(a\in {\mathcal{A}}\) and φX, we have
https://static-content.springer.com/image/art%3A10.1007%2Fs00233-012-9432-0/MediaObjects/233_2012_9432_Equj_HTML.gif
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=adx, for some xX. □

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

Suppose that 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:
$$\bigl(\alpha.(e+{J})\bigr).\bigl(\beta.(e+{J})\bigr)=(\alpha\beta).(e+{J}) \quad (\alpha,\beta\in \mathfrak{A}). $$
Define
$$\theta:{\mathfrak{A}}\longrightarrow {\mathcal{L}},\qquad \theta(\alpha)= \alpha.(e+J), $$
it is clear that 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].
Consider the pseudo-unital Banach \(\mathcal{A}/J\)-bimodule 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 x1X such that
$$D(l)=l.x_1-x_1. l\quad(l\in\mathcal{L}). $$
Let \(\widetilde{D}=D-\mathrm{ad}_{x_{1}}\) and Y be the closed linear span of the set
$$\bigl\{(a+{J}).\widetilde{D}(b+{J}).(c+{J}):a,b,c\in {\mathcal{A}} \bigr\}. $$
Then \(\widetilde{D}:{\mathcal{A}/J}\longrightarrow X\) is a derivation with \(\widetilde{D}|_{\mathcal{L}}=0\). Since 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
$$\alpha\circ y:=\bigl(\alpha.(e+{J})\bigr).y,\qquad y\circ\alpha:=\alpha\circ y \quad (\alpha \in{\mathfrak{A}},\ y\in Y). $$
For each \(\alpha \in \mathfrak{A}\) and \(a\in \mathcal{A}\), since e+J is an identity for \(\mathcal{A}/J\), we have
https://static-content.springer.com/image/art%3A10.1007%2Fs00233-012-9432-0/MediaObjects/233_2012_9432_Equp_HTML.gif
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
https://static-content.springer.com/image/art%3A10.1007%2Fs00233-012-9432-0/MediaObjects/233_2012_9432_Equq_HTML.gif
By continuity and linearity, we have
$$\bigl(\alpha.(e+{J})\bigr).y=y.\bigl(\alpha.(e+{J})\bigr) \quad (\alpha\in { \mathfrak{A}},\ y\in Y). $$
This shows that the compatible actions are well-defined and Y is a commutative Banach \({\mathcal{A}}/J\)-\(\mathfrak{A}\)-module. Also,
https://static-content.springer.com/image/art%3A10.1007%2Fs00233-012-9432-0/MediaObjects/233_2012_9432_Equs_HTML.gif
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 x2Y such that \(\widetilde{D}=\mathrm{ad}_{x_{2}}\). Consequently, \(D=\mathrm{ad}_{x_{1}+x_{2}}\) for x1+x2X. □

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
$$\frac{\mathcal{A}}{J}\cong \Bbb {M}_{n_1}(\Bbb{C}) \oplus \cdots\oplus \Bbb {M}_{n_k}(\Bbb{C}), $$
for some\(n_{1},\ldots,n_{k}\in \Bbb{N}\), where\(\Bbb{M}_{n}(\Bbb{C})\)is ann×nmatrix 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

Suppose that \(\mathcal{B}\) is a finite-dimensional, amenable Banach algebra and 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 xF such that for all \(a\in {\mathcal{B}}\),
https://static-content.springer.com/image/art%3A10.1007%2Fs00233-012-9432-0/MediaObjects/233_2012_9432_Equu_HTML.gif
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 \(a\in {\mathcal{A}}\).

Let Y,Z and W be commutative Banach \(\mathcal{A}/J\)-\(\mathfrak{A}\)-modules, and f:YZ and g:ZW be \(\mathcal{A}/J\)-\(\mathfrak{A}\)-module homomorphisms, then the short, exact sequence
$$\varSigma :0\to Y\to Z\to W\to 0, $$
  1. (i)

    is admissible if there is an \(\mathfrak{A}\)-module homomorphism h:WZ such that gh=idW.

     
  2. (ii)

    splits if there is an \(\mathcal{A}/J\)-\(\mathfrak{A}\)-module homomorphism h:WZ such that gh=idW.

     

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
$$\varPi:0\longrightarrow \operatorname{ker} \widetilde{\omega}\stackrel{i}{\longrightarrow} { \mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\,{\mathcal{A}}\stackrel{\widetilde{ \omega}}{\longrightarrow} {\mathcal{A}/J}\longrightarrow 0. $$
  1. (i)

    If\(\mathcal{A}/J\)has an identity, thenΠis admissible.

     
  2. (ii)

    \(\mathcal{A}\)is super module amenable if and only ifΠsplits 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 \(\widetilde{\omega}\).

(ii) Since \({\mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\,{\mathcal{A}}\) is a commutative Banach \(\mathfrak{A}\)-module, \({\mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\,{\mathcal{A}}\) is a commutative Banach \({\mathcal{A}/J}\)-\(\mathfrak{A}\)-module with the given actions on \(\mathfrak{A}\) and the following actions on \(\mathcal{A}/J\),
$$(a+{J})\circ x=x.a,\qquad x\circ (a+{J})=a.x \quad (a\in {\mathcal{A}},\ x\in { \mathcal{A}}\,\widehat{\otimes}_{\mathfrak{A}}\,{\mathcal{A}}). $$
By Theorem 3.5 in [8], super module amenability of \(\mathcal{A}\) implies that \(\mathcal{A}\) has a module diagonal 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

Recall that a semigroup S is an inverse semigroup if for each sS there is a unique element sS such that sss=s and sss=s. If we denote the set of idempotents of S by ES, then ES is a semilattice with the following order
$$e\leq f\quad\Longleftrightarrow\quad ef=e \quad (e,f\in E_S). $$
An inverse semigroup whose idempotents are in the center is called a Clifford semigroup. It is easy to see that ES is a commutative subsemigroup of S. In particular l1(ES) could be regarded as a subalgebra of l1(S) [7]. Consequently, l1(S) is a Banach algebra and a Banach l1(ES)-module with compatible actions. It is possible to consider arbitrary actions of l1(ES) on l1(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 ES is finite.

Selivanov showed that for any locally compact group G, the Banach algebra L1(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 ES 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

Letl1(S) be a pseudo-unital Banachl1(ES)-module. Ifl1(S)/Jis a commutative Banachl1(ES)-module andJis finite-dimensional, thenl1(S) is super module amenable if and only ifSis finite.

Proof

Since ES is a finite semilattice, l1(ES) is a finite dimensional amenable Banach algebra, by Theorem 8 of [4]. From Lemma 3.7, it follows that l1(ES) is super amenable. Since each semigroup algebra has the approximation property [11] and J is complemented in l1(S), so l1(S)/J has the approximation property. Suppose that l1(S) is super module amenable, from Corollary 3.6 it follows that l1(S)/J is finite-dimensional and so is l1(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 l1(S) is amenable and so it is super amenable. Also super amenability of l1(ES) implies that l1(ES) has an identity, hence super module amenability of l1(S) follows from Theorem 3.3. □

Ghahremani, Loy and Willis in [5] showed that if G is a locally compact group, then L1(G)∗∗ is amenable if and only if G is finite.

Now by Lemma 3.7, we conclude that L1(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, thenl1(S)∗∗is super module amenable if and only ifSis finite.

Proof

If l1(S)∗∗ is super module amenable, then so is (l1(S)/J)∗∗l1(S)∗∗/J⊥⊥. Since l1(S)/J is a commutative l1(ES)-module, from Proposition 3.1 it follows that l1(S)/J is super module amenable. Therefore l1(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

Letl1(S) be a pseudo-unital Banachl1(ES)-module with commutative compatible actions. Then the following statements are equivalent:
  1. (i)

    Sis finite.

     
  2. (ii)

    l1(S) is super module amenable.

     
  3. (iii)

    l1(S)∗∗is super module amenable.

     
  4. (iv)

    l1(S) is module Arens regular.

     
In the case wherel1(S) is unital, then the above statements are equivalent to the following:
  1. (v)

    l1(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 ES is finite, we conclude that l1(S) is Arens regular if and only if S is finite [9]. Also J=0 implies that J=l1(S), it follows that l1(S) is Arens regular if and only if it is module Arens regular. This shows that (i) and (iv) are equivalent. Now let l1(S) be unital, then equivalently of (ii) and (v) follows from Proposition 3.9. □

It should be noted that the results of super-module amenability and module Arens regularity of l1(S) were only known for the very specific compatible actions δe.δs=δs, δs.δe=δse (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 ES is finite. If l1(ES) acts on l1(S) by the following compatible actions
$$\delta_e.\delta_s=\delta_s. \delta_e=\delta_{es} \quad (e\in E_S,\ s\in S), $$
then l1(S) is a pseudo-unital Banach l1(ES)-module and the above four conditions are equivalent.
We finish by an example. Let \(S=(\Bbb{N},\vee)\) be the inverse semigroup of positive integers with maximum operation. Then S is infinite and each element of S is an idempotent. Since l1(S) is commutative, then it is a commutative Banach l1(ES)-module with the above compatible actions. Let X be a commutative Banach l1(S)-l1(ES)-module and D be a module derivation from l1(S) into X. Then as in the classical case, without loss of generality we may assume that X is a pseudo-unital Banach ł1(S)-bimodule. Since δ1 is an identity for l1(S), we have
$$D(\varphi)=D(\varphi*\delta_1)=\varphi.D(\delta_1)=0 \quad \bigl(\varphi\in {l^1(S)}\bigr), $$
and it follows that 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.

Copyright information

© Springer Science+Business Media, LLC 2012