Generalized injectivity of Banach modules
In this paper, we study the notion of $\phi$-injectivity in the special case that $\phi=0$. For an arbitrary locally compact group $G$, we characterize the 0-injectivity of $L^{1(G)$ as a left $L^{1(G)$ module. Also, we show that $L^{1(G)^{**$ and $L^{p(G)$ for $1<p<\infty$ are 0-injective Banach $L^{1(G)$ modules.
1502.04506.tex
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\title[Generalized injectivity of Banach modules]{Generalized injectivity of Banach modules}

\author[M. Fozouni]{Mohammad Fozouni}

\address{Department of Mathematics,  Gonbad Kavous University, P.O. Box 163, Gonbad-e Kavous, Golestan, Iran.}

\email{\textcolor[rgb]{0.00,0.00,0.84}{fozouni@gonbad.ac.ir}}

\subjclass[2010]{Primary: 46M10, Secondary: 43A20, 46H25.}

\keywords{Banach algebra, injective module, character, $\phi$-injective module, locally compact group.}

%\date{Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.
%\newline \indent $^{*}$ Corresponding author}

\begin{abstract}
In this paper,  we study the notion of $\phi$-injectivity in the special case that $\phi=0$. For an arbitrary locally compact group $G$,  we characterize the 0-injectivity of $L^{1}(G)$ as a left $L^{1}(G)$ module. Also, we show that $L^{1}(G)^{**}$ and $L^{p}(G)$ for $1<p<\infty$  are 0-injective Banach $L^{1}(G)$ modules.
\end{abstract} \maketitle
\noindent
\section{introduction}
The homological properties of Banach modules such as injectivity, projectivity, and flatness was first introduced and investigated by Helemskii; see \cite{Helemskii2, Helemskii}. White in \cite{white} gave a quantitative version of these concepts, i.e., he introduced the concepts of $C$-injective, $C$-projective, and $C$-flat Banach modules for a positive real number $C$. Recently Nasr-Isfahani and Soltani Renani introduce a version of these homological concepts based on a character of a Banach algebra $A$ and they showed that every injective (projective, flat) Banach module is a character injective (character projective, character flat respectively) module but the converse is not valid in general. With use of these new homological concepts, 
they gave a new characterization of $\phi$-amenability of  Banach algebra $A$ such that $\phi\in \Delta(A)$ and a necessary condition for $\phi$-contractibility of $A$; see \cite{Nasr}.

%In this paper, we study the notion of  $\phi$-injectivity in the special case that $\phi=0$.
%In this paper we only focus on injectivity and introduce the concept of 0-injective Banach module. We show that each injective Banach module is an injective module but the converse is not valid in general. Also, we show that this concept is different from character injectivity. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{preliminaries}
Let $A$ be a Banach algebra and $\Delta(A)$ denote the character space of $A$, i.e., the space of all non-zero homomorphisms from $A$ onto $\mathbb{C}$. We denote by \textbf{A-mod} and \textbf{mod-A}  the category of all Banach left $A$-modules and all Banach right $A$-modules respectively.  In the case that $A$ has an identity we denote by $\textbf{A-unmod}$ the category of all Banach left unital modules.   For $E, F\in \textbf{A-mod}$, let $_{A}B(E,F)$ be the space of all bounded linear left $A$-module morphisms from $E$ into $F$.

For each Banach space $E$, $B(A,E)$; the Banach algebra consisting of all bounded linear operator from $A$ into $E$, is in $\textbf{A-mod}$ with the following module action:
\begin{equation*}
(a\cdot T)(b)=T(ba)\quad (T\in B(A,E), a, b\in A).
\end{equation*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\label{dfn: inj mod} Let $A$ be a Banach algebra and $J\in \textbf{A-mod}$. We say that $J$ is injective if  for each $F, E\in \textbf{A-mod}$ and  admissible monomorphism $T:F\rightarrow E$   the induced map $T_{J}:\ _{A}B(E,J)\rightarrow\ _{A}B(F,J)$ defined by $T_{J}(R)=R\circ T$ is onto.
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Suppose that $\phi\in \Delta(A)$. For $E\in \textbf{A-mod}$, put
\begin{align*}
\ I(\phi,E)&=\textrm{span}\{a\cdot \xi-\phi(\xi)a : a\in A, \xi\in E\},\\
\ _{\phi}B(A^{\sharp},E)&=\{T\in B(A^{\sharp},E) : T(ab-\phi(b)a)=a\cdot T(b-\phi(b)e^{\sharp}), \quad (a, b\in A)\}.
 \end{align*}
It is clear that $I(\phi,E)=\{0\}$ if and only if the module action of $E$ is given by $a\cdot x=\phi(a)x$ for all $a\in A$ and $x\in E$.

Obviously, $\ _{\phi}B(A^{\sharp},E)$ is a Banach subspace of $B(A^{\sharp},E)$. On the other hand, for each $b\in \ker(\phi)$, if $T\in\ _{\phi}B(A^{\sharp},E) $, then $T(ab)=a\cdot T(b)$ for all $a\in A$. Therefore, we conclude that $\ _{\phi}B(A^{\sharp},E)$ is a Banach left $A$-submodule of $B(A^{\sharp},E)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55

Note that if  $E, F\in \textbf{A-mod}$ and $\rho:E\rightarrow F$ is a left $A$-module homomorphism, we can extend the module actions of $E$ and $F$  from $A$ into $A^{\sharp}$ and $\rho$ to a left $A^{\sharp}$-module homomorphism in the following way:
\begin{align*}
&(a,\lambda)\cdot e=a\cdot e+\lambda e\hspace{0.5cm}(a\in A, \lambda \in \mathbb{C}, e\in E)\\
&(a,\lambda)\cdot f=a\cdot f+\lambda f\hspace{0.5cm}(a\in A, \lambda \in \mathbb{C}, f\in F).
\end{align*}
So, $\rho((a, \lambda)\cdot e)=a\cdot \rho(e)+\lambda \rho(e)=(a, \lambda)\cdot \rho(e)$.

For  Banach spaces $E$ and $F$, $T\in B(E,F)$  is admissible if and only if there exists $S\in
B(F,E)$ such that $T\circ S\circ T=T$.

The following definition of a \emph{$\phi$-injective} Banach module,  introduced  by Nasr-Isfahani and Soltani Renani in \cite{Nasr}.
\begin{definition}\label{dfn: char inj mod} Let $A$ be a Banach algebra, $\phi\in \Delta(A)$ and $J\in \textbf{A-mod}$. We say that $J$ is $\phi$-injective if  for each $F, E\in \textbf{A-mod}$ and  admissible monomorphism $T:F\rightarrow E$  with $I(\phi,E)\subseteq \textrm{Im}T$, the induced map $T_{J}$ is onto.
\end{definition}
By Definition \ref{dfn: inj mod} and \ref{dfn: char inj mod}, one can easily check that each injective module is  $\phi$-injective,  although  by \cite[Example 2.5]{Nasr}, the converse is not valid. In \cite{EFL}, the authors with use of the semigroup algebras, gave two good examples  of $\phi$-injective  Banach modules which they are not injective.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $E, F$ be in $\textbf{A-mod}$. An operator $T\in\ _{A}B(E,F)$ is called a \emph{retraction} if there exists an $S\in\ _{A}B(F,E)$ such that $T\circ S=Id_{F}$. In this case $F$ is called a retract of $E$. Also, an operator $T\in\ _{A}B(E,F)$ is called a \emph{coretraction} if there exists an $S\in\ _{A}B(F,E)$ such that  $S\circ T=Id_{E}$.

 For $E\in \textbf{A-mod}$, let $\ _{\phi}\Pi^{\sharp}:E\rightarrow\ _{\phi}B(A^{\sharp},E)$ be defined by $_{\phi}\Pi^{\sharp}(x)(a)=a\cdot x$ for all $a\in A^{\sharp}$ and $x\in E$. 

\begin{theorem}\cite[Theorem 2.4]{Nasr}\label{Th: Nasr} Let $A$ be a Banach algebra and $\phi\in \Delta(A)$. For $J\in \emph{\textbf{A-mod}}$ the following statements are equivalent.
\begin{enumerate}

  \item $J$ is $\phi$-injective.
  \item $\ _{\phi}\Pi^{\sharp}\in \ _{A}B(J,\ _{\phi}B(A^{\sharp},J))$ is a coretraction.

\end{enumerate}
\end{theorem}
%This paper is indeed a continuation of \cite{EFL2}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{0-injectivity of Banach modules}
In this section, we give the definition of  a \emph{0-injective} Banach left $A$-module and show that this class of Banach modules are strictly larger than the class of injective Banach modules.

For each $E\in \textbf{A-mod}$ define
  \begin{equation*}
  _{0}B(A^{\sharp},E)=\{T\in B(A^{\sharp},E) : T(ab)=a\cdot T(b)  \text{ for all }a, b\in A \}.
  \end{equation*}
  Clearly, $\ _{0}B(A^{\sharp},E)$ is a Banach left $A$-submodule of $B(A^{\sharp},E)$. It is well-known that $E^{*}$ is in $\textbf{mod-A}$ with the following module action:
  \begin{equation*}
  (f\cdot a)(x)=f(a\cdot x)\hspace{0.5cm}(a\in A, x\in E, f\in E^{*}).
  \end{equation*}
%========================================================
 \begin{definition} Let $A$ be a Banach algebra and $E\in \textbf{A-mod}$. We say that $E$ is (left) 0-injective if  for each $F, K\in \textbf{A-mod}$ and admissible monomorphism $T: F\rightarrow K$ for which $A\cdot K=\textrm{span}\{a\cdot k : a\in A, k\in K\}\subseteq \textrm{Im}T$, the induced map $T_{J}$ is onto.
 
 Similarly, one can define the concept of (right) 0-injective  $A$-module. We say that $E\in \textbf{A-mod}$ is 0-flat if $E^{*}\in \textbf{mod-A}$ is (right) 0-injective.
\end{definition}
Clearly, each injective module is 0-injective.

We use of the following characterization of 0-injectivity  in the sequel without giving the reference.
\begin{proposition} Let $A$ be a Banach algebra and $E\in\emph{\textbf{A-mod}}$. Then $E$ is 0-injective if and only if $\ _{0}\Pi^{\sharp}$ is a coretraction.
\end{proposition}
\begin{proof} Suppose $E\in \textbf{A-mod}$ is 0-injective. Take $F=E$, $K=\ _{0}B(A^{\sharp},E)$ and
$T=\ _{0}\Pi$. Then $A\cdot
K\subseteq\textrm{Im}(_{0}\Pi)$ and $a\cdot
T=\ _{0}\Pi(T(a))$  for each $a\in A$ and $T\in K$.
Hence, for the identity map $I_{E}\in
_{A}\hspace{-0.1cm}B(F,E)=_{A}\hspace{-0.1cm}B(E,E)$, there exists
$\rho\in_{A}\hspace{-0.1cm}B(K, E)=_{A}\hspace{-0.1cm}B(
_{0}B(A^{\sharp},E),E)$ such that $\rho\circ\: _{0}\Pi=\rho\circ
T=I_{E}$.

Conversely, let $_{0}\rho:\ _{0}B(A^{\sharp},E)\longrightarrow E$ be  a left $A$-module morphism and a left inverse for the canonical
morphism $_{0}\Pi$. Suppose that $F, K\in
\textbf{A-mod}$ and $T:F\longrightarrow K$ is an admissible
monomorphism such that  $A\cdot K\subseteq\mathrm{Im}T$. Let
$W\in_{A}\hspace{-0.1cm}B(F,E)$ and define the map
$R:K\longrightarrow\ _{0}B(A^{\sharp},E)$ by $$ R(k)(a)=W\circ
T^{\prime}(a\cdot k) \quad (k\in K, a\in A^{\sharp}),$$ where
$T^{\prime}\in B(K,F)$ satisfies $T\circ T^{\prime}\circ T=T$. We
show that $R$ is well defined, i.e., $R(k)\in\ _{0}B(A^{\sharp},E)$
for each $k\in K$. So, we will show that $R(k)(ab)=a\cdot R(k)(b)$
for each $a, b\in A$. By assumption $A\cdot K\subseteq
\mathrm{Im}T$ and so there exist $f\in F$ such that
$b\cdot k=T(f)$. Therefore
\begin{align*}
a\cdot R(k)(b)&=a\cdot W\circ T^{\prime}(b\cdot k)=a\cdot W\circ
T^{\prime}(T(f))\\
&=a\cdot W(f)=W(a\cdot f)\\
&=W\circ
T^{\prime}(T(a\cdot f))=W\circ T^{\prime}(ab\cdot k)\\
&=R(k)(ab).
\end{align*}
Moreover, for each
$b\in A^{\sharp}$ we have
\begin{align*}
R(a\cdot k)(b)&=W\circ T^{\prime}(b\cdot(a\cdot k))=W\circ T^{\prime}(ba\cdot k)\\
&=R(k)(ba)=(a\cdot R(K))(b).
\end{align*}
It follows that $R(a\cdot k)=a\cdot R(k)$. Now, take $S=\
_{0}\rho\circ R\in\ _{A}B(K, E)$. Since $R\circ T=\ _{0}\Pi\circ W$,
we conclude that $S\circ T=W$, which completes the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now,  we  give a sufficient condition for 0-inectivity which provide for us  a large class of Banach algebras $A$ such that they are 0-injective in $\textbf{A-mod}$.

Recall that by \cite[Corollary 2.2.8(i)]{Ramsden}, if $A\in \textbf{A-mod}$ is injective, then $A$ has a right identity. Moreover, the converse is not valid in general even in the case that $A$ has an identity; see Example \ref{Ex}.

\begin{proposition}\label{Prop: sufiicient for 0-inj} Let $A$ be a Banach algebra. If $A$ has an identity, then $A\in \emph{\textbf{A-mod}}$ is 0-injective.
\end{proposition}
\begin{proof} Let $e$ be the identity of $A$. Define $\rho:$$_{0}B(A^{\sharp},A)\rightarrow A$ by $\rho(T)=T(e)$ for all $T\in\ _{0}B(A^{\sharp},A)$. It is obvious that $\rho$ is a left inverse for $_{0}$$\Pi^{\sharp}$, because for each $a\in A$, we have
\begin{align*}
\rho\circ\ _{0}\Pi^{\sharp}(a)=(_{0}\Pi^{\sharp}(a))(e)=ea=a.
\end{align*}
Also, $\rho$ is a left $A$-module morphism, because for each $a\in A$ and $T\in\  _{0}B(A^{\sharp},A)$ we have
\begin{align*}
&\rho(a\cdot T)=(a\cdot T)(e)=T(ea)=T(a)\\
&a\cdot \rho(T)=a\cdot T(e)=T(ae)=T(a).
\end{align*}
Therefore, $A\in \textbf{A-mod}$ is 0-injective.
\end{proof}
For each locally compact group $G$, let $M(G)$ be the Banach algebra consisting of all complex regular Borel measure of $G$ and let $L^{\infty}(G)$ be the space of all measurable complex-valued functions on $G$ which they are essentially bounded; see \cite{Dales} for more details.

The group $G$ is said to be \emph{amenable} if there exists an $m\in L^{\infty}(G)^{*}$ such that $m\geq 0$, $m(1)=1$ and $m(L_{x}f)=m(f)$ for each $x\in G$ and $f\in L^{\infty}(G)$, where $L_{x}f(y)=f(x^{-1}y)$.


As an application of the above theorem we give the following example which shows the difference between 0-injectivity and injectivity.
%=============================================================
\begin{example}\label{Ex} Let $G$ be a non-amenable locally compact group. Then by \cite[Theorem 3.1.2]{Ramsden}, $M(G)\in \textbf{M(G)-mod}$ is not injective, but it  is 0-injective. 
\end{example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The following proposition is 0-injective version of \cite[Corollary 2.3]{Nasr}.
By \cite[Proposition VII.1.35]{Helemskii}, if $E\in \textbf{A-unmod}$, each retract of $E$ is injective. For 0-injective Banach modules we have the following proposition.
\begin{proposition}\label{retract} Let $A$ be a Banach algebra and let $E\in \emph{\textbf{A-mod}}$ be 0-injective. Then each retract of $E$ is also 0-injective.
\end{proposition}
\begin{proof} Let $F\in \textbf{A-mod}$ be a retract of $E$. Also, let $T\in\ _{A}B(E,F)$ and $S\in\ _{A}B(F,E)$ be such that $T\circ S=I_{F}$.

Since $E\in \textbf{A-mod}$ is 0-injective, there exists $_{E}\rho^{\sharp}\in\ _{A}B(_{0}B(A^{\sharp},E),E)$ for which $_{E}\rho^{\sharp}\circ\ _{E}\Pi^{\sharp}(x)=x$ for all $x\in E$.

Now, define the map $_{F}\rho^{\sharp}:\ _{0}B(A^{\sharp}, F)\rightarrow F$ by
\begin{equation*}
\ _{F}\rho^{\sharp}(W)=T\circ \ _{E}\rho^{\sharp}(S\circ W)\quad(W\in\ _{0}B(A^{\sharp}, F)).
\end{equation*}
It is straightforward to check that $_{F}\rho^{\sharp}$ is a left $A$-module morphism. On the other hand, for each $y\in F$ we have
\begin{align*}
_{F}\rho^{\sharp}\circ\ _{F}\Pi^{\sharp}(y)&=\ _{F}\rho^{\sharp}(\ _{F}\Pi^{\sharp}(y))\\
&=T\circ \ _{E}\rho^{\sharp}(S\circ \ _{F}\Pi^{\sharp}(y))\\
&=T\circ \ _{E}\rho^{\sharp}(\ _{E}\Pi^{\sharp}(S(y)))\\
&=T\circ S(y)=y.
\end{align*}
Therefore, $F\in \textbf{A-mod}$ is 0-injective.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%+==theorem======================================================================================================
%%%%+==============================================================================================================
%%%%+==============================================================================================================
Now, we try to characterize the 0-injectivity of $L^{1}(G)$ in $\mathbf{L^{1}(G)}$\textbf{-mod}. First we give the following lemma. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{Nec for 0-inj} Let $A$ be a Banach algebra and $E\in \emph{\textbf{A-mod}}$. If $E$ is 0-injective, then
\begin{equation*}
\ _{0}B(A^{\sharp},E)=\{T :  T=R_{x} \text{ on } A \text{ for some } x\in E\},
\end{equation*}
where $R_{x}a=a\cdot x$ for all $a\in A$.
\end{lemma}
\begin{proof}
Let $E\in \textbf{A-mod}$ be 0-injective. So, there exists$\ _{0}\rho^{\sharp}\in\ _{A}B(_{0}B(A^{\sharp},E),E)$ with $_{0}\rho^{\sharp}\circ\ _{0}\Pi^{\sharp}(x)=x$ for all $x\in E$.

Let $T$ be an element of $\ _{0}B(A^{\sharp},E)$. Hence
\begin{align*}
T(b)=\ _{0}\rho^{\sharp}\circ\ _{0}\Pi^{\sharp}(T(b))&=\ _{0}\rho^{\sharp}(\ _{0}\Pi^{\sharp}(T(b)))\\
&=\ _{0}\rho^{\sharp}(b\cdot T)\\
&=b\cdot \ _{0}\rho^{\sharp}(T).
\end{align*}
Take $x_{0}=\ _{0}\rho^{\sharp}(T)$. So, $T=R_{x_{0}}$ on $A$ and this completes the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%
Recall that $E\in \textbf{A-mod}$ is faithful in $A$, if for each $x\in E$,  the relation $a\cdot x=0$ for all $a\in A$, implies $x=0$.
\begin{theorem}\label{0-inj of L1} Let $G$ be a locally compact group. Then $L^{1}(G)\in \mathbf{L^{1}(G)}$\emph{\textbf{-mod}} is 0-injective if and only if $G$ is discrete.
\end{theorem}
\begin{proof} Let $G$ be a discrete group. Then $L^{1}(G)$ is unital and so the result follows from Proposition \ref{Prop: sufiicient for 0-inj}.

Conversely, let $G$ be  non-discrete. So, $L^{1}(G)\neq M(G)$. Suppose that $\mu\in M(G)\setminus L^{1}(G)$. Since $L^{1}(G)$ is an ideal of $M(G)$, the operator $T_{\mu}$ defined by
\begin{equation*}
T_{\mu}((f, \lambda))=f\cdot \mu\quad((f,\lambda)\in L^{1}(G)^{\sharp}),
\end{equation*}
 is in $\ _{0}B(L^{1}(G)^{\sharp}, L^{1}(G))$, but it is not of the form $R_{x}$ for some $x\in L^{1}(G)$, because $M(G)$ is  faithful in $L^{1}(G)$. Therefore, by Lemma \ref{Nec for 0-inj}, $L^{1}(G)$ in $\mathbf{L^{1}(G)}$\textbf{-mod} is not 0-injective.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Consider the Banach algebra $A=A_{\phi}(X)$ \cite[Example 2.5]{Nasr} with the product defined by $ab=\phi(a)b$ for all $a, b\in A$ where $X$ is a Banach space with $\dim(X)\geq 2$. By \cite[Example 2.5]{Nasr}, $A_{\phi}(X)\in \mathbf{A_{\phi}(X)}$\textbf{-mod} is $\phi$-injective. 

%By use of the following theorem and the Banach algebra $A_{\phi}(X)$, we can give an example of a Banach modules which it is $\phi$-injective but the are not 0-injective.
%\begin{theorem}
%\emph{$A_{\phi}(X)\in \mathbf{A_{\phi}(X)}$\textbf{-mod}} is 0-injective if and only if $\dim(X)=2$. 
%\end{theorem}
%\begin{proof}
%To see this one can easily check that $_{0}B(A_{\phi}(X)^{\sharp},A_{\phi}(X))=B(A_{\phi}(X)^{\sharp},A_{\phi}(X))$. Therefore the result follows from \cite[Proposition 3.6]{Ramsden Paper} and this fact that every injective module is 0-injective.
%\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Recall that a Banach algebra $A$ is  left 0-amenable if for every Banach $A$-bimodule $X$ with $a\cdot x=0$ for all $a\in A$ and $x\in X$, every continuous derivation $D:A\rightarrow X^{*}$ is inner,  or equivalently, $H^{1}(A, X^{*})=0$ where  $H^{1}(A, X^{*})$ denotes the first cohomology group of $A$ with coefficients in $X^{*}$; see \cite{Nasr-Studia} for more details.

Now, we investigate the relation between 0-injectivity and 0-amenability.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $E, F\in \textbf{A-mod}$. Suppose that  $Z^{1}(A\times E, F)$ denotes the Banach space of all continuous bilinear maps $B: A\times E\longrightarrow F$ satisfying 
\begin{equation*}
a\cdot B(b, \xi)-B(ab, \xi)+B(a, b\cdot \xi)=0\quad (a, b\in A, \xi\in E).
\end{equation*}
 Define $\delta_{0}: B(E, F)\longrightarrow Z^{1}(A\times E, F)$ by $(\delta_{0}T)(a, \xi)=a\cdot T(\xi)-T(a\cdot \xi)$ for all $a\in A$ and $\xi \in E$. Then we have
 \begin{equation*}
 \mathrm{Ext}_{A}^{1}(E, F)=Z^{1}(A\times E, F)/\textrm{Im}\delta_{0}.
 \end{equation*}
By \cite[Proposition VII.3.19]{Helemskii2}, we know that $\mathrm{Ext}_{A}^{1}(E, F)$ is topologically isomorphic to $H^{1}(A, B(E, F))$ where $B(E, F)$ is a Banach $A$-bimodule with the following module actions:
\begin{equation*}
(a\cdot T)(\xi)=a\cdot T(\xi),\quad (T\cdot a)(\xi)=T(a\cdot \xi)\hspace{0.5cm}(a\in A, \xi\in E, T\in B(E,F)).
\end{equation*}
 
 To see further details about $\mathrm{Ext}_{A}^{1}(E, F)$; see \cite{Helemskii}.
 

\begin{lemma}\label{Lemma}
Let  $E\in \emph{\textbf{A-mod}}$. If $\mathrm{Ext}_{A}^{1}(F,E)=\{0\}$ for all $F\in \emph{\textbf{A-mod}}$ with $A\cdot F=0$, then $E\in \emph{\textbf{A-mod}}$ is 0-injective.
\end{lemma}
\begin{proof}
 To show this,  let $K, W\in \textbf{A-mod}$ and $T:K\rightarrow W$ be an admissible monomorphism with $A\cdot W\subseteq \textrm{Im}T$. We claim that the induced map $T_{E}$  is onto.

We know that the short complex  $0\rightarrow K\xrightarrow{T} W\xrightarrow{q} \frac{W}{\textrm{Im}T}\rightarrow 0$ is admissible where $q$ is the quotient map. But for all $a\in A$ and $x\in W$, $a\cdot (x+\textrm{Im}T)=\textrm{Im}T$, because $A\cdot W\subseteq \textrm{Im}T$. Therefore, by assumption $\textrm{Ext}_{A}^{1}(\frac{W}{\textrm{Im}T},E)=\{0\}$. Now, by \cite[III Theorem 4.4]{Helemskii}, the  complex 
\begin{equation*}
0\rightarrow\ _{A}B(\frac{W}{\textrm{Im}T},E)\rightarrow\ _{A}B(W,E)\xrightarrow{T_{E}}\ _{A}B(K,E)\rightarrow \textrm{Ext}_{A}^{1}(\frac{W}{\textrm{Im}T},E)\rightarrow \cdots,
\end{equation*}
is exact. Therefore, $T_{E}$ is onto.

\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Recall that if $E, F$ be two Banach spaces and $E\widehat{\otimes}F$ denotes the projective tensor product space, then $(E\widehat{\otimes}F)^{*}$ is isomorphic to $B(E, F^{*})$ as two Banach spaces with the pairing
\begin{equation*}
<Tx, y>=T(x\otimes y)\hspace{0.5cm}(x\in E, y\in F, T\in (E\widehat{\otimes}F)^{*}).
\end{equation*}
Also, note that $E\widehat{\otimes}F$ is  isometrically isomorphic to $F\widehat{\otimes}E$ as two Banach spaces. 
\begin{theorem}\label{Th: 0-amenable} Let $A$ be a Banach algebra. Then $A$ is left 0-amenable if and only if each $J\in \emph{\textbf{mod-A}}$ is 0-flat.
\end{theorem}
\begin{proof} 
Suppose that $A$ is left 0-amenable. We show that $\textrm{Ext}_{A}^{1}(E,J^{*})=\{0\}$ for all $E\in\textbf{A-mod}$ with $A\cdot E=0$. We have
\begin{equation*}
\textrm{Ext}_{A}^{1}(E, J^{*})=H^{1}(A,B(E,J^{*}))=H^{1}(A, (E\widehat{\otimes}J)^{*})=\{0\},
\end{equation*}
because $E\widehat{\otimes}J\in \textbf{mod-A}$ has the module action, $a\cdot z=0$ for all $z\in E\widehat{\otimes}J$. Therefore,  by Lemma \ref{Lemma}, $J^{*}\in \textbf{A-mod}$ is 0-injective.

Conversely, let $J\in \textbf{mod-A}$ be 0-flat. So, for Banach right $A$-module $\mathbb{C}$ with module action $\lambda\cdot a=0$ for all $a\in A$ and  $\lambda\in \mathbb{C}$ we have
\begin{align*}
H^{1}(A, J^{*})=H^{1}(A, B(J, \mathbb{C}))&=H^{1}(A, B(J, \mathbb{C}^{*}))\\
&=H^{1}(A, (J\widehat{\otimes}\mathbb{C})^{*})\\
&=H^{1}(A, (\mathbb{C}\widehat{\otimes}J)^{*})\\
&=H^{1}(A, B(C,J^{*}))\\
&=\textrm{Ext}_{A}^{1}(\mathbb{C},J^{*})\\
&=0.
\end{align*}
Hence, if we take $J$ a left $A$ module with module action $a\cdot x=0$ for all $a\in A$ and $x\in J$, then the above relation implies that $A$ is 0-amenable.
\end{proof}
By \cite[Corollary 4.7]{Dales. Polyakov}, we know that $L^{1}(G)^{**}\in \mathbf{L^{1}(G)}$\textbf{-mod} is injective if and only if $G$ is an amenable group. Also, if $1<p<\infty$ by \cite[Theorem 9.6]{DDPR}, $L^{p}(G)\in \mathbf{L^{1}(G)}$\textbf{-mod} is injective if and only if $G$ is an amenable group.
\begin{corollary} Let $G$ be a locally compact group, $1<p<\infty$ and $E\in \mathbf{L^{1}(G)}$\emph{\textbf{-mod}} be  $L^{p}(G)$ or $L^{1}(G)^{**}$. Then $E\in \mathbf{L^{1}(G)}$\emph{\textbf{-mod}} is 0-injective.
\end{corollary}
\begin{proof}
Since $L^{1}(G)$ has a bounded approximate identity by \cite[Proposition 3.4 (i)]{Nasr-Studia}, we know that $L^{1}(G)$ is 0-amenable. So, by Theorem \ref{Th: 0-amenable} we conclude the result. The second part follows similarly, because for each $1<p<\infty$ we know that $L^{q}(G)^{*}=L^{p}(G)$ where $q$ satisfies the relation $q^{-1}+p^{-1}=1$.
\end{proof}
%%%%%%%%%%%%%%
\begin{remark}
In general, by \cite[Proposition 3.4 (i)]{Nasr-Studia}, if $A$ is a Banach algebra with a bounded approximate identity, then each $E\in \textbf{mod-A}$ is 0-flat.
\end{remark}
%==========================================================
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%====================Bibligraphy++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%=============================++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
\bibliographystyle{amsplain}
\begin{thebibliography}{99}
\bibitem{Dales}  H.G. Dales, \textit{Banach Algebras and Automatic Continuity}, Clarendon Press, Oxford, 2000.

\bibitem{Dales. Polyakov}  H.G. Dales, M.E. Polyakov, \textit{Homological properties of modules over group algebras}, Proc. London Math. Soc. 89 (2004), 390--426.

\bibitem{DDPR} H.G. Dales, M. Daws, H.L. Pham,  P. Ramsden, \textit{Multi-norms and the injectivity of $L^{p}(G)$}, J. London Math. Soc. (2012) 86 (3), 779--809.

\bibitem{EFL}  M. Essmaili, M. Fozouni, J. Laali, \textit{Hereditary properties of character injectivity with application to semigroup algebras}. Ann. Funct. Anal. 6 (2015), no 2, 162--172.

%\bibitem{EFL2} ----------,  \textit{$\phi$-injectivity and character injectivity of Banach modules}, arXiv: 1502.04506 [math.FA].

\bibitem{Helemskii2}  A.Ya. Helemskii, \textit{Banach and Locally Convex Algebras}, Clarendon Press, Oxford, 1993.

\bibitem{Helemskii} ----------, \textit{The Homology of Banach and Topological Algebras}, Kluwer Academic
Publishers Group, Dordrecht, 1989.

\bibitem{Nasr-Studia} R. Nasr-Isfahani, S. Soltani Renani, \textit{Character contractibility of Banach algebras and
homological properties of Banach modules}, Studia Math. 202 (2011), 205--225.

\bibitem{Nasr} ----------, \textit{Character injectivity and projectivity of Banach modules}, Quart. J. Math. (2014) 65 (2), 665--676.

\bibitem{Ramsden Paper} P. Ramsden, \textit{Homological properties of modules over semigroup algebras}, J. Funct. Anal. 258 (2010) 3988--4009.

\bibitem{Ramsden} ----------, \textit{Homological properties of semigroup algebras}, Ph. D. Thesis, University of Leeds, 2008.

\bibitem{white} M.C. White, \textit{Injective modules for uniform algebras}, Proc. London Math. Soc. (3) 73 (1996), 155--184.
\end{thebibliography}

\end{document}

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% End of journal.tex
%------------------------------------------------------------------------------

1502.04506.extracted_bib
{Dales H.G. Dales, Banach Algebras and Automatic Continuity, Clarendon Press, Oxford, 2000.
1502.04506.extracted_bib
{Dales. Polyakov H.G. Dales, M.E. Polyakov, Homological properties of modules over group algebras, Proc. London Math. Soc. 89 (2004), 390--426.
1502.04506.extracted_bib
{DDPR H.G. Dales, M. Daws, H.L. Pham, P. Ramsden, Multi-norms and the injectivity of $L^{p(G)$, J. London Math. Soc. (2012) 86 (3), 779--809.
1502.04506.extracted_bib
{EFL M. Essmaili, M. Fozouni, J. Laali, Hereditary properties of character injectivity with application to semigroup algebras. Ann. Funct. Anal. 6 (2015), no 2, 162--172. %
1502.04506.extracted_bib
{EFL2 ----------, $\phi$-injectivity and character injectivity of Banach modules, arXiv: 1502.04506 [math.FA].
1502.04506.extracted_bib
{Helemskii2 A.Ya. Helemskii, Banach and Locally Convex Algebras, Clarendon Press, Oxford, 1993.
1502.04506.extracted_bib
{Helemskii ----------, The Homology of Banach and Topological Algebras, Kluwer Academic Publishers Group, Dordrecht, 1989.
1502.04506.extracted_bib
{Nasr-Studia R. Nasr-Isfahani, S. Soltani Renani, Character contractibility of Banach algebras and homological properties of Banach modules, Studia Math. 202 (2011), 205--225.
1502.04506.extracted_bib
{Nasr ----------, Character injectivity and projectivity of Banach modules, Quart. J. Math. (2014) 65 (2), 665--676.
1502.04506.extracted_bib
{Ramsden Paper P. Ramsden, Homological properties of modules over semigroup algebras, J. Funct. Anal. 258 (2010) 3988--4009.
1502.04506.extracted_bib
{Ramsden ----------, Homological properties of semigroup algebras, Ph. D. Thesis, University of Leeds, 2008.
1502.04506.extracted_bib
{white M.C. White, Injective modules for uniform algebras, Proc. London Math. Soc. (3) 73 (1996), 155--184.