\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*}
 <F.\alpha,x> =<F,\alpha.x>\qquad &,&\qquad <\alpha.F,x>=<F,x.\alpha>\vspace{0.3cm}\\
<F.a,x>=<F,a.x>\qquad &,& \qquad <a.F,x>=<F,x.a>
\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. The authors will be asked to supply the title of the paper, abstract, keywords and AMS classification, and to %suggest of the Editorial Board whose area lies closest to the subject.
%Every manuscript considered for publication is passed to referees for peer review, and then it is subject to final approval by the %Editorial Board.
%After the paper has been accepted, the author(s) are asked to provide the source in LaTeX format together with all %accompanying files, according to the guidelines in preparation manuscript below.
%The manuscripts must be in their final form, essential changes at the proof stage will be considered only exceptionally.
%The Editors cannot be held responsible for possible loss of the manuscript. 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}
