\documentclass[11pt,twoside]{article} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float} \usepackage{lipsum} % \usepackage[utf8]{inputenc} \usepackage{newunicodechar} \usepackage{afterpage} \usepackage[labelfont=bf]{caption} \usepackage[nottoc,notlof,notlot]{tocbibind} %\renewcommand\bibname{References} \def\bibname{\Large \bf References} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[LE,RO]{\thepage} \thispagestyle{empty} %\afterpage{\lhead{new value}} \fancyhead[CE]{A. Hemmatzadeh, H. Pourmahmood Aghababa and M.H. Sattari} \fancyhead[CO]{ Module bounded approximate amenability of Banach algebras} %\topmargin=-1.6cm \textheight 17.5cm% \textwidth 12cm % \topmargin 8mm % \oddsidemargin 20mm % \evensidemargin 20mm % \footskip=24pt % \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} %\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$} \makeatletter \def\th@newremark{\th@remark\thm@headfont{\bfseries}} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\long\def\symbolfootnote[#1]#2{\begingroup% %\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup} \def \thesection{\arabic{section}} \begin{document} %\baselineskip 9mm %\setcounter{page}{} \thispagestyle{plain} {\noindent Journal of Mathematical Extension \\ Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\ ISSN: 1735-8299\\ URL: http://www.ijmex.com\\ \vspace*{9mm} \begin{center} {\Large \bf Module bounded approximate amenability \ of Banach algebras\\} %{\bf Do You Have a Subtitle? \\ If so, Write It Here} \let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)} {\bf A. Hemmatzadeh}\vspace*{-2mm}\\ \vspace{2mm} {\small Department of Mathematics, Azarbijan Shahid Madani University, Tabriz, Iran } \vspace{2mm} {\bf H. Pourmahmood Aghababa}\vspace*{-2mm}\\ \vspace{2mm} {\small Department of Mathematics, University of Tabriz, Tabriz, Iran} \vspace{2mm} {\bf M.H. Sattari $^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\ \vspace{2mm} {\small Department of Mathematics, Azarbijan Shahid Madani University, Tabriz, Iran} \vspace{2mm} %{\bf M.H. Sattari $^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\ %\vspace{2mm} {\small Department of Mathematics, Azarbijan Shahid Madani University, Tabriz, Iran} \vspace{2mm} \end{center} \vspace{4mm} {\footnotesize \begin{quotation} {\noindent \bf Abstract.} In this study we continue an investigation of the notion of module approximate amenability of a Banach algebra $ \mathcal{A} $ which is a module over another Banach algebra $ \mathfrak{A}$. In fact we introduce the class of module boundedly approximately amenable Banach algebras $ (m.b.app.am.) $ . It is shown that the class of module boundedly approximately amenable Banach algebra is different from the class of amenable Banach algebras. Also, we show that for an inverse semigroup $ S $ with the set of idompotent $ E $, $ l^{1}(S) $ is module boundedly approximately amenable as $ l^{1}(E)$-module if and only if $ S $ is amenable. Further examples are given of $ l^{1}$-semigroup Banach algebras which are module boundedly approximately amenable but are not amenable. %****\underline{examole 3.4.ii ra check kon} ********* \end{quotation} \begin{quotation} \noindent{\bf AMS Subject Classification:} MSC 43A07; MSC 46H25. \noindent{\bf Keywords and Phrases:} module boundedly approximately amenable, module derivation, boundedly approximately inner. \end{quotation}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{intro} % It is advised to give each section and subsection a unique label. %\begin{Left} The concept of approximate amenability was introduced by Ghahramani and Loy in 2004 \cite{gahremani2004}. They showed that the class of approximately amenable Banach algebras is larger than the class of amenable Banach algebras. Also, they proved that the group algebra $ L^{1}(G) $ is approximately amenable if and only if $ G $ is amenable, but this fails to be true for any discrete semigroup $ S $. In fact for any semigroup $ S $ just approximately amenability of $ l^{1}(S) $ implies the amenability of $ S $ \cite{gahremani2008}. Also, they introduced the class of boundedly approximately amenable Banach algebras. %\cite[ {Corollary 3.4 }]{zhang 2009} Ghahramani and Read built a boundedly approximately amenable Banach algebra which has no right bounded approximate identity \cite[Corollary 3.2]{gahremani2012}, and so it is not amenable. %\end{Left} Amini considered a Banach algebra $ \mathcal{A} $ over another Banach algebra $ \mathfrak{A} $ as an $ \mathfrak{A}$-module and introduced the concept of module amenability of Banach algebras \cite{amini2004}. He showed that under some natural conditions, for an inverse semigroup $ S $ with the set of idompotent $ E $, $ l^{1}(S) $ is $ l^{1}(E)$-module amenable if and only if $ S $ is amenable. Amini defined a bounded virtual diagonal for $ \mathcal{A} $ and proved that existing this diagonal implies the module amenability of $ \mathcal{A}$. Yazdanpanah and Najafi defined the module approximate amenability of Banach algebras \cite{Yazdanpanah 2009}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Pourmahmood and Bodaghi investigated the notions of module approximate amenability and module approximate contractibility for Banach algebras \cite{pourmahmood2}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% They showed that the classes of module approximately amenable and module approximately contractible Banach algebras are the same. They defined the unital Banach algebra $ \mathcal{B}=\mathcal{A}\oplus \mathfrak{A}^{\#} $ as $ \mathfrak{A}^{\#}$-module unitization of $ \mathcal{A}$ which also is a $ \mathfrak{A}^{\#}$-module with compatible actions and proved that the module approximate amenability (contractibility) of $ \mathcal{A}$ and $ \mathcal{B}$ is equivalent. Similar to module amenability, in approximate version for an inverse semigroup $ S $ with the set of idempotent $ E $ they concluded that $ l^{1}(S) $ is $ l^{1}(E)$-module approximately amenable if and only if $ S $ is amenable. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this paper we consider $ \mathcal{A}$ as an $ \mathfrak{A}$-module Banach algebra and introduce the bounded version of $ \mathfrak{A}$-module approximate amenability of $ \mathcal{A}$. %Our main reference is \cite{pourmahmood2}, that in most cases we adapt almost the same proofs. Here we show that the module bounded approximate amenability of $ \mathcal{A} $ and $ \mathcal{B}$ are equivalent. Also, we prove that the existence of a net in $ (\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B})^{**} $ is equivalent to module bounded approximate amenability of $ \mathcal{B}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Also, we get that, for an inverse semigroup $ S $ with the set of idempotent $ E$, the equivalence relation between amenability of $ S $ and module approximate amenability of $ l^{1}(S) $ (as an $ l^{1}(E)$-module) is true in boundedly version.\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Throughout the paper, we shall use the abbreviation $ m.b.app.am. $ for module boundedly approximately amenable, $ b.a.i. $ for bounded approximate identity, $ m.b.r.a.i. $ for multiplier-bounded right approximate identity and $ m.b.l.a.i. $ for multiplier-bounded left approximate identity. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %The JME is abstracted and indexed in the following databases:\\ %Mathematical Reviews(MathSciNet)\\ %Zentralblatt Math\\ %Scientific Information (SID)\\ %JME is also indexed in the Islamic World Science Citation Center (ISC) and has been granted the Scientific Research Rank by Islamic Azad University. \section{Notations and preliminaries %Some notations and preliminary results of module approximate amenability (contractibility) } \label{sec:2} %\subsection{Subsection Title} %\label{subsec:1} We first recall some definitions . Let $ \mathcal{A}$ be a Banach algebra, and $ X $ be a Banach $ \mathcal{A}$-bimodule. A bounded linear map $ D: \mathcal{A}\rightarrow X $ is called a derivation if \begin{eqnarray*} D(a \cdot b)=a \cdot D(b)+D(a) \cdot b \ \ \ \ \ \ \ \ \ ( a, b\in \mathcal{A}). \end{eqnarray*} For each $ x\in X $, we define the map $ ad_{x} : \mathcal{A}\rightarrow X $ by \begin{equation} ad_{x}(a)=a\cdot x-x \cdot a \ \ \ \ \ \ \ \ (a\in X). \end{equation} It is easy to see that $ ad_{x} $ is a derivation. Derivations of this form are called $ inner\ derivations$. A derivation $ D: \mathcal{A}\rightarrow X $ is said to be boundedly approximately inner if there exists a net $ (\xi_{i}) \subset X $ such that \begin{equation*} D(a)=\lim_i\,ad_{\xi_i}(a) \ \ \, \ \ \ ( a\in \mathcal{A}) \end{equation*} and \begin{equation*} \exists L>0:\,\sup\| ad_{\xi_i}(a)\| \leq L\| a\| \ \ \ \ ( a\in \mathcal{A}). \end{equation*} A Banach algebra $ \mathcal{A}$ is boundedly approximately amenable if every bounded derivation $D: \mathcal{A}\rightarrow X^{*}$ %($ D: \mathcal{A}\rightarrow X$) is boundedly approximately inner, for each Banach $ \mathcal{A} $-bimodule $ X $, where $X^{*}$ denotes the first dual of $ X $ which is a Banach $ \mathcal{A} $-bimodule in the canonical way. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $ \mathcal{A} $ and $ \mathfrak{A} $ be Banach algebras such that $ \mathcal{A} $ is a Banach $ \mathfrak{A}$-bimodule with compatible actions as follows: $$\alpha \cdot (ab)=(\alpha\cdot a)b, \qquad (ab)\cdot \alpha=a(b\cdot \alpha)\qquad (a,b\in \mathcal{A},\,\alpha\in\mathfrak{A}).$$ Let $X$ be a left Banach $\mathcal{A}$-module and a Banach $\mathfrak{A}$-bimodule with the following compatible actions: $$\alpha\cdot (a \cdot x)=(\alpha \cdot a)\cdot x,\ \ (a\cdot x)\cdot \alpha=a\cdot (x \cdot \alpha),\ \ a \cdot (\alpha \cdot x)=(a\cdot \alpha)\cdot x, $$ for all $x\in X $, $ a\in \mathcal{A}$ and $\alpha\in\mathfrak{A}$ then $ X $ is called a left Banach $\mathcal{A}$-$\mathfrak{A}$-module, right and $\mathcal{A}$-$\mathfrak{A}$-bimodule are defined similarly. Moreover, if $ \alpha.x=x.\alpha $ for all $ \alpha\in\mathfrak{A} $ and $ x\in X $, then $X $ is called a commutative Banach $\mathcal{A}$-$\mathfrak{A}$-module. If $ X $ is a (commutative) Banach $\mathcal{A}$-$\mathfrak{A}$-module, then $X^{*}$ is too, where the actions of $\mathcal{A}$ and $\mathfrak{A}$ on $X^{*}$ are defined as usual: \begin{eqnarray*} =\qquad &,&\qquad <\alpha.F,x>=\vspace{0.3cm}\\ =\qquad &,& \qquad = \end{eqnarray*} for all $ \alpha\in\mathfrak{A} $, $ a\in \mathcal{A} $،, $ x\in X $ and $ F\in X^* $. Note that, in general, $ \mathcal{A} $ is not an $\mathcal{A}$-$\mathfrak{A}$-module because $ \mathcal{A} $ does not satisfy in the compatibility condition $ a.(\alpha.b) = (a.\alpha).b $ for all $ \alpha \in \mathfrak{A} $ and $ a, b \in \mathcal{A} $. But when A is a commutative Banach $\mathfrak{A}$-module and acts on itself by multiplication, it is an $\mathcal{A}$-$\mathfrak{A}$-module. We give some examples of commutative Banach $ \mathcal{A}$-$ \mathfrak{A}$-modules. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{example}\label{1-3-3} \begin{description} \item[(i)] Every Banach space is a commutative Banach $\mathbb{C}$-$\mathbb{C}$-module. \item[(ii)] \ \ Suppose that $ G $ is a locally compact group. Also, $ \mathcal{A}=L^{1}(G)$, $\mathfrak{A}=l^{1}(G)$ and $ X= M(G) $ are the group algebra, Banach algebra of discrete measures and measure algebra of $ G$, respectively. Then $ M(G) $ is a Banach $ L^{1}(G)$-$l^{1}(G)$-module by the convolution action, which also is commutative as $l^{1}(G)$-module $ iff $ $ G $ is abelian. % \item[(iii)] \ \ Suppose that $ S $ is an inverse semigroup with the set of idempotents $ E $ . %Set $ \mathcal{A}= l^{1}(E)$, $ \mathfrak{A}=\mathbb{C} $ % and $ X = l^{1}(S) $ . Then $ l^{1}(S) $ is a $l^{1}(E)$-$\mathbb{C}$-module, such that %$ l^{1}(E) $ acts on $ l^{1}(S) $ by the convolution. Then $ l^{1} (S) $ is a commutative %$l^{1} (E)$-$\mathbb{C}$-module if and only if $ E $ lies in the center of $ S $. \end{description} \end{example} %##################################################################### Let $ \mathcal{A} $ and $ \mathfrak{A} $ be Banach algebras such that $ \mathcal{A} $ is a Banach $ \mathfrak{A}$-bimodule with compatible actions and $ X $ be a Banach $\mathcal{A}$-$\mathfrak{A}$-module. A ($\mathfrak{A}-$)module derivation is a bounded map $ D: \mathcal{A}\rightarrow X $ such that \begin{eqnarray*} D(a\pm b)&=&D(a)\pm D(b)\vspace{0.3cm}\\ D(a \cdot b)&=&a \cdot D(b)+D(a) \cdot b \end{eqnarray*} and \begin{equation*} D(\alpha \cdot a)=\alpha \cdot D(a), \qquad D(a \cdot \alpha)=D(a) \cdot \alpha\qquad (\alpha\in\mathfrak{A},\, a\in \mathcal{A}) \end{equation*} Although $ D $ is not necessarily $ \mathbb{C}$-linear, but still its boundedness implies its norm continuity. When $ X $ is a commutative $ \mathfrak{A} $-bimodule, each $x \in X$ defines an $inner$ module derivation as follows \begin{equation} ad_{x}(a) = a\cdot x - x \cdot a \ \ \ \ (a \in A). \end{equation} Remark that if $ \mathcal{A}$ is a left (right) essential $\mathfrak{A}$-module, then every $\mathfrak{A}$-module derivation is also a derivation \cite{pourmahmood2}, in fact, it is $ \mathbb{C}$-linear. %%%%%%%%%%%%%%%%%%%%%%%######################################################### \begin{definition}\label{def 1.2.3} Let $ \mathcal{A}$ and $\mathfrak{A}$ be Banach algebras and $ \mathcal{A}$ be an $\mathfrak{A}$-bimodule with compatible actions. Then %\item[(i)] $ \mathcal{A}$ is module boundedly approximately amenable ($m.b.app.am. $) as an $\mathfrak{A}$-module if for any commutative Banach $ \mathcal{A}$-$\mathfrak{A}$-module $X$, each module derivation $ D:\mathcal{A}\rightarrow X^{*}$ is boundedly approximately inner; % \item[(ii)] %$ \mathcal{A}$ is module boundedly approximately contractible ($m.b.app.con.$) as an $\mathfrak{A}$-module if for any %commutative Banach $ \mathcal{A}$-$\mathfrak{A}$-module $X$, each module derivation $ D:\mathcal{A}\rightarrow X$ is boundedly approximately inner; %\item[(ii)] %$ \mathcal{A} $ is module boundedly uniformly approximately amenable ($m. b.u.app.am $) as an $\mathfrak{A}$-module, when each module derivation $ D: \mathcal{A}\rightarrow X$ is $ b.u.app.inner.$ \end{definition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Note that a left Banach $ \mathfrak{A}$-module $X$ is called left $\mathfrak{A}-$essential if the linear span of $ \mathfrak{A}\cdot X= \{\alpha\cdot x \ :\ \alpha\in \mathfrak{A} , \ x \in X \} $ is dense in $ X $. Right essential $\mathfrak{A}$-modules and two-sided essential $ \mathfrak{A}$-bimodules are defined similarly. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pro1-2-3} Let $ \mathcal{A}$ be $ b.app.am. $ that is essential as one-sided Banach $\mathfrak{A}$-module. Then $ \mathcal{A}$ is $m.b.app.am.$. \end{proposition} \begin{proof} According to descriptions above Definition \ref{def 1.2.3} and our assumptions any module derivation is a derivation, so we conclude our proof. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Let $ X $ be a commutative Banach $ \mathcal{A}$-$\mathfrak{A}$-module and %$ \mathcal{A}$ % be $ b.app.am. (con.) $ that is essential as one-sided $\mathfrak{A}$-module. So any $\mathfrak{A}$-module derivation $ D: \mathcal{A}\rightarrow X^{*} $ ($ D: \mathcal{A}\rightarrow X $) is a derivation. Therefore it is $ b.app.inner.$. Hence $ \mathcal{A}$ is %$m.b.app.am.(con.)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %The proof is similar to the proof of %\cite [\lr{Proposition 2.2}]{pourmahmood2} %with considring that a bound obtained from bounded approximate amenability, works for %module bounded approximate amenability. \end{proof} We will give Example \ref{exam 6-3-3} -$ (i) $ to show that the converse is not true in general. % Let $ \mathcal{A}$ be an essential Banach left (or right) $\mathfrak{A}$-module, $ X $ a commutative Banach $ \mathcal{A}$-$\mathfrak{A}$-module, and $ D: \mathcal{A}\rightarrow X^{*} $ a module derivation. \underline{By****** the above discussion}, $ D $ is a derivation, Since every derivation $ D^{\prime}: \mathcal{A}\rightarrow X^{*} $ which $ X $ is a Banach $ \mathcal{A}$-module, is boundedly approximately inner, we conclude that $ D $ is also boundedly approximately inner.The argument is the same in contractible case. % In the next section we give an example to show that the converse of above proposition does not hold in general. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $ \mathcal{A} \widehat{\otimes}\mathcal{A} $ be the projective tensor product of $ \mathcal{A} $ which is a Banach $ \mathcal{A}$-bimodule and a Banach $ \mathfrak{A} $-bimodule. Now consider the module projective tensor product $ \mathcal{A}\widehat{\otimes}_{\mathfrak{A}}\mathcal{A} $ which is the quotient space $(\mathcal{A}\widehat{\otimes}\mathcal{A})/I_\mathcal{A} $ where $ I_\mathcal{A} $ is the closed linear span of $ \big\{a.\alpha\otimes b-a\otimes\alpha.b:\;\;\alpha\in\mathfrak{A},\,a,b\in \mathcal{A}\big\}$. Also, consider the closed ideal $ J_\mathcal{A} $ of $ \mathcal{A} $ generated by the elements $(a.\alpha)b-a(\alpha.b)$ for $ a,b\in \mathcal{A}$ and $\alpha\in\mathfrak{A}$. It follows that $ I_\mathcal{A} $ and $ J_\mathcal{A} $ are both $ \mathcal{A}$-submodules and $ \mathfrak{A}$-submodules of $ (\mathcal{A}\widehat{\otimes}\mathcal{A})$ and $ \mathcal{A} $, respectively. Both of the quotients $ \mathcal{A}\widehat{\otimes}_{\mathfrak{A}}\mathcal{A} $ and $ \mathcal{A}/J_{\mathcal{A}} $ are $ \mathcal{A}$-modules and $ \mathfrak{A}$-modules. Also, $(\mathcal{A}\widehat{\otimes}_{\mathfrak{A}}\mathcal{A})$ is a $\mathcal{A}$-$\mathfrak{A}$-module if $ \mathcal{A} $ is a $ \mathcal{A}$-$\mathfrak{A}$-module. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Moreover, when $ \mathcal{A}$ acts on $ {A}/J_{\mathcal{A}} $ canonically, then $ \mathcal{A}/J_{\mathcal{A}} $ is a Banach $ \mathcal{A}$-$ \mathfrak{A}$-module Consider $ \omega_\mathcal{A} : \mathcal{A}\widehat{\otimes}\mathcal{A}\longrightarrow \mathcal{A} $ defined by $ \omega_\mathcal{A}(a\otimes b)=ab $, $ (a,b \in \mathcal{A}) $ and extended by linearity. Then both $ \omega$ and its second conjugate $ \omega^{**} $ are $ \mathcal{A}$-module homomorphisms. We define $\tilde{\omega}_\mathcal{A}:\ (\mathcal{A}\widehat{\otimes}_{\mathfrak{A}}\mathcal{A})= (\mathcal{A}\widehat{\otimes}\mathcal{A})/I_{\mathcal{A}}\longrightarrow \mathcal{A}/J_{\mathcal{A} }\ $ by \begin{equation*} \tilde{\omega}_\mathcal{A}(a\widehat{\otimes}b+I_{\mathcal{A}})=ab+J_{\mathcal{A}}. \ \ \ , \ \ \ (a,b \in \mathcal{A}). \end{equation*} We denote by $ \square $ the first Arens product on $ \mathcal{A}^{**}$, the second dual of $ \mathcal{A}$. We assume that $ \mathcal{A}^{**}$ is equipped with the first Arens product. % The canonical images of $ a \in \mathcal{A} $ and $ \mathcal{A} $ will be denoted by $ \hat{a} $ and $ \hat{\mathcal{A}} $, respectively. For a Banach algebra $ \mathfrak{A} $, its unitization, denoted by $ \mathfrak{A}^\#$, is the Banach algebra $\mathfrak{A}\oplus \mathbb{C} $ with the multiplication \begin{equation*} (u,\alpha)(v,\beta)=(uv+\beta u+\alpha v ,\alpha \beta)\qquad (u,v \in \mathfrak{A},\ \alpha,\beta\in \mathbb{C} ). \end{equation*} Let $ \mathcal{A} $ be a Banach algebra and a Banach $ \mathfrak{A}$-bimodule with compatible actions and let $\mathcal{B}=(\mathcal{A}\oplus\mathfrak{A}^\#, \bullet) $, where the multiplication $ \bullet $ is defined through \begin{equation*} (a,u)\bullet(b,v)=(ab+a \cdot v+u \cdot b,uv)\qquad (\ a,b\in \mathcal{A},\ \,u,v \in \mathfrak{A}^\#). \end{equation*} $\mathcal{B} $ is called the module unitization of $\mathcal{A}$. Consider the module actions of $ \mathfrak{A}^\# $ on $\mathcal{B} $ as follows: \begin{equation*} u \cdot (a,v)=(u \cdot a,uv),\;\;(a,v)\cdot u=(a \cdot u,vu)\qquad (\ a \in \mathcal{A}, \ \ u,v\in\mathfrak{A}^\#). \end{equation*} Then $\mathcal{B} $ is a unital Banach algebra and a Banach $ \mathfrak{A}^\# $-bimodule with compatible actions. We can investigate when ($ \textrm{Ker}\,\tilde{\omega}_{\mathcal{B}}\subset$) $ \mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B} $ is a commutative Banach $ \mathcal{B}$-$\mathfrak{A}^\#$-module (in other words when it is a commutative Banach $ \mathfrak{A}^\#$-module). Actually $ \mathcal{B} $ must be a commutative Banach $ \mathfrak{A}^\#$-module with compatible actions (so $ J_{\mathcal{B}}=0$), then $ \mathfrak{A} $ must be a commutative Banach algebra and $\mathcal{A}$ be a commutative as an $ \mathfrak{A}$-module. % ( in other words $ \mathcal{A} $ be a $ \mathcal{A}$-$\mathfrak{A}$-module). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pro3-2-3} Let $ \mathcal{A}$ be a Banach algebra and an $\mathfrak{A}$-bimodule with compatible actions. Then the following are equivalent: \begin{description} \item[(i)] $ \mathcal{A}$ is $\mathfrak{A}^\#$-module boundedly approximately amenable ; \item[(ii)] $ \mathcal{B}$ is $\mathfrak{A}^\#$-module boundedly approximately amenable ; \end{description} If, in addition $ \mathcal{A}$ is a left or right essential $ \mathfrak{A}$-module, then $ (i) $ and $ (ii) $ are equivalent to \begin{description} \item[(iii)] $ \mathcal{A}$ is $\mathfrak{A}$-module boundedly approximately amenable. \end{description} \end{proposition} \begin{proof} Since every $\mathfrak{A}^\#$-module derivation on $ \mathcal{B}$ reduces to a $\mathfrak{A}^\#$-module derivation from $ \mathcal{A}$, by vanishing on $\mathfrak{A}^\#$, the proposition can be proved in essentially the same way as \cite [{Theorem 3.1 }]{pourmahmood2}. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % We need the following lemma from \cite [\lr{Lemma 2.1 }]{amini2004}, the proof is similar and so omitted. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We need the following lemma: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \begin{lemma}\label{lem2.2.3} % Let $ \mathcal{A}$ be a Banach algebra with a $ r.b.a.i. $ and be an $ \mathfrak{A}$-module with compatible actions, also $ X $ be a commutative Banach $ \mathcal{A}$- $ \mathfrak{A}$-module such that $ X\cdot \mathcal{A}=\{0\}$. Then every module derivation % $ D: \mathcal{A}\rightarrow X $ is $ b.app.inner. $. % \end{lemma} % \begin{proof} % Suppose that $ D: \mathcal{A}\rightarrow X $ is a module derivation and $ (e_{\alpha}) $ is a $ r.b.a.i. $ with the bound $ K $, then we have % $ D(ab)=a \cdot D(b) $ and % \begin{equation*} %D(a)= D(\lim_{\alpha} ae_{\alpha})=\lim_{\alpha} D ( ae_{\alpha})=\lim_{\alpha} ad_{D(e_{\alpha})}(a) \ \ \ (a \in \mathcal{A}) %\end{equation*} % and $ \| ad_{D(e_{\alpha})}(a) \| = \|a\cdot D(e_{\alpha})\| \leq K \|a\| \|D \| $ %\end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{lem3.2.3} If $ \mathcal{A}$ has a bounded approximate identity, then it is module boundedly approximately amenable $ iff $ every $\mathfrak{A}$-module derivation $ D: \mathcal{A}\rightarrow X^{*} $ is boundedly approximately inner for each commutative $\mathcal{A}$-pseudo-unital Banach $ \mathcal{A}$-$\mathfrak{A}$-module $ X$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ Bounded approximate module amenability of Banach algebras} \label{sec:3} In this section we provide some equivalent conditions for the module bounded approximate amenability in terms of diagonal for $ \mathcal{B}$ with results related to the existence of bounded approximate identity for $ \mathcal{A}$. It is shown that if $ \mathcal{A}^{**}$ is $m.b.app.am.$, so is $ \mathcal{A}$ when $ \mathcal{A} $ is a Banach $ \mathcal{A} $-$ \mathfrak{A}$-module and $ (\mathcal{A}\widehat{\otimes}_{\mathfrak{A}}\mathcal{A}) $ is commutative as $ \mathcal{A} $-$ \mathfrak{A}$-module. Finally, the $ l^{1}(E) $-module bounded approximate amenability of $ l^{1}(S) $ and $ l^{1}(S)^{**} $ are characterized where $ S $ is an inverse semigroups with the set of idempotent elements $ E$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Note that Example 6.1 in \cite{gahremani2004} is a non-amenable Banach algebra that is boundedly approximately amenable \cite[{Remark 5.2}]{gahremani2008}. So two notions 'bounded approximate amenability' and 'amenability' do not coincide. Since these are the special cases of module bounded approximate amenability and module amenability with $ \mathfrak{A}$=$\mathbb{C}$, respectively, then module bounded approximate amenability $ and $ module amenability are different notions. %%######################################################### % \begin{proposition}\label{prop 4-2-3} %Let $\mathfrak{A}$ be a commutative Banach algbra and $ \mathcal{A} $ be a commutative Banach % $\mathfrak{A}$-bimodule (so $\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B} $ is a commutative $\mathcal{A}$-$ \mathfrak{A}^\#$-module) which is $ m.b.app.am.$. Then the followings are equivalent.% % Let $ \mathcal{A} $ be a Banach algebra and a Banach $\mathfrak{A}$-bimodule with compatible actions which is $ m.b.app.con.$ as a ${\mathfrak{A}^\#}$-module. Let also $\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B} $ be commutative as Banach $ \mathfrak{A}^\#$-module. Then either of the following equivalent conditions hold: %\begin{description} %\item[(i)] There exists a net % $ (m^{\prime}_i)\subset (\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B}) $ and $ L^{\prime}>0 $ % such that for all %$ b\in \mathcal{B}$, % $b.m^{\prime}_i-m^{\prime}_i.b \longrightarrow0 $ %and %$\tilde{\omega}_\mathcal{B}(m^{\prime}_i)= 1 $ and $\| b.m^{\prime}_i-m^{\prime}_i.b\|\leq L^{\prime}\| b\|$ for each $ i $; %\item[(ii)] There exists a net %$ (m_i)\subset (\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B}) $ and $ L>0 $ such that for all $ b\in B $, $ b.m_i-m_i.b\longrightarrow 0$ and $\tilde{\omega}_B (m_i)\rightarrow 1$ and $\| b.m_i-m_i.b\|\leq L\| b\|$ for each $ i$. %\end{description} %\end{proposition} % \begin{proof} %********************** %Existance of the nets $ (m_{i}) $ and equivalence of them are similar to \cite [\lr{Thorem 3.3 }]{pourmahmood2}. %\end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If, in addition, $ \mathcal{A}$ be left or right essential as an $ \mathfrak{A}$-module, then the above proposition is yield in $ \mathfrak{A} $-module case. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Now we prove a proposition % similar to the Theorem \ref{prop 1-4-2-3} for $ m.b.app.am. $ Banach algebras, as follows: \begin{theorem}\label{prop 12-2-3} Let $ \mathcal{A} $ be a Banach algebra and a Banach $\mathfrak{A}$-bimodule with compatible actions. Let also $ \mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B} $ be commutative as a $ \mathfrak{A}^\#$- module. Then the following are equivalent: \begin{description} \item[(i)] $\mathcal{B} $ is $ m.b.app.am. $ as a $ {\mathfrak{A}^\#}$-module; \item[(ii)] \ \ There exist a net $ (M_i)\subset (\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B})^{**} $ and $ L>0 $ such that for all $ b\in B $, $ b.M_i-M_i.b\longrightarrow 0$, $\| b.M_i-M_i.b\|\leq L\| b\|$, $\tilde{\omega}_\mathcal{B}^{**}(M_i)\rightarrow 1_{\mathcal{B}}$ and $\tilde{\omega}_\mathcal{B}^{**}(M_i) $ is bounded; \item[(iii)] \ \ There exist a net $ (M_i)\subset (\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B})^{**} $ and $ L>0 $ such that for all $ b\in \mathcal{B}$, $b.M_i-M_i.b \longrightarrow0 $, $\| b.M_i-M_i.b\|\leq L\| b\|$ and $\tilde{\omega}_\mathcal{B}^{**}(M_i)= 1_{\mathcal{B}} $. \end{description} \end{theorem} % % \begin{proof} $ (i)\Longrightarrow (iii)$: Let $F=1{\otimes}_{\mathfrak{A}^\#}1$. It is straightforward to check that the inner derivation $D_{F}:\mathcal{B} \rightarrow (\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B})^{**}$ satisfies $D_{F}(\mathcal{B})\subset \ker \tilde{\omega}_\mathcal{B}^{**} = (\ker \tilde{\omega}_\mathcal{B})^{**}$, and so there exists a net $ (N_i)\subset (\ker \tilde{\omega}^{**}_\mathcal{B})$ and a constant $k>0$ such that $D_{N_{i}}(b)\longrightarrow D_{F}(b)$ and $\| D_{N_{i}}(b)\|\leq k\| b\|$ for all $ b\in \mathcal{B}$. Letting $M_{i}=F-N_{i}$ for all $i$, we have \begin{equation*} \tilde{\omega}_\mathcal{B}^{**}(M_i)=\tilde{\omega}_\mathcal{B}^{**}(F)-\tilde{\omega}_\mathcal{B}^{**}(N_i)=1_\mathcal{B}-0=1_\mathcal{B}, \end{equation*} \begin{equation*} b.M_i-M_i.b= D_{F}(b)- D_{N_{i}}(b)\longrightarrow 0 \end{equation*} and \begin{eqnarray*} \| b.M_i-M_i.b\| &\leq & \parallel D_{F}(b)\parallel + \parallel D_{N_{i}}(b) \parallel \\ &\leq & (\parallel D_{F}\parallel +k) \parallel b \parallel. \end{eqnarray*} Therefore (iii) holds for $L=\parallel D_{F}\parallel +k$ \\ $ (iii)\Longrightarrow (ii)$: is obvious. \\ $ (ii)\Longrightarrow (i)$: It is similar to \cite [{Theorem 3.3 }]{pourmahmood2}, with this additional notion that $ \sup_{i}\|\tilde{\omega}_\mathcal{B}^{**}(M_{i}) \| <\infty $. So we have \begin{eqnarray*} \| ad_{f_{i}} (b)\| &\leq & \| F\| \| b \cdot M_{i}- M_{i} \cdot b \| + \| D(b) \| \|\tilde{\omega}_\mathcal{B}^{**}(M_{i}) \| \\ &\leq & \| D \| \| b \| L + \| D \| \| b \| \sup_{i}\|\tilde{\omega}_\mathcal{B}^{**}(M_{i}) \| , \end{eqnarray*} for all $ i $ and $ b \in \mathcal{B}$. So $\|ad_{f_{i}}(b)\| \leq K\| b\| $ for all $ b \in \mathcal{B}$, where $ K= \| D \| ( L + \sup_{i}\|\tilde{\omega}_\mathcal{B}^{**}(M_{i}) \|)$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 Remark that by using \cite [{Lemma 3.1 }]{Jabbari 2017} we can conclude that when $ \mathcal{A}$ is $ m.b.app.am.$ as a commutative Banach $ \mathfrak{A} $-module, it has left and right approximate identity. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %************************************************************************************* %Consider \chi as the canonical embedding of $ \mathcal{A} $ into $ \mathcal{A}^{**} $. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{remark}\label{remark 3.3} %Considring the relations $(ii)$ and $(iii)$ in Theorem \ref{them 9.2.3} and the isomorphisims %$ \big(\mathfrak{A}^\#\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A}\big) \cong \mathcal{A} $ %and %$ \big(\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathfrak{A}^\#\big) \cong \mathcal{A}$, %shows that the nets $(u_{i} \cdot b_{i}) $, $(a_{i}\cdot v_{i}) \subset \mathcal{A} $ %are $ m.b.l.a.i. $ and $ m.b.r.a.i. $ for $ \mathcal{A} $ respectively. %\end{remark} %************************************************************************************************************ %Let %$ \iota $ %show the canonical embedding of $ \mathcal{A}$ into $ \mathcal{A}^{**}$. We have the following analogue of Theorem \ref{them 9.2.3}. %############################################################### \begin{theorem}\label{them10-2-3} Suppose that $ \mathcal{A}$ is a Banach algebra and a Banach $\mathfrak{A}$-bimodule with compatible actions which is $ m.b.app.am. $. Also, let $\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B} $ be commutative as Banach $ \mathfrak{A}^\# $-module. Then exist a constant $ L>0 $, nets $ (m_{i})\subset\big(\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A}\big)^{**} $ and $ (a_{i}), ( b_{i})\subset \mathcal{A}^{**} $ such that for all $ a \in \mathcal{A} $ we have \begin{description} \item[(i)] $\tilde{\omega}^{**}_\mathcal{A}(m_i) = a_{i}+ b_{i}$; \item[(ii)] $ b_{i}\cdot a \longrightarrow a, \ \| b_{i} \cdot a \| \leq L \|a \| $ for all $ i $; \item[(iii)] $ a \cdot a_{i} \longrightarrow a, \ \| a\cdot a_{i} \| \leq L \|a \|$ for all $ i $; \item[(iv)] $a \cdot m_{i} -m_{i} \cdot a + a_{i}\otimes a - a \otimes b_{i} \longrightarrow 0, $ \\ $\|a \cdot m_{i} -m_{i} \cdot a + a_{i}\otimes a - a \otimes b_{i} \| \leq L \|a \|$ for all $ i $. \end{description} \end{theorem} \begin{proof} By Theorem \ref{prop 12-2-3} there is a net $ (M_{i}) \subset (\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B})^{**} $ and a constant $L>0$ satisfying $b.M_i-M_i.b \longrightarrow0 $, $\| b.M_i-M_i.b\|\leq L\| b\|$ and $\tilde{\omega}_\mathcal{B}^{**}(M_i)= 1_{\mathcal{B}} $ for all $ b\in \mathcal{B}$. Following \begin{eqnarray*} (\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B})^{**}&=&((\mathcal{A}\oplus \mathfrak{A}^\#)\widehat{\otimes}_{\mathfrak{A}^\#}(\mathcal{A}\oplus \mathfrak{A}^\#))^{**}\\ &=&(\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A})^{**}\oplus (\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathfrak{A}^\#)^{**}\oplus (\mathfrak{A}^\#\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A})^{**}\oplus (\mathfrak{A}^\#\widehat{\otimes}_{\mathfrak{A}^\#}\mathfrak{A}^\#)^{**}, \end{eqnarray*} we can write \begin{equation*} M_{i}^{\prime}=m_{i}-(a_{i}\otimes_{\mathfrak{A}^\#} 1_{ \mathfrak{A}^\#})- (1_{ \mathfrak{A}^\#}\otimes_{\mathfrak{A}^\#} b_{i})+(t_{i}\otimes_{\mathfrak{A}^\#} 1_{ \mathfrak{A}^\#})\ , \end{equation*} for some $ (m_{i})\subset\big(\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A}\big)^{**}$, $ (a_{i}),( b_{i})\subset \mathcal{A}^{**}$, and $ (t_{i})\subset(\mathfrak{A}^\#)^{**}$. Applying $\tilde{\omega}_\mathcal{B}^{**}(M_i)= 1_{\mathcal{B}} $ yields \begin{equation*} \tilde{\omega}_\mathcal{A}^{**}(m_i)-a_{i} - b_{i} +t_{i} =1_{\mathcal{B}} = (0,1) \in (\mathcal{A}\oplus \mathfrak{A}^\# ). \end{equation*} This follows that $ \tilde{\omega}_\mathcal{A}^{**}(m_{i})-a_{i} - b_{i}= 0$, and $t_{i}= 1$, for all $i$. Also, we have %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{eqnarray}\label{eqna 3-30} a \cdot M_{i}^{\prime}-M_{i}^{\prime} \cdot a &=&\big((a \cdot m_{i}-m_{i} \cdot a) +(a_{i}\otimes_{\mathfrak{A}^\#} a)-(a\otimes_{\mathfrak{A}^\#} b_{i})\big) \nonumber \\ &+ & (1_{ \mathfrak{A}^\#}\otimes_{\mathfrak{A}^\#} b_{i} a-1_{ \mathfrak{A}^\#}\otimes_{\mathfrak{A}^\#} a)\nonumber \\ &+ & (-a a_{i}\otimes_{\mathfrak{A}^\#} 1_{ \mathfrak{A}^\#} +a \otimes_{\mathfrak{A}^\#} 1_{ \mathfrak{A}^\#})\longrightarrow 0, \end{eqnarray} for all $ a \in \mathcal{A}$. Hence %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{equation*} \big((a \cdot m_{i}-m_{i} \cdot a) +(a_{i}\otimes_{\mathfrak{A}^\#} a)-(a\otimes_{\mathfrak{A}^\#} b_{i})\big) \longrightarrow 0, \end{equation*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{equation*}\label{equa3-a} (1_{ \mathfrak{A}^\#}\otimes_{\mathfrak{A}^\#} b_{i} a-1_{ \mathfrak{A}^\#}\otimes_{\mathfrak{A}^\#} a)\longrightarrow 0, \end{equation*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{equation*}\label{equa3-4} (-a a_{i}\otimes_{\mathfrak{A}^\#} 1_{ \mathfrak{A}^\#} +a \otimes_{\mathfrak{A}^\#} 1_{ \mathfrak{A}^\#})\longrightarrow 0, \end{equation*} \\ we may conclude \begin{equation*} 1_{ \mathfrak{A}^\#}\otimes_{\mathfrak{A}^\#}( b_{i} a- a)\longrightarrow 0 \ \ \ \ \ \ \Longrightarrow \ \ \ \ \ b_{i} a\longrightarrow a, \end{equation*} \begin{equation*}\ (a a_{i} -a )\otimes_{\mathfrak{A}^\#} 1_{ \mathfrak{A}^\#}\longrightarrow 0 \ \ \ \ \ \Longrightarrow \ \ \ \ \ a a_{i} \longrightarrow a, \end{equation*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for all $ a \in \mathcal{A}$. The left side of \ref{eqna 3-30} is bounded by $ L \| a\| $ for all $ i $ and $ a \in \mathcal{A}$, then we get \begin{equation*} \| a \cdot m_{i}-m_{i} \cdot a +a_{i}\otimes a -a\otimes b_{i} \| < L \| a\|, \end{equation*} \begin{equation*} \ \ \ \ \| b_{i} a\| \leq L\| a\|, \end{equation*} \begin{equation*} \ \ \ \ \ \| a a_{i} \| \leq L\| a\|. \end{equation*} \end{proof} % %******************************************************************************************************** %Note that % like the Remark \ref{remark 3.3}% %applying %the isomorphisims %$ \big(\mathfrak{A}^\#\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A}^{**}\big) \cong \mathcal{A}^{**} $ %and %$ \big(\mathcal{A}^{**}\widehat{\otimes}_{\mathfrak{A}^\#}\mathfrak{A}^\#\big) \cong \mathcal{A}^{**}$, yields that %the nets $(u_{i}\cdot b_{i}), (a_{i} \cdot v_{i} ) $ are % $ m.b.l.a.i $ and $ m.b.r.a.i. $ for $ \mathcal{A}$ in $ \mathcal{A}^{**}$ respectively. %********************************************************************************************************** \begin{theorem}\label{them11-2-3} Suppose that $ \mathcal{A}$ is a Banach algebra and a Banach $\mathfrak{A}$-bimodule with compatible actions which is $ m.b.app.am. $ and has both $ m.b.l.a.i. $ and $ m.b.r.a.i.$. Also $\mathcal{B}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{B} $ is commutative as Banach $ \mathfrak{A}^\# $-module. Then $\mathcal{A}$ has a $ b.a.i.$. %Suppose that $\mathfrak{A}$ is a commutative Banach algebra and $ \mathcal{A} $ is a Banach $ \mathcal{A}$-$\mathfrak{A}$-module ($\equiv $ commutative Banach $\mathfrak{A}$-module) that is module boundedly approximately amenable has both multiplier-bounded left and right approximate identities . Then $\mathcal{A}$ has a bounded approximate identity.% \end{theorem} \begin{proof} Let $ (f_{\gamma})$ and $ (e_{\beta}) $ be left and right multiplier-bounded approximate identities for $ \mathcal{A}$, respectively. So there is $K>0 $ such that \begin{equation}\label{eq3-1} \| a\cdot e_\beta\|\leq K\| a\|, \quad \| f_\gamma \cdot a\|\leq K\| a\| \end{equation} for all $ a\in \mathcal{A}$ and for all $\beta, \gamma$. From this relation and projective tensor norm we have % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{eqnarray*}\label{eq3-1-1} \| f_\gamma \cdot m\|_{\widehat{\otimes}} =\Big\| \sum_{n=1}^{\infty} f_\gamma \cdot a_n\otimes b_n\Big\|_{\widehat{\otimes}} \leq K\sum_{n=1}^{\infty}\| a_n\|\,\| b_n\| \ \ \ \ \ \ \ \ (m\in \mathcal{A}\widehat{\otimes}\mathcal{A}) \end{eqnarray*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for any representation $ m= \sum_{n=1}^{\infty} a_n\otimes b_n$, and so $ \| f_\gamma \cdot m\|_{\widehat{\otimes}}\leq K\| m\|_{\widehat{\otimes}}$. By passing to the quotient we have $ \| f_\gamma \cdot m\|_{\widehat{\otimes}_{\mathfrak{A}^\#}}\leq K\| m\|_{\widehat{\otimes}_{\mathfrak{A}^\#}} $ for all $ m \in \big(\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A}\big) $ and all $ \gamma $, where the index $ \widehat{\otimes}_{\mathfrak{A}^\#} $ in the norm, denotes the norm on $ \mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A} $ that from now on, it will be omitted. According to Goldestine's Theorem for any $ T\in(\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A})^{**} $ there exists a net $(m_j) \subseteq \mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A}$ such that $m_j{\overset{w^*}\longrightarrow}T $ and $ \sup_{j} \| m_j \| \leq \| T \|$. \\ Using this and the $ \omega^{*}$-continuity of the left module action of $ \mathcal{A}$ on $ (A\widehat{\otimes}_{\mathfrak{A}^\#}A)^{**} $ yield \begin{equation*} f_\gamma \cdot m_j {\overset{w^*}\longrightarrow} f_\gamma \cdot T, \quad \| f_\gamma \cdot m_{j}\|\leq K\| m_{j}\| \leq K \| T\|. \end{equation*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% So $ \| f_\gamma \cdot T\| \leq K \| T \| $. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Now the latter inequality and $\omega^{*}$-continuity of right module action of $\mathfrak{ A} $ (as a subset of $ \mathfrak{ A}^{**}$) on $ (\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A})^{**}$ we have $ \| T\cdot e_\beta\|\leq K\| T\| $ for each $ T\in (\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A})^{**}$. So we have obtained the following inequalities: %\begin{equation}\label{eq3-2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\| f_\gamma \cdot m\|\leq K\| m\|, \quad \| m\cdot e_\beta\|\leq K\| m\| \qquad m\in \mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A}, %\end{equation} %\begin{equation}\label{eq3-3} %\| f_\gamma \cdot T\|\leq K\| T\|, \quad \| T\cdot e_\beta\|\leq K\| T\| \qquad T\in (\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A})^{**}. %\end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% By the same argument we have \begin{equation}\label{eq3-2-1} \| m\cdot e_\beta\|\leq K\| m\|, \quad \| T\cdot e_\beta\|\leq K\| T\| , \end{equation} for all $ m\in \mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A} $ and $ T\in (\mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A})^{**} $. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Similar relations hold for $\mathfrak{A}^\#\widehat{\otimes}_{\mathfrak{A}^\#}\mathcal{A}$, %$ \mathcal{A}\widehat{\otimes}_{\mathfrak{A}^\#}\mathfrak{A}^\#$ and %$ \mathfrak{A}^\#\widehat{\otimes}_{\mathfrak{A}^\#}\mathfrak{A}^\# $. % Let the nets $(a_i) $ and $(b_i) $ and the constant $ L $ satisfy in the previous theorem. Suppose, on the contrary, that the net $ ( f_\gamma) $ is unbounded. According to Theorem \ref{them10-2-3}-(iv) for every $i $ and $ \gamma $ we have \begin{equation*} \| f_\gamma \cdot m_i-m_i \cdot f_\gamma - f_\gamma \otimes b_i + a_i \otimes f_\gamma \| \leq L\| f_\gamma\|. \end{equation*} Applying \eqref{eq3-2-1} gives \begin{equation*} \|\big( f_\gamma \cdot m_i-m_i \cdot f_\gamma - f_\gamma \otimes b_i + a_i\otimes f_\gamma \big) \cdot e_{\beta}\| \leq K L\| f_\gamma\|, \end{equation*} for all $i$, $ \beta $ and $ \gamma $. Utilizing this relation, the triangle inequality and left-multiplier boundedness of the net $ (f_{\gamma}) $ we get \begin{equation}\label{eq3-32} \begin{array}{ll} \| f_\gamma \| \| b_{i}\cdot e_{\beta}\| & \leq KL \|f_\gamma \| + \|f_\gamma \cdot (m_i \cdot e_{\beta})\| + \|m_i \cdot (f_\gamma \cdot e_{\beta})\| \\ & + \| a_{i} \cdot (f_\gamma \cdot e_{\beta})\| \vspace{0.1cm} \\ & \leq K L \| f_\gamma\| + 2K\| m_i\| \| e_{\beta} \|+K \| a_i \| \| e_\beta\|, \end{array} \end{equation} for all $ i$, $ \beta $ and $ \gamma $. So we have %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $$\begin{array}{ll} \| b_{i}\cdot e_{\beta} \| \leq K L +\frac{1}{\| f_\gamma\|}\big( 2K\| m_i\| \| e_{\beta} \| \!\!\! & +K \| a_i \| \| e_\beta\| \vspace{0.1cm} \big). \end{array}$$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% For fixed $ i $ and $ \beta $, our assumption regarding unboundedness of $ (f_{\gamma}) $ implies: $$ \| b_{i} \cdot e_{\beta}\| \leq K L.$$ Taking limits with respect to $ i$, according to Theorem \ref{them10-2-3}, we obtain $ \| e_{\beta} \| \leq KL $ for each $ \beta $. Using $(e_{\beta})$ as a right approximate identity and $ (f_{\gamma})$ as a $ m.b.l.a.i $ and then, applying the latter inequality we find out \begin{equation*} \| f_{\gamma}\|= \lim_{\beta}\| f_{\gamma} \cdot e_{\beta} \| \leq\lim_{\beta}K \| e_{\beta} \| \leq K^{2}L \end{equation*} for all $ \gamma $. This contradicts our assumption that the net $ (f_{\gamma}) $ is unbounded. A similar argument shows that the net $ (e_{\beta}) $ is also bounded. Therefore, $\mathcal{A}$ has a bounded approximate identity. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{remark}\label{remark 3.3.1} %We can conclude that a $ m.b.app.am. $ Banach algebra $ \mathcal{A} $ with the assumptions of Theorem \ref{them10-2-3} has a $ b.a.i $ in $ \mathcal{A}^{**}$. %\end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %********************************************************************************************** %\begin{remark}\label{remark 11.2.3} %Since the concepts of $ m.b.app.am. $ and $ m.b.app.con. $ are not equivalent in the case $ \mathfrak{A}=\mathbb{C}$ \cite{gahremani2008}. Then the $ \mathfrak{A} $-module boundedly virtual diagonal and $\mathfrak{A}$-module boundedly approximately diagonal are not equievalent. %\end{remark} %Now we can bring the next corollary. %\begin{corollary}\label{coro11-2-3} %Suppose that $ \mathcal{A} $ is a Banach algebra and a commutative Banach $ \mathfrak{A}$-module.Then $ \mathcal{A} $ is $ m.b.app.am. $ and has $ m.b.l.a.i. $ and $ m.b.r.a.i. $ if and only if it is $ m.b.app.con. $. %\end{corollary} %\begin{proof} %According to theorem \ref{them11-2-3} if $ \mathcal{A} $ is $ m.b.app.am. $ and has both left and right %multiplier-bounded approximate identities, then it has a $ b.a.i$. In the proof of theorem \ref{prop 12-2-3} by using these assumptions we obtained a boundedly $ \mathfrak{A} $-module approximate diagonal that is equivalent with module boundedly approximately contractibility of $ \mathcal{A} $ %\end{proof} \begin{remark}\label{re 2-14} Ghahramani and Read made a Banach algebra $ \mathcal{A} $ which was $ b.app.am $, but they proved that $ \mathcal{A}\oplus \mathcal{A}^{op} $ is not $ app.am$ \cite[{Theorem 4.1}]{gahremani2012}. So the direct sum of two $ m.b.app.am. $ Banach algebras is not necessarily $ m.b.app.am.$. \end{remark} %********************************************************************************************************* %Since the existence of approximate identity for $ \mathcal{A} $ in $ \mathfrak{A} $ impliese that every module derivation is a derivation. So similary to \cite[\lr{ Proposition 2-1}]{amini2004} we have the following proposition. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pro17-2-3} Suppose that $ \mathcal{A} $ is a Banach $ \mathcal{A} $-$ \mathfrak{A}$-module and $ (\mathcal{A}\widehat{\otimes}_{\mathfrak{A}}\mathcal{A}) $ is a commutative Banach $ \mathcal{A} $-$ \mathfrak{A}$-module. If $ \mathcal{A}^{**} $ is $ m.b.app.am. $, so is $ \mathcal{A}$. \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} Consider Theorem \ref{prop 12-2-3} for $ \mathcal{B}^{**}$, since the role of $ \mathcal{B}^{**}$ for $ \mathcal{A}^{**}$ is the same as the role of $ \mathcal{B}$ for $ \mathcal{A}$. We follow the notations of \cite [{Proposition 3.7}]{pourmahmood2}, the proof is similar to the proof of this proposition with these additional assumptions: \begin{description} \item[(i)] $ \tilde{\omega}_\mathcal{B^{**}}^{**}(\theta_{j})$ is bounded; \item[(ii)] $ \| b\cdot \theta_{j} - \theta_{j} \cdot b \|< L \| b \| $ for all $ b \in \mathcal{B^{**}}$ and $ L>0$. \end{description} Since $ \Omega_{u} $ is a bounded mapping, $ T $ is canonical embedding and $ \Omega_{u} $ and $ T $ and their adjoints are $ \mathcal{B}$-$ \mathfrak{A^{\# }}$-module homomorphisms, then for $ M_{j}=T^{*}\big(\Omega^{**}_{u} (\theta_{j}) \big)$ exists a $ C>0 $ such that $\| b \cdot M_{j} - M_{j}\cdot b \|\leq C \| b\|$, for all $ b \in \mathcal{B}$. Actually $ \lambda $ and its adjoint are $ \mathcal{B}$-$ \mathfrak{A^{\# }}$-module homomorphisms, so according to the proof of \cite [{Proposition 3.7}]{pourmahmood2} we have $ \tilde{\omega}_\mathcal{B}^{**}(M_{j})= \lambda^{**} \big(\tilde{\omega}_\mathcal{B^{**}}^{**}(\theta_{j})\big)$. Moreover $ \lambda $ and its adjoint are continuous, so $ \tilde{\omega}_\mathcal{B}^{**}(M_{j}) $ is bounded. \end{proof}\\ %\begin{proof} %(i) For $ p=1 $ is like to Lemma \ref{lem17-2-3} and for $ p\neq 1 $ the proof is the same. %\ \ (ii) Let $ (e_{\alpha_{i }})_{{\alpha}_{i } \in J_{i}} $ be a $ r.a.i $ for $ \mathcal{A}_i $. %Set $ \alpha = (\alpha_{i})_{i \in I} $ and consider $J =\prod_{i\in I} J_{i}$. % Then $ (e_{\alpha})_{\alpha \in J} $ is a $ r.a.i $ for $ \mathcal{A} $. %\end %Applying theorem %\ref{them10-2-3} and lemma \ref{lem19-2-3} and corollary \ref{coro11-2-3} implies the following corollary. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{proposition}\label{pro22-2-3} %Suppose that $\mathcal{A}=l^p-\oplus_{i\in I} \mathcal{A}_i $. Then we have %\begin{description} %\item[(1)] %Let $ \mathcal{A}$ is $ m.b.app.am. $ if each of $ \mathcal{A}_i $ be $ m.b.app.con. $, then $ \mathcal{A}$ is $ m.b.app.con. $ %\item[(2)] %Let each $ \mathcal{A}_i $ is $ m.b.app.am. $. if $ \mathcal{A}$ is $ m.b.app.con. $, then each of $ \mathcal{A}_i $ is $ m.b.app.con. $ %\end{description} %\end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We can get $ m.b.app.am. $ version of Johnson's Theorem for inverse semigroups. For an inverse semigroups $ S $ with the set of idempotent elements $ E$, in fact $ E $ is a commutative subsemigroup of $ S $, so $ l^{1}(E) $ is a commutative subalgebra of $ l^{1}(S)$. Suppose that $ l^{1}(E) $ acts on $ l^{1}(S)$ and its second dual with trivial left action $ \delta_{e} \cdot \delta_{s} = \delta_{s} $ and the right action $ \delta_{s} \cdot \delta_{e} =\delta_{se} = \delta_{s} \ast \delta_{e}$ for all $ e \in E$ and $ s \in S$. So $ l^{1}(S)$ is a Banach $ l^{1}(E) $-module with compatible actions \cite{amini2004}. Hence the closed ideal $ J_{l^{1}(S)} $ of $ \mathcal{A}= l^{1}(S)$ is the closed linear span of $ \{ \delta_{set} - \delta_{st} \ \ : s, t \in S, e \in E\}$. Now consider the equivalence relation $ \approx $ on $ S $ as $ s\approx t $ if and only if $ \delta_{s} - \delta_{t} \in J_{l^{1}(S)}$, for all $ s,t \in S$. We can bring our intended propositions. The next proposition holds because the m.am version (\cite[{Theorem 3.1}]{amini2004}) and m.app.am version (\cite[{Theorem 3.9}]{pourmahmood2}) hold. %****************************************************************************** %\begin{definition} %Suppose that $ S $ is an inverse semigroup with the set of idompotent elements $ E $ which is a subsemigroup of $ S $. In fact $ E $ is a semmilattice. Also $ l^{1}(E) $ is a commutative subalgebra of $ l^{1}(S) $. %\end{definition} %************************* %Now we need the following result of \cite[{ Proposition 3.2}]{amini2010}. The proof is similar, so omitted. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{proposition}\label{pro24-2-3} %Let $ \mathcal{A} $ be a $ m.b.app.am.(con.) $ as an $ \mathfrak{A}$-module with trivial left action. Also $ J^{\prime} $ be a closed ideal of $ \mathcal{A} $ such that Also $ J_{\mathcal{A} } \subseteq J^{\prime}$. if $ \frac{\mathcal{A}}{J^{\prime}} $ has an identity then it is a $ b.app.am.(con.) $ as a Banach algebra. %\end{proposition} %Now we can bring our intended propositions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{them17-2-3} Let $ S $ be an inverse semigroup with the idempotent elements set $ E $. Then $ l^{1}(S) $ is $ m.b.app.am. $ as $ l^{1}(E) $-module $ iff$ $ S $ is amenable. \end{proposition} %\begin{proof} %Suppose that $ l^{1}(S) $ is $ m.b.app.am. $ as $ l^{1}(E) $-module. by applying \cite[{Theorem 3.9}]{pourmahmood2}, $ S $ is amenable. % %Conversely if $ S $ is amenable. Then by \cite[{Theorem 3.1}]{amini2004}, $ l^{1}(S) $ % is module amenable and so % % is $ m.b.app.am.$. %\end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Applying both results (\cite[{Theorem 2.11}]{pourmahmood1}) and (\cite[{Theorem 3.10}]{pourmahmood2}) yields the following proposition. \begin{proposition}\label{them18-2-3} Suppose that $ S $ is an inverse semigroup with the set of idompotent elements $ E $. Then $ l^{1}(S)^{**} $ is $ m.b.app.am. $ as $ l^{1}(E) $-module $ iff $ $\frac{S}{\approx} $ is finite. \end{proposition} %\begin{proof} %Let $ l^{1}(S)^{**} $ be $ m.b.app.am. $ as $ l^{1}(E) $-module. So it is $ app.m.am $. Hence, by \cite[{Theorem 3.10}]{pourmahmood2}, $\frac{S}{\approx} $ is finite. % %Conversely if $\frac{S}{\approx} $ is finite, then by using \cite[{Theorem 2.11}]{pourmahmood1}, $ l^{1}(S)^{**} $ %is $ m.b.app.am.$ as $ l^{1}(E) $-module. %\end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{proposition}\label{prop26-2-3} % Let $ S=M(G,I) $ be a Brandt semigroup. Then the following are equivalent. %\begin{description} %\item[(i)] %$ l^{1}(S) $ is amenable. %\item[(ii)] %$ l^{1}(S) $ is approximately amenable %\item[(iii)] %$ l^{1}(S) $ is boundedly approximately amenable %\item[(iv)] %$ I $ is finite and $ G $ is amenable. %\end{description} %\end{proposition} %\begin{proof} %{\cite[ {Theorem 4.5 }]{Yazdanpanah 2009}} %\end{proof} %******************************************************************************************* \section{Examples} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{example}\label{2-3-3} Let $ (\mathcal{A}_n) $ be a sequence of amenable Banach algebras. According to \cite[{Remark 5.2}]{gahremani2008} the Banach algebra $ \mathcal{C}={c}_{0}-\oplus_{n=1}^{\infty}\mathcal{A}_{n}^\# $ is $ b.app.am$. Then %by proposition \ref{pro1-2-3},% $ \mathcal{C} $ is $ m.b.app.am. $ as $ \mathbb{C}$-module. If their amenability constant $ M(\mathcal{A}_n) $ (the infimum of the norms of virtual diagonals of $ \mathcal{A}_n$) tends to $ \infty $, then $ \mathcal{C} $ is not amenable. \end{example} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{example}\label{3-3-3} %\begin{description} Suppose that $ K( l^{1}) $ is the Banach algebra of all compact operators on $ l^{1}$. According to \cite[{ Lemma 2.4 }]{gahremani2012} the Banach algebra $ \mathcal{A}^{(n)}=(K(l^1), \| . \| _{n}) $ has a $ l.b.a.i $ with the bound $ 1 $ but the smallest bound of any $ r.b.a.i$ in $ \mathcal{A}^{(n)} $ is $ n+1$. Thus the Banach algebra $\mathcal{A}={c}_{0}-\oplus_{n=1}^{\infty}\mathcal{A}^{(n)}$ has a $ l.b.a.i $ but has no $ m.b.r.a.i $. We can consider $ \mathcal{A}={c}_{0}-\oplus_{n=1}^{\infty}\mathcal{A}^{(n)} $ as a (commutative) Banach $ \mathbb{C}$-module which is $ m.b.app.am.$ but has no $ b.a.i $. (so according to \cite [{Proposition 2.2 }]{amini2004} is not $ m.am$). %\end{description} \end{example} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In the next example we see some Banach algebras that are $ m.b.app.am. $ but are not $ b.app.am $ in the classical case. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{example}\label{exam 6-3-3} \begin{description} \item[(i)] Suppose that $ \mathcal{C} $ is the bicyclic semigroup in two generators, then by \cite{amini2010}, % page 10% $\frac{\mathcal{C}}{\approx} \simeq \mathbb{Z}$. So $ \frac{\mathcal{C}}{\approx} $ is infinite. Applying Proposition \ref{them18-2-3}, $ l^{1}({\mathcal{C}})^{**} $ is not $ m.b.app.am. $ as $ l^{1}(E)$-module. $ \mathcal{C} $ is amenable semigroup {\cite[{Examples}]{Duncan1978}}. So by Proposition \ref{them17-2-3}, $ l^{1}({\mathcal{C}}) $ is $ m.b.app.am. $ as $ l^{1}(E)$-module. However according to \cite[{Theorem}]{zhang 2009}, $ l^{1}({\mathcal{C}}) $ is not $ b.app.am.$. \item[(ii)] Suppose that $ G $ is a group and $ I $ is a non-empty set and $ S=M(G,I) $ is the $ Brandt\ inverse\ semigroup $ corresponding to the group $ G $ and the index set $ I $. It is shown in {\cite[{Example 3.2 }]{pourmahmood1}} that $ \frac{S}{\approx} $ is trivial group. According to Proposition \ref{them18-2-3} $ l^{1}(S)^{**}$ is $m.b.app.am $. Therefore $ l^{1}(S) $ is $ m.b.app.am $ as $ l^{1}(E)$-module by Proposition \ref{pro17-2-3}. However we can get from {\cite[ {Theorem 4.5 }]{pourabbas}}that $ l^{1}(S) $ is $ b.app.am $ $ iff$ $ l^{1}(S) $ is amenable $ iff$ $ I $ is finite and $ G $ is amenable. \end{description} \end{example} % \cite [{Thorem 3.1 }]{pourmahmood۴} %%%%%%%%%%%%*************************************************************************************** %\begin{theorem} %\end{theorem} %\begin{definition} %\end{definition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %{\normalfont (See \cite{RefJ2})} %\begin{definition} %{\it Enter definitions in this environment. } %\end{definition} %\begin{theorem} {\normalfont (See \cite{RefJ2})} %Here, theorems can be stated. %\end{theorem} %\begin{proof} %\end{proof} %\begin{definition} %{\it Here is another definiton. } %\end{definition} %\begin{corollary} %Corollary in here. %\end{corollary} %\begin{remark} %{\it Remark in here.} %\end{remark} %\begin{example} %Check out our amazing numerical example. Check results in Table \ref{tab1}. %\end{example} % For two-column wide figures use %\begin{figure*} % Use the relevant command to insert your figure file. % For example, with the graphicx package use %\includegraphics[width=0.75\textwidth]{example.eps} %figure caption is below the figure %\caption{Please write your figure caption here} %\label{fig:2} % Give a unique label %\end{figure*} %Manuscripts of papers intended for publication should be prepared according to the instruction and submitted through the electronic Journal system by understanding that they have not been published, submitted or accepted for publication elsewhere. %The manuscript should be written in English language. %Authors are invited to submit their papers as PDF and Tex files using the Journal System for online manuscript submission and %management. 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Therefore the authors should keep a copy of all submitted files. % \vspace{4mm}\noindent{\bf Acknowledgements}\\ \noindent If you'd like to thank anyone, place your comments here. % BibTeX users please use one of %\bibliographystyle{spbasic} % basic style, author-year citations %\bibliographystyle{spmpsci} % mathematics and physical sciences %\bibliographystyle{spphys} % APS-like style for physics %\bibliography{} % name your BibTeX data base % Non-BibTeX users please use \begin{center} \begin{thebibliography}{99} % Enter references in alphabetical order and according to the following format. % \bibitem{amini2004} M. Amini, {\it Module amenability for semigroup algebras}, Semigroup Forum. 69 ,no. 2 (2004), 243-254. \bibitem{amini2010} M. Amini, A. Bodaghi and D. Ebrahimi Bagha, {\it Module amenability of the second dual and module topological center of semigroup algebras}, Semigroup Forum 80, no. 2 (2010), 302-312. \bibitem{Jabbari 2017} A. Bodaghi and A. Jabbari, {\it Module pseudo amenability of Banach algebras}, An. Stiint. Univ. Al. I. Cuza. Iasi. Mat. (N. S) 63, no.3 (2017), 449-461. %\bibitem{Ghahramani 2009} % Y. Choi, F. Ghahramani, Y. Zhang, {\it Approximate and pseudo-amenability of various classes of Banach algebras}, J. Funct. Anal. 256 (2009), 3158-3191. \bibitem{Duncan1978} J. Duncan, I. Namioka, {\it Amenability of inverse semigroups and their semigroup algebras}, Proc. R. Soc. Edinb. A 80 (1978), 309-321 \bibitem{gahremani2004} F. Ghahramani and R. J. Loy, {\it Generalized notions of amenability}, J. Funct. Anal. 208, no. 1 (2004), 229-260 \bibitem{gahremani2008} F. Ghahramani, R. J. Loy and Y. Zhang, {\it Generalized notions of amenability, II}, J. Funct. Anal. 254, no. 7 (2008), 1776-1810. \bibitem{gahremani2012} F. Ghahramani and C. J. Read, {\it Approximate identities in approximate amenability}, J. Funct. Anal. 262, no. 9 (2012), 3929-3945. \bibitem{zhang 2009} F. Gheoraghe and Y. Zhang, {\it A note on the approximate amenability of semigroup algebras}, Semigroup Forum. 79, no. 2 (2009), 349-354. \bibitem{pourabbas} M. Maysami Sadr and A. Pourabbas, {\it Approximate amenability of Banach cat- egory algebras with application to semigroup algebras}, Semigroup Forum 79, no.1(2009), 55-64. %\bibitem{pourmahmood3} %A. R. Medghalchi and H. Pourmahmood-Aghababa, {\it Figa-Talamanca-Herz algebras for restricted inverse semigroups and Clifford semigroups}, J. Math. Anal. Appl. 395 (2012) 473-485. \bibitem{pourmahmood1} H. Pourmahmood-Aghababa, {\it (Super) module amenability, module topological centre and semigroup algebras}, Semigroup Forum 81,no. 2 (2010), 344-356. \bibitem{pourmahmood2} H. Pourmahmood-Agababa and A. Bodaghi, {\it Module approximate amenability of Banach Algebras}, Bulletin of the Iranian Mathematical Society Vol. 39, no. 6 (2013), 1137-1158. %\bibitem{khedri 2} %H. Pourmahmood-Aghababa, F.khedri, M.h. Sattari, {\it ‌Bounded Approximate Character Contractibility of Banach Algebras}, Mediterr. J. Math. 17, 5 (2020). https://doi.org/10.1007/s00009-019-1429-4 %\bibitem{Runde} %V. Runde, {\it Lectures on amenability}, Lecture Notes in Mathematics, Vol. %1774, Springer, Berlin (2002). %\bibitem{pourmahmood۴} %H. Pourmahmood-Aghababa, L. Y. Shi and Y. J. Wu, {\it Generalized notions of character amenability}, %Acta Math. Sin. (Engl. Ser.) 29 (2013) 1329-1350. %\bibitem{amini2008} %D. Rezavand, R.Amini, M. Sattari, M.H., Ebrahimi Bagha, {\it %Module Arens regularity for semigroup %algebras}, Semigroup, Forum 77 (2008), 300-305. \bibitem{Yazdanpanah 2009} T. Yazdanpanah and H. Najafi, {\it Module Approximate Amenability for Semi-group Algebras}, J. Applied Sciences. 9 (2009) 1482-1488. %\bibitem{RefJ1} % Format for Journal Reference %F. Author and S. Author, Instructions for authors, {\it Journal Name}, Volume (year), pp-pp. %\bibitem{RefJ2} % Format for Journal Reference %F. Author, S. Author and T. Author, Article title should be written here, {\it Journal Name}, Volume (year), pp-pp. % Format for books %\bibitem{RefB} %F. Author, {\it Book Title Should Be Written Here}, pp-pp, Publisher, place (year) % etc \end{thebibliography} \end{center} {\small \noindent{\bf A. Hemmatzadeh} \noindent Department of Mathematics \noindent Ph.D. Student \noindent Azarbaijan Shahid Madani University \noindent Tabriz, Iran \noindent E-mail: a.hemmatzadeh@azaruniv.ac.ir}\\ {\small \noindent{\bf H. Pourmahmood Aghababa } \noindent Department of Mathematics \noindent Associate Professor of Mathematics \noindent University of Tabriz \noindent Tabriz, Iran \noindent E-mail: pourmahmood@gmail.com, h-p-aghababa@tabriz.ac.ir}\\ {\small \noindent{\bf M.H. Sattari } \noindent Department of Mathematics \noindent Assistant Professor of Mathematics \noindent Azarbaijan Shahid Madani University \noindent Tabriz, Iran \noindent E-mail: sattari@azaruniv.ac.ir} \end{document}