# Generalized notions of character amenability

Luo Yi Shi  Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA<EMAIL_ADDRESS>,  YU Jing Wu  Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA<EMAIL_ADDRESS>and  You Qing Ji  Department of Mathematics
Jilin University
Changchun 130012
P.R. CHINA<EMAIL_ADDRESS>
###### Abstract.

In this paper the concepts of character contractibility, approximate character
amenability (contractibility) and uniform approximate character amenability
(contractibility) are introduced. We are concerned with the relations among
the generalized concepts of character amenability for Banach algebra. We prove
that approximate character amenability and approximate character
contractibility are the same properties, as are uniform approximate character
amenability and character amenability, as are uniform approximate character
contractibility and character contractibility. For commutative Banach algebra,
we prove that character contractibility and contractibility are the same
properties. Moreover, general theory for those concepts is developed.

###### Key words and phrases:

Character amenability; Approximately inner; Derivation

###### 2000 Mathematics Subject Classification:

46H20(46H25 47B47)

##  1\. Introduction

The concept of amenability for Banach algebras was first introduced by B. E.
Johnson in [14]. Suppose that $A$ is a Banach algebra and that $E$ is a Banach
$A$-bimodule, then $E^{*}$, the dual of $E$, has a natural Banach $A$-bimodule
structure defined by

$(a\cdot f)(x)=f(x\cdot a),(f\cdot a)(x)=f(a\cdot x),a\in A,x\in E,f\in
E^{*}.$

Such a Banach $A$-bimodule $E^{*}$ is called a dual $A$-bimodule. A derivation
$D:A\rightarrow E^{*}$ is a continuous linear map such that $D(ab)=a\cdot
D(b)+D(a)\cdot b$ for all $a,b\in A$. Given $f\in E^{*}$, the inner derivation
$\delta_{f}:A\rightarrow E^{*}$, is defined by $\delta_{f}(a)=a\cdot f-f\cdot
a$. According to Johnson’s original definition, a Banach algebra $A$ is
amenable if every derivation from $A$ into the dual $A$-bimodule $E^{*}$ is
inner for all Banach $A$-bimodules $E$. As a complement to this notion, a
Banach algebra $A$ is contractible if every derivation from $A$ into every
Banach $A$-bimodule is inner [4, 12].

Ever since its introduction, the concept of amenability has occupied an
important place in the research of Banach algebras, operator algebras and
harmonic analysis. For example, an early result of Johnson [20] shows that the
amenability of the group algebra $L^{1}(G)$, for G a locally compact group, is
equivalent to the amenability of the underlying group G. Results of Connes and
Haagerup show that a $C^{*}$-algebra is amenable if and only if it is nuclear
[20]. However it has been realized that amenability is essentially a
finiteness condition, and in many instances is too restrictive. As for
contractibility, it is even conjectured in [12] that a contractible Banach
algebra must be finite dimensional (see also [21]). For this reason by
relaxing some of the constrains in the definition of amenability new concepts
have been introduced. The most notable are the concepts of Connes amenability
[11, 13], weak amenability [2, 5] and character amenability [17, 18]. More
recently, F. Ghahramani and R. J. Loy have introduced and studied the concepts
of approximate amenability (contractibility) and uniform approximate
amenability (contractibility) for Banach algebras [9, 10]. In this paper we
introduce the generalized concepts of character amenability (see Definition
1.2-1.7). We are concerned with the relations among those concepts and shall
develop general theory for them.

######  Definition 1.1.

A derivation $D:A\rightarrow E$ is approximately inner, if there exists a net
$\\{\xi_{i}\\}\subset E$ such that
$D(a)=\lim\limits_{i}(a\cdot\xi_{i}-\xi_{i}\cdot a)$ for all $a\in A$, the
limit being in norm.

Note that $\\{\xi_{i}\\}$ in the above is not necessarily bounded. The
stronger assumption, that $D$ is in the uniform closure of the inner
derivations, has been well studied in the $C^{*}$-algebra case with the
restriction to the single Banach $A$-bimodule $E=A$ (see [1, 15]). The case of
semigroup algebras is considered in [3] for the Banach $A$-bimodule $A^{*}$.

Let $A$ be a Banach algebra and $\sigma(A)$ be the set of all non-zero
multiplicative linear functionals on $A$. If $\varphi\in\sigma(A)\cup\\{0\\}$
and $E$ is an arbitrary Banach space, then $E$ can be viewed as a Banach left
or right $A$-module by the following actions. For $a\in A,x\in E$:

$None$ $a\cdot x=\varphi(a)x,$ $None$ $x\cdot a=\varphi(a)x.$

If the left action of $A$ on $E$ is given by (2.1), then it is easily verified
that the right action of $A$ on the dual $A$-module $E^{*}$ is given by
$f\cdot a=\varphi(a)f$ for all $f\in E^{*},a\in A$. Throughout, by a
$(\varphi,A)$-bimodule $E$, we mean that $E$ is a Banach $A$-bimodule for
which the left module action is given by (2.1). $(A,\varphi)$-bimodule is
defined similarly by (2.2).

######  Definition 1.2.

Let $A$ be a Banach algebra and $\varphi\in\sigma(A)\cup\\{0\\}$. $A$ is
approximately $\varphi$-amenable, if every derivation $D$ from $A$ into the
dual $A$-bimodule $E^{*}$ is approximately inner for all
$(\varphi,A)$-bimodules $E$.

######  Definition 1.3.

A Banach algebra $A$ is approximately right character amenable, if for every
$\varphi\in\sigma(A)\cup\\{0\\}$ and every $(\varphi,A)$-bimodule $E$, every
derivation $D$ from $A$ into the dual $A$-bimodule $E^{*}$ is approximately
inner.

Approximately left character amenability is defined similarly by considering
$(A,\varphi)$-bimodules $E$.

$A$ is approximately character amenable if it is both approximately left and
right character amenable.

######  Definition 1.4.

Let $A$ be a Banach algebra and $\varphi\in\sigma(A)\cup\\{0\\}$. $A$ is
$\varphi$-contractible, if every derivation $D:A\rightarrow E$ is inner, for
all $(A,\varphi)$-bimodules $E$.

######  Definition 1.5.

A Banach algebra $A$ is left character contractible, if for every
$\varphi\in\sigma(A)\cup\\{0\\}$ and every $(\varphi,A)$-bimodule $E$, every
derivation $D:A\rightarrow E$ is inner.

Right character contractibility is defined similarly by considering
$(A,\varphi)$-bimodules $E$.

$A$ is character contractible if it is both left and right character
contractible.

######  Definition 1.6.

Let $A$ be a Banach algebra and $\varphi\in\sigma(A)\cup\\{0\\}$. $A$ is
approximately $\varphi$-contractible, if every derivation $D:A\rightarrow E$
is approximately inner for all $(A,\varphi)$-bimodules $E$.

######  Definition 1.7.

A Banach algebra $A$ is approximately left character contractible, if for
every $\varphi\in\sigma(A)\cup\\{0\\}$ and every $(\varphi,A)$-bimodule $E$,
every derivation $D:A\rightarrow E$ is approximately inner.

Approximately right character contractible is defined similarly by considering
$(A,\varphi)$-bimodules $E$.

$A$ is approximately character contractible if it is both approximately left
and right character contractible.

Any statement about approximate left character amenability (contractibility)
turns into an analogous statement about approximate right character
amenability (contractibility, respectively) by simply replacing $A$ by its
opposite algebra. Approximate right character amenability (contractibility) of
$A$ is equivalent to approximate $\varphi$-amenability
($\varphi$-contractibility) for all $\varphi\in\sigma(A)$ together with
approximate $0$-amenability ($0$-contractibility, respectively).

The qualifier uniform on the above definitions will indicate that the
convergence of the net is uniform over the unit ball of $A$. Similarly $w^{*}$
will indicate that convergence is in the appropriate $w^{*}$-topology. Let
$CC$ denote character contractibility, $UACC$ denote uniform approximate
character contractibility, $ACC$ denote approximate character contractibility,
$CA$ denote character amenability, $UACA$ denote uniform approximate character
amenability and $ACA$ denote approximate character amenability. Clearly the
relations among the various character amenability are as follows:

$Contractibility\Rightarrow CC\Rightarrow UACC\Rightarrow ACC$

$\Downarrow\ \ \ \ \not\Uparrow\ \ \ \ \ \ \ \ \ \ \ \ \ \ \Downarrow\ \ \ \ \
\ \ \ \ \ \ \Downarrow\ \ \ \ \ \ \ \ \Downarrow$

$Amenability\Rightarrow CA\Rightarrow UACA\Rightarrow ACA$

In this paper, we prove that $ACA\Leftrightarrow ACC$ (Theorem 5.2),
$UACA\Leftrightarrow CA$ (Theorem 5.7), $UACC\Leftrightarrow CC$ (Theorem
5.9). Moreover, in Section 6, Example 1 shows that $ACA\nRightarrow UACA$.
Example 2 shows that $CA\nRightarrow Amenability$ and $CA\nRightarrow CC$.
Thus $ACC\nRightarrow UACC$. For commutative Banach algebra, we obtain that
$Contractibility\Leftrightarrow CC$ (Theorem 5.11). For non-commutative Banach
algebra, Example 3 shows that $CC\nRightarrow Amenability$.

In this paper, the second dual $A^{**}$ of a Banach algebra $A$ will always be
equipped with the first Arens product [7] which is defined as follows. For
$a,b\in A,f\in A^{*}$ and $m,n\in A^{**}$, the elements $f\cdot a$ and $m\cdot
f$ of $A^{*}$ and $mn\in A^{**}$ are defined by

$(f\cdot a)(b)=f(ab),(m\cdot f)(b)=m(f\cdot b),mn(f)=m(n\cdot f),$

respectively. With this multiplication, $A^{**}$ is a Banach algebra and $A$
is a subalgebra of $A^{**}$. Moreover, for all $m,n\in A^{**}$ and
$\varphi\in\sigma(A)$, $(mn)(\varphi)=m(\varphi)n(\varphi)$. Consequently,
each $\varphi\in\sigma(A)$ extends uniquely to some element $\varphi^{**}$ of
$\sigma(A^{**})$. The kernel of $\varphi^{**}$, $\ker(\varphi^{**})$, contains
$\ker\varphi$ in the same sense that $A^{**}$ naturally contains $A$. Since
each of these ideals has codimension 1, the theory of second dual shows that
$\ker\varphi$ is $w^{*}$-dense in $\ker(\varphi^{**})$ and that
$\ker\varphi^{**}=(\ker\varphi)^{**}$. For further details the reader is
referred to [7].

The organization of the paper is as follows. In Section 2 we characterize
(uniform) approximate character amenability in three different ways. In
Section 3 we are concerned with hereditary properties of (uniform) approximate
character amenability.

In Section 4 we characterize character contractibility and (uniform)
approximate character contractibility in two different ways.

Section 5 is devoted to the relations among generalized notions of character-
amenability. We prove that approximate character amenability and approximate
character contractibility are the same properties, as are uniform approximate
character amenability (contractibility) and character amenability
(contractibility, respectively). For commutative Banach algebra, character
contractibility and contractibility are the same properties.

Section 6 gives three examples. The first example shows that there exists a
Banach algebra which is approximately character amenable but not uniformly
approximately character amenable. The second example shows that there exists a
Banach algebra which is character amenable but neither amenable nor character
contractible. The last example shows that there exists a Banach algebra which
is character contractible but not amenable.

##  2\. Characterization of approximately character amenability

In this section, we first characterize (uniform) approximate
$\varphi$-amenability in three different ways and then characterize (uniform)
approximate character amenability in these different ways.

Suppose that $A$ is a Banach algebra, we make $E=A$ into a Banach $A$-bimodule
as usual by $a\cdot b=ab,b\cdot a=ba,$ for all $a\in A,b\in E$. Then
$A^{*},A^{**}$ are dual $A$-bimodules and the module actions are given by
$(a\cdot f)(b)=f(ba),(f\cdot a)(b)=f(ab);(a\cdot m)(f)=m(f\cdot a),(m\cdot
a)(f)=m(a\cdot f)$ for all $a,b\in A,f\in A^{*},m\in A^{**}$. If we take
$A^{**}$ with the first Arens product, then $am=a\cdot m,ma=m\cdot a$ for all
$a\in A,m\in A^{**}$. Let $\varphi\in\sigma(A)$, a net
$\\{m_{\alpha}\\}\subset A^{**}$ is called an approximate $\varphi$-mean, if
$m_{\alpha}(\varphi)=1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$. The
following proposition characterizes approximate $\varphi$-amenable in terms of
approximately $\varphi$-mean. The corresponding result characterizing
$\varphi$-amenability of a Banach algebra was obtained in [16, Theorem 1.1].

######  Proposition 2.1.

Let $A$ be a Banach algebra and $\varphi\in\sigma(A)$. Then the following are
equivalent:

$(i)$ $A$ is (uniformly) approximately $\varphi$-amenable;

$(ii)$ There exists a net $\\{m_{\alpha}\\}\subset A^{**}$ such that
$m_{\alpha}(\varphi)=1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$, for all $a\in A$ (uniformly
on the unit ball of $A$, respectively);

$(iii)$ There exists a net $\\{m_{\alpha}\\}\subset A^{**}$ such that
$m_{\alpha}(\varphi)\rightarrow 1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$, for all $a\in A$ (uniformly
on the unit ball of $A$, respectively);

$(iv)$ Give $(\ker\varphi)^{**}$ a dual $A$-bimodule structure by taking the
right action to be $m\cdot a=\varphi(a)m$ for $m\in A^{**}$ and taking the
left action to be the natural one. Then any continuous derivation
$D:A\rightarrow(\ker\varphi)^{**}$is (uniformly, respectively) approximately
inner.

Proof. $(i)\Rightarrow(iv)$ and $(ii)\Rightarrow(iii)$ is clear. Therefore, in
order to establish the proposition it suffices to show the implications
$(i)\Rightarrow(ii)$, $(iii)\Rightarrow(i)$ and $(iv)\Rightarrow(ii)$.

$(i)\Rightarrow(ii)$ We first define an action of $A$ on $E=A^{*}$ by

$a*f=\varphi(a)f,f*a=f\cdot a,~{}a\in A,f\in E,$

then $E$ is $(\varphi,A)$-bimodule and $A^{**}$ is a dual $A$-bimodule and
module actions are defined by $a*m=a\cdot m,m*a=\varphi(a)m$, for all $a\in
A,m\in A^{**}$.

We know that $\varphi\in A^{*}$, and $a\cdot\varphi=\varphi\cdot
a=\varphi(a)\varphi$. Therefore
$\mathbb{C}\varphi=\\{\lambda\varphi,\lambda\in\mathbb{C}\\}$ is a closed
submodule of $A^{*}$ and $A^{*}/\mathbb{C}\varphi$ is a $(\varphi,A)$-bimodule
for which the module actions are given by $a\cdot[f]=\varphi(a)[f],[f]\cdot
a=[f*a]$, for all $a\in A,[f]\in A^{*}/\mathbb{C}\varphi$.

Choose any $m\in A^{**}$ with $m(\varphi)=1$, and define a derivation
$D:A\rightarrow A^{**}$ by $D(a)=a\cdot m-\varphi(a)m$, then $D(a)\in\\{n\in
A^{**},n(\varphi)=0\\}=\\{\mathbb{C}\varphi\\}^{\bot}\cong(A^{*}/\mathbb{C}\varphi)^{*}$.
Since $A$ is approximately $\varphi$-amenable, it follows that there exists a
net $\\{n_{\alpha}\\}\subset\\{\mathbb{C}\varphi\\}^{\bot}$ such that
$D(a)=\lim\limits_{\alpha}(a\cdot n_{\alpha}-\varphi(a)n_{\alpha})$. Set
$m_{\alpha}=m-n_{\alpha}$, then $m_{\alpha}(\varphi)=1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$.

$(iii)\Rightarrow(i)$ Let $\\{m_{\alpha}\\}\subset A^{**}$ such that
$m_{\alpha}(\varphi)\rightarrow 1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$. Let $E$ be a
$(\varphi,A)$-bimodule. Also, let $D:A\rightarrow E^{*}$ be a continuous
derivation, and let $D^{{}^{\prime}}=D^{*}|_{E}:E\rightarrow A^{*}$ and
$g_{\alpha}=(D^{{}^{\prime}})^{*}(m_{\alpha})\in E^{*}$. Then, for all $a,b\in
A$ and $x\in E$,

$\displaystyle D^{{}^{\prime}}(x\cdot a)(b)$ $\displaystyle=$ $\displaystyle
D(b)(x\cdot a)$ $\displaystyle=$ $\displaystyle D(ab)(x)-D(a)(b\cdot x)$
$\displaystyle=$ $\displaystyle(D^{{}^{\prime}}(x)\cdot
a)(b)-D(a)(x)\varphi(b),$

and hence $D^{{}^{\prime}}(x\cdot a)=D^{{}^{\prime}}(x)\cdot
a-D(a)(x)\varphi.$ This implies that

$\displaystyle(a\cdot g_{\alpha})(x)$ $\displaystyle=$ $\displaystyle
g_{\alpha}(x\cdot a)$ $\displaystyle=$
$\displaystyle(D^{{}^{\prime}})^{*}(m_{\alpha})(x\cdot a)$ $\displaystyle=$
$\displaystyle m_{\alpha}(D^{{}^{\prime}}(x\cdot a))$ $\displaystyle=$
$\displaystyle m_{\alpha}(D^{{}^{\prime}}(x)\cdot
a)-D(a)(x)m_{\alpha}(\varphi)$ $\displaystyle=$ $\displaystyle(a\cdot
m_{\alpha})(D^{{}^{\prime}}(x))-D(a)(x)m_{\alpha}(\varphi).$

It follows that

$\displaystyle||(a\cdot g_{\alpha})(x)-\varphi(a)g_{\alpha}(x)+D(a)(x)||$
$\displaystyle\leq$ $\displaystyle||\varphi(a)g_{\alpha}(x)-(a\cdot
m_{\alpha})(D^{{}^{\prime}}(x))||$ $\displaystyle+$
$\displaystyle||D(a)(x)-D(a)(x)m_{\alpha}(\varphi)||.$

Hence,

$\displaystyle\lim\limits_{\alpha}||(a\cdot
g_{\alpha})(x)-\varphi(a)g_{\alpha}(x)+D(a)(x)||$ $\displaystyle\leq$
$\displaystyle\lim\limits_{\alpha}\\{||\varphi(a)m_{\alpha}(D^{{}^{\prime}}(x))-(a\cdot
m_{\alpha})(D^{{}^{\prime}}(x))||$ $\displaystyle+$
$\displaystyle||\varphi(a)g_{\alpha}(x)-\varphi(a)m_{\alpha}(D^{{}^{\prime}}(x))||$
$\displaystyle+$ $\displaystyle||D(a)(x)-D(a)(x)m_{\alpha}(\varphi)||\\}.$

Thus, for each $a\in A$, $D(a)=\lim\limits_{\alpha}\varphi(a)g_{\alpha}-a\cdot
g_{\alpha}$. Combining this with the equation $g_{\alpha}\cdot
a=\varphi(a)g_{\alpha}$, we obtain $D(a)=\lim\limits_{\alpha}g_{\alpha}\cdot
a-a\cdot g_{\alpha}$ for all $a\in A$. Set $f_{\alpha}=-g_{\alpha}$, then
$D(a)=\lim\limits_{\alpha}a\cdot f_{\alpha}-f_{\alpha}\cdot a$ for all $a\in
A$. Since $D$ was arbitrary, it follows that $A$ is approximately
$\varphi$-amenable.

$(iv)\Rightarrow(ii)$ Choose $b\in A$ with $\varphi(b)=1$. Then
$Da=ab-\varphi(a)b$, $a\in A$, defines a derivation from A into
$(\ker\varphi)^{**}$. By $(iv)$, D is approximately inner, it follows that
there exists a net $\\{n_{\alpha}\\}\subset(\ker\varphi)^{**}$ such that
$D(a)=\lim\limits_{\alpha}a\cdot n_{\alpha}-\varphi(a)n_{\alpha}$ for all
$a\in A$.

Set $m_{\alpha}=b-n_{\alpha}$. Then $m_{\alpha}(\varphi)=1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$.

The proof in the case of uniform approximate $\varphi$-amenability is similar.
$\blacksquare$

######  Proposition 2.2.

For a Banach algebra $A$ and $\varphi\in\sigma(A)$, the following are
equivalent:

$(i)$ There exists a net $\\{m_{\alpha}\\}\subset A^{**}$ such that
$m_{\alpha}(\varphi)\rightarrow 1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$ ;

$(ii)$ There exists a net $\\{n_{\beta}\\}\subset A$ such that
$\varphi(n_{\beta})\rightarrow 1$ and
$||an_{\beta}-\varphi(a)n_{\beta}||\rightarrow 0$ for all $a\in A$.

Proof. It suffices to show that $(ii)\Rightarrow(i)$.

Suppose that $(ii)$ holds. Take $\varepsilon>0$ and finite sets $F\subset A$,
$\Phi\subset A^{*}$. Then there exists $\alpha$ such that

$|(a\cdot
m_{\alpha}-\varphi(a)m_{\alpha})(f)|<\frac{\varepsilon}{3},|m_{\alpha}(\varphi)|>1-\varepsilon,a\in
F,f\in\Phi.$

By Goldstine’s theorem, there exists $b_{\alpha}\in A$ such that

$|f(b_{\alpha})-m_{\alpha}(f)|<\frac{\varepsilon}{3K},f\in\Phi\cup\Phi\cdot
F\cup\\{\varphi\\},$

where $K=\sup\\{|\varphi(a)|,a\in F\\}$.

Thus, for any $f\in\Phi$ and $a\in F$,

$\displaystyle|f(ab_{\alpha}-\varphi(a)b_{\alpha})|$ $\displaystyle\leq$
$\displaystyle|f(ab_{\alpha})-(a\cdot m_{\alpha})(f)|$ $\displaystyle+$
$\displaystyle|(a\cdot m_{\alpha})(f)-\varphi(a)m_{\alpha}(f)|$
$\displaystyle+$
$\displaystyle|\varphi(a)m_{\alpha}(f)-\varphi(a)f(b_{\alpha})|$
$\displaystyle\leq$ $\displaystyle|(f\cdot a)(b_{\alpha})-m_{\alpha}(f\cdot
a)|$ $\displaystyle+$ $\displaystyle|(a\cdot
m_{\alpha})(f)-\varphi(a)m_{\alpha}(f)|$ $\displaystyle+$
$\displaystyle|\varphi(a)m_{\alpha}(f)-\varphi(a)f(b_{\alpha})|$
$\displaystyle<$ $\displaystyle\varepsilon.$

Then there exists a net $\\{b_{\lambda}\\}\subset A$ such that for every $a\in
A$, $ab_{\lambda}-\varphi(a)b_{\lambda}\rightarrow
0,\varphi(b_{\lambda})\rightarrow 1$ weakly in $A$.

Finally, for each finite set $F\subset A$, say
$F=\\{a_{1},a_{2},\cdots,a_{n}\\}$,

$(a_{1}b_{\lambda}-\varphi(a_{1})b_{\lambda},a_{2}b_{\lambda}-\varphi(a_{2})b_{\lambda},\cdots,a_{n}b_{\lambda}-\varphi(a_{n})b_{\lambda},\varphi(b_{\lambda}))\rightarrow(0,0,\cdots,0,1)$

weakly in $A^{n}\oplus\mathbb{C}.$ Thus

$(0,0,\cdots,0,1)\in\overline{co}^{weak}\\{(a_{1}b_{\lambda}-\varphi(a_{1})b_{\lambda},a_{2}b_{\lambda}-\varphi(a_{2})b_{\lambda},\cdots,a_{n}b_{\lambda}-\varphi(a_{n})b_{\lambda},\varphi(b_{\lambda}))\\}$.

The Hahn-Banach theorem now gives that for each $\varepsilon>0$, there exists
$u_{\varepsilon,F}\in co\\{b_{\lambda}\\}$, such that
$||au_{\varepsilon,F}-\varphi(a)u_{\varepsilon,F}||<\varepsilon,|\varphi(u_{\varepsilon,F})-1|<\varepsilon$
for all $a\in F$. Therefore, there exists a net $\\{n_{\beta}\\}\subset A$
such that $\varphi(n_{\beta})\rightarrow 1$ and
$||an_{\beta}-\varphi(a)n_{\beta}||\rightarrow 0$ for all $a\in A$.
$\blacksquare$

Let $A$ be a Banach algebra, $\\{e_{\alpha}\\}$ be a net of $A$. We call
$\\{e_{\alpha}\\}$ a right approximate identity for $A$, if
$||ae_{\alpha}-a||\rightarrow 0$ for all $a\in A$. Left (two-sided)
approximate identity for $A$ is defined similarly. We call $\\{e_{\alpha}\\}$
a bounded right (left, two-sided, respectively) approximate identity for $A$,
if it is a bounded net. The next proposition characterizes approximate
$\varphi$-amenability in terms of the existence of right approximate identity
for $\ker\varphi$. The corresponding result characterizing
$\varphi$-amenability of a Banach algebra was obtained in [16, Proposition
2.1].

######  Lemma 2.3.

Let $A$ be a Banach algebra and $\varphi\in\sigma(A)$. If the ideal
$I_{\varphi}=\ker\varphi$ has a right approximate identity, then $A$ is
approximately $\varphi$-amenable.

Proof. Choose $u_{0}\in A$ such that $\varphi(u_{0})=1$. Then
$a_{0}=u_{0}^{2}-u_{0}\in I_{\varphi}$. Let $\\{b_{\alpha}\\}$ be a right
approximate identity for $I_{\varphi}$.

Set $m_{\alpha}=u_{0}-u_{0}b_{\alpha}\in A$. Then, for any $b\in I_{\varphi}$,

$||b(u_{0}-u_{0}b_{\alpha})||=||bu_{0}-bu_{0}b_{\alpha}||\rightarrow 0.$

Furthermore,

$||u_{0}(u_{0}-u_{0}b_{\alpha})-(u_{0}-u_{0}b_{\alpha})||=||u_{0}^{2}-u_{0}^{2}b_{\alpha}-u_{0}+u_{0}b_{\alpha}||=||a_{0}-a_{0}b_{\alpha}||\rightarrow
0.$

It follows that, $\varphi(m_{\alpha})=1$ and
$||am_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$, for all $a\in A$. Thus
$A$ is approximately $\varphi$-amenable by Proposition 2.1/2.2. $\blacksquare$

######  Lemma 2.4.

Suppose that $A$ is approximately $\varphi$-amenable for some
$\varphi\in\sigma(A)$ and that $A$ has a right approximate identity. Then
$I_{\varphi}$ has a right approximate identity.

Proof. Choose $u_{0}\in A$ such that $\varphi(u_{0})=1$ and
$A=\mathbb{C}u_{0}\oplus I_{\varphi}$. Let
$n_{\beta}=\lambda_{\beta}u_{0}+b_{\beta}$ be a right approximate identity for
$A$, where $b_{\beta}\in I_{\varphi}$ and $\lambda_{\beta}\rightarrow 1$.
Since $A$ is approximately $\varphi$-amenable, it follows from Proposition
2.1/2.2 that there exists a net
$m_{\alpha}=\lambda_{\alpha}u_{0}+b_{\alpha}\in A$ such that
$||am_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$, where $b_{\alpha}\in
I_{\varphi}$ and $\lambda_{\alpha}\rightarrow 1$.

Set $e_{\alpha,\beta}=b_{\beta}-b_{\alpha}$. Notice that
$||be_{\alpha,\beta}-b||\rightarrow 0,$ for any $b\in I_{\varphi}$. In fact,
for any $b\in I_{\varphi}$, $||bn_{\beta}-b||\rightarrow 0$ and
$||bm_{\alpha}||\rightarrow 0$. It follows that
$||bu_{0}+bb_{\beta}-b||\rightarrow 0$ and $||bu_{0}+bb_{\alpha}||\rightarrow
0$. Thus $||be_{\alpha,\beta}-b||\rightarrow 0$ for all $b\in I_{\varphi}$.
$\blacksquare$

The following result follows immediately from Lemma 2.3/2.4 and we omit its
proof.

######  Proposition 2.5.

Let $A$ be a Banach algebra with a right approximate identity and
$\varphi\in\sigma(A)$. Then $A$ is approximate $\varphi$-amenable if and only
if $I_{\varphi}$ has a right approximate identity.

If $A$ is a Banach algebra and $A\widehat{\otimes}A$ denotes the projective
product [20], then the corresponding diagonal operator is defined as

$\triangle:A\widehat{\otimes}A\rightarrow A,a\otimes b\rightarrow ab.$

Then $A\widehat{\otimes}A$ becomes a Banach $A$-bimodule through

$a\cdot(b\otimes c)=ab\otimes c,(b\otimes c)\cdot a=b\otimes ca$

for all $a,b,c\in A$. By [20, Theorem 2.2.4], $A$ is amenable if and only if
there is a net $\\{m_{\alpha}\\}\subset A\widehat{\otimes}A$ such that $a\cdot
m_{\alpha}-m_{\alpha}\cdot a\rightarrow 0,a\triangle m_{\alpha}\rightarrow a$
for all $a\in A$. Such a net is called an approximate diagonal for $A$. The
following proposition characterizes approximate $\varphi$-amenability in terms
of approximate diagonal for $A$.

######  Proposition 2.6.

Let $A$ be a Banach algebra and $\varphi\in\sigma(A)$. Then the following are
equivalent:

$(i)$ $A$ is (uniformly) approximately $\varphi$-amenable;

$(ii)$ There exists a net $\\{M_{\alpha}\\}\subset(A\widehat{\otimes}A)^{**}$
such that $(\bigtriangleup^{**}M_{\alpha})(\varphi)=1$ and $||a\cdot
M_{\alpha}-\varphi(a)M_{\alpha}||\rightarrow 0$ for all $a\in A$ (uniformly on
the unit ball of $A$, respectively);

$(iii)$ There exists a net $\\{M_{\alpha}\\}\subset(A\widehat{\otimes}A)^{**}$
such that $(\bigtriangleup^{**}M_{\alpha})(\varphi)\rightarrow 1$ and
$||a\cdot M_{\alpha}-\varphi(a)M_{\alpha}||\rightarrow 0$ for all $a\in A$
(uniformly on the unit ball of $A$, respectively).

Proof. $(i)\Rightarrow(ii)$ Choose $u_{0}\in A$ such that $\varphi(u_{0})=1$.
Then $A=\mathbb{C}u_{0}\oplus I_{\varphi}$, where $I_{\varphi}=\ker\varphi$.
We define an action of $A$ on $X=A\widehat{\otimes}A$ by

$a\cdot(b\otimes c)=ab\otimes c,(b\otimes c)\cdot a=\varphi(a)b\otimes
c,~{}~{}a,b,c\in A.$

Then the dual $A$-bimodule $(A\widehat{\otimes}A)^{*}$ is a
$(\varphi,A)$-bimodule.

We know that $\varphi\otimes\varphi\in(A\otimes A)^{*}$, and
$a\cdot(\varphi\otimes\varphi)=(\varphi\otimes\varphi)\cdot
a=\varphi(a)\varphi\otimes\varphi$. Therefore
$\mathbb{C}\varphi\otimes\varphi$ is a closed submodule of $(A\otimes A)^{*}$
and $(A\otimes A)^{*}/\mathbb{C}\varphi\otimes\varphi$ is a
$(\varphi,A)$-bimodule for which the module actions are given by
$a\cdot[f]=\varphi(a)[f],[f]\cdot a=[f*a]$, for all $a\in A,[f]\in(A\otimes
A)^{*}/\mathbb{C}\varphi\otimes\varphi$.

Moreover, define a derivation $D:A\rightarrow(A\widehat{\otimes}A)^{**}$ given
by $D(a)=au_{0}\otimes u_{0}-\varphi(a)u_{0}\otimes u_{0}$. Then
$D(a)\in\ker(\varphi\otimes\varphi)\subset\\{\mathbb{C}\varphi\otimes\varphi\\}^{\bot}\cong[(A\otimes
A)^{*}/\mathbb{C}\varphi\otimes\varphi]^{*}$. Since $A$ is approximately
$\varphi$-amenable, it follows that there exists a net
$\\{N_{\alpha}\\}\subset\\{\mathbb{C}\varphi\otimes\varphi\\}^{\bot}$ such
that $D(a)=\lim\limits_{\alpha}a\cdot N_{\alpha}-\varphi(a)N_{\alpha}$ for all
$a\in A$.

Set $M_{\alpha}=u_{0}\otimes u_{0}-N_{\alpha}$, then
$(\bigtriangleup^{**}M_{\alpha})(\varphi)=1$ and $||a\cdot
M_{\alpha}-\varphi(a)M_{\alpha}||\rightarrow 0$ for all $a\in A$.

$(ii)\Rightarrow(iii)$ It is clear.

$(iii)\Rightarrow(i)$ Now suppose that $(iii)$ holds. Let $E$ be a
$(\varphi,A)$-bimodule and let $D:A\rightarrow E^{*}$ be a derivation. For
each $\alpha$, set $f_{\alpha}(x)=M_{\alpha}(\mu_{x})$, where for $a,b\in
A,x\in E,\mu_{x}(a\otimes b)=(D(a)\cdot b)(x)$. Then, with
$(m_{\alpha}^{\nu})\subset A\widehat{\otimes}A$ converging $w^{*}$ to
$M_{\alpha}$, and noting that for $m\in A\widehat{\otimes}A$,

$\mu_{x\cdot a}(m)=(\mu_{x}\cdot a)(m)-(D(a)\cdot\bigtriangleup(m))(x),$

we have

$\displaystyle(a\cdot f_{\alpha})(x)$ $\displaystyle=$ $\displaystyle
f_{\alpha}(x\cdot a)$ $\displaystyle=$ $\displaystyle M_{\alpha}(\mu_{x\cdot
a})$ $\displaystyle=$ $\displaystyle\lim\limits_{\nu}\mu_{x\cdot
a}(m_{\alpha}^{\nu})$ $\displaystyle=$ $\displaystyle M_{\alpha}(\mu_{x}\cdot
a)-\lim\limits_{\nu}[D(a)\cdot\bigtriangleup(m_{\alpha}^{\nu})](x)$
$\displaystyle=$ $\displaystyle(a\cdot
M_{\alpha})(\mu_{x})-\lim\limits_{\nu}\varphi(\bigtriangleup(m_{\alpha}^{\nu}))D(a)(x)$
$\displaystyle=$ $\displaystyle(a\cdot
M_{\alpha})(\mu_{x})-(\bigtriangleup^{**}M_{\alpha})(\varphi)D(a)(x).$

Thus, for $a\in A$ and $x\in E$,

$\displaystyle||D(a)(x)-(f_{\alpha}\cdot a-a\cdot f_{\alpha})(x)||$
$\displaystyle\leq$
$\displaystyle||D(a)(x)-(\bigtriangleup^{**}M_{\alpha})(\varphi)D(a)(x)||$
$\displaystyle+$ $\displaystyle||(a\cdot M_{\alpha})(\mu_{x})-(f_{\alpha}\cdot
a)(x)||$ $\displaystyle\leq$
$\displaystyle|1-(\bigtriangleup^{**}M_{\alpha})(\varphi)|\cdot||x||\cdot||D(a)||$
$\displaystyle+$ $\displaystyle||(a\cdot
M_{\alpha})-\varphi(a)M_{\alpha}||\cdot||\mu_{x}||,$

whence $D(a)=\lim\limits_{\alpha}(f_{\alpha}\cdot a-a\cdot f_{\alpha})$ for
all $a\in A$. Set $g_{\alpha}=-f_{\alpha}$, then
$D(a)=\lim\limits_{\alpha}(a\cdot g_{\alpha}-g_{\alpha}\cdot a)$ for all $a\in
A$. Since $D$ was arbitrary, it follows that $A$ is approximately
$\varphi$-amenable.

The proof in the case of uniformly approximately $\varphi$-amenable is
similar. $\blacksquare$

The following proposition characterizes (uniform) approximate $0$-amenability
in terms of right approximate identity for $A$.

######  Proposition 2.7.

Banach algebra $A$ is (uniform) approximate $0$-amenability if and only if $A$
has a (bounded, respectively) right approximate identity.

Proof. Suppose that $E$ is a $(\varphi,A)$-bimodule and $D:A\rightarrow E^{*}$
is a derivation. Then $D(b)\cdot a=0$, for all $a,b\in A$. If
$\\{e_{\alpha}\\}$ is (bounded) right approximate identity for $A$, then
$D(a)=\lim\limits_{\alpha}D(ae_{\alpha})=\lim\limits_{\alpha}a\cdot
D(e_{\alpha}).$ This implies that $A$ is (uniformly) approximately
$0$-amenable.

The converse is clear from [9, Lemma 2.1,Lemma 2.2, Theorem 4.2].
$\blacksquare$

Note that (uniform) approximate right character amenability of $A$ is
equivalent to (uniform, respective) approximate $\varphi$-amenability for all
$\varphi\in\sigma(A)$ together with (uniform, respectively) approximate
$0$-amenability. Any statement about (uniform) approximate right character
amenability turns into an analogous statement about (uniform, respectively)
approximate left character amenability by simply replacing $A$ by its opposite
algebra. Then standard arguments of Proposition 2.1, 2.5, 2.6 and 2.7 apply,
we have the following theorems.

######  Theorem 2.8.

For a Banach algebra $A$ the following are equivalent:

$(i)$ $A$ is (uniformly) approximately character amenable;

$(ii)$ $A$ has (bounded, respectively) both left and right approximate
identities, and for any $\varphi\in\sigma(A)$, there exist nets
$\\{m_{\alpha}\\},\\{m_{\alpha}^{{}^{\prime}}\\}\subset A^{**}$ such that
$m_{\alpha}(\varphi)=1,m_{\alpha}^{{}^{\prime}}(\varphi)=1(or~{}m_{\alpha}(\varphi)\rightarrow
1,m_{\alpha}^{{}^{\prime}}(\varphi)\rightarrow 1)$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||+||m_{\alpha}^{{}^{\prime}}\cdot
a-\varphi(a)m_{\alpha}^{{}^{\prime}}||\rightarrow 0$ for all $a\in A$
(uniformly on the unit ball of $A$, respectively);

$(iii)$ $A$ has (bounded, respectively) both left and right approximate
identities, and for any $\varphi\in\sigma(A)$, there exist nets
$\\{m_{\alpha}\\},\\{m_{\alpha}^{{}^{\prime}}\\}\subset A$, such that
$\varphi(m_{\alpha})=1,\varphi(m_{\alpha}^{{}^{\prime}})=1$ (or
$\varphi(m_{\alpha})\rightarrow 1,\varphi(m_{\alpha}^{{}^{\prime}})\rightarrow
1)$ and
$||am_{\alpha}-\varphi(a)m_{\alpha}||+||m_{\alpha}^{{}^{\prime}}a-\varphi(a)m_{\alpha}^{{}^{\prime}}||\rightarrow
0$ for all $a\in A$ (uniformly on the unit ball of $A$, respectively);

$(iv)$ $A$ has (bounded, respectively) both left and right approximate
identities, and for any $\varphi\in\sigma(A)$, there exist nets
$\\{M_{\alpha}\\},\\{M_{\alpha}^{{}^{\prime}}\\}\subset(A\widehat{\otimes}A)^{**}$
such that $(\bigtriangleup^{**}M_{\alpha})(\varphi)=1$,
$(\bigtriangleup^{**}M_{\alpha}^{{}^{\prime}})(\varphi)=1(or~{}(\bigtriangleup^{**}M_{\alpha})(\varphi)\rightarrow
1,(\bigtriangleup^{**}M_{\alpha}^{{}^{\prime}})(\varphi)\rightarrow 1)$ and
$||a\cdot M_{\alpha}-\varphi(a)M_{\alpha}||+||M_{\alpha}^{{}^{\prime}}\cdot
a-\varphi(a)M_{\alpha}^{{}^{\prime}}||\rightarrow 0$ for all $a\in A$
(uniformly on the unit ball of $A$, respectively).

######  Theorem 2.9.

A Banach algebra $A$ is left (right) approximately character amenable if and
only if $\ker\varphi$ has a left (right, respectively) approximate identity
for every $\varphi\in\sigma(A)\cup\\{0\\}$.

##  3\. Hereditary properties of approximate character amenability

In this section, we are concerned with hereditary properties of (uniform)
approximate character amenability.

######  Proposition 3.1.

Suppose that $A$ is (uniformly) approximately character amenable
(contractible) and $\Phi:A\rightarrow B$ is a continuous epimorphism. Then $B$
is (uniformly, respectively) approximately character amenable (contractible,
respectively).

Proof. The standard argument, [9, Proposition 2.2] applies. $\blacksquare$

######  Proposition 3.2.

Suppose that $A$ is (uniformly) approximately character amenable
(contractible), and $J$ is a closed two-sided ideal of $A$. Then $A/J$ is
(uniformly, respectively) approximately character amenable (contractible,
respectively). If $J$ is character amenable (contractible) and $A/J$ is
(uniformly) approximately character amenable (contractible), then $A$ is
(uniformly, respectively)approximately character amenable (contractible,
respectively).

Proof. The standard argument, [19, Proposition 1.30] applies. $\blacksquare$

######  Proposition 3.3.

Let $A$ be a Banach algebra and $J$ a weakly complemented ideal of $A$. Let
$\varphi\in\sigma(A)$ satisfy $\varphi|_{J}\neq 0$. If $A$ is (uniformly)
approximately $\varphi$-amenable, then J is (uniformly, respectively)
approximately $\varphi|_{J}$-amenable.

Proof. Since $A$ is approximately $\varphi$-amenable, it follows from
Proposition 2.1 that there exists a net $\\{m_{\alpha}\\}\subset A^{**}$, such
that $m_{\alpha}(\varphi)=1$ and $||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$.

Since $J$ weakly complemented of $A$, there exist a closed subspace $X$ of
$A^{*}$ such that $A^{*}=J^{\bot}\oplus X$. That is to say, there exists $K>0$
such that for any $F\in A^{*}$, $F=x_{F}+y_{F}$, where $x_{F}\in
J^{\bot},y_{F}\in X$, and $||x_{F}||\leq K||F||,||y_{F}||\leq K||F||$. If, in
addition, $a\in J$, then $x_{F}\cdot a=0$. Thus
$||\varphi(a)m_{\alpha}(x_{F})||\rightarrow 0$ for all $a\in J,$ and uniformly
for $F\in A^{*}$ and $||F||\leq 1$. Choose $a=u_{0}\in J$ with
$\varphi(u_{0})=1$, then $||m_{\alpha}(x_{F})||\rightarrow 0$ uniformly for
$||F||\leq 1$.

Set $n_{\alpha}(f)=m_{\alpha}(y_{F})$ for $f\in J^{*}$, where $F$ is any
extension of $f$. Notice that $||a\cdot
n_{\alpha}-\varphi(a)n_{\alpha}||\rightarrow 0,$ and
$n_{\alpha}(\varphi|_{J})\rightarrow 1$ for all $a\in J$. To see this, for
$a,b\in J,f\in J^{*}$ and $F$ is a extension of $f$,

$(y_{F}\cdot a)(b)=y_{F}(ab)=f(ab),~{}y_{F\cdot a}(b)=f(ab).$

It follows that there is a $x\in J^{\bot}$ such that $y_{F}\cdot a=x+y_{F\cdot
a}$ and $||y_{F\cdot a}||\leq K||y_{F}\cdot a||,||x||\leq K||y_{F}\cdot a||$.
Then, for any $f\in J^{*}$ and $F$ an extension of $f$,

$\displaystyle|a\cdot n_{\alpha}(f)-\varphi(a)n_{\alpha}(f)|$ $\displaystyle=$
$\displaystyle|n_{\alpha}(f\cdot a)-\varphi(a)m_{\alpha}(y_{F})|$
$\displaystyle=$ $\displaystyle|m_{\alpha}(y_{F\cdot
a})-\varphi(a)m_{\alpha}(y_{F})|$ $\displaystyle\leq$
$\displaystyle|m_{\alpha}(y_{F}\cdot
a)-\varphi(a)m_{\alpha}(y_{F})|+|m_{\alpha}(x)|$ $\displaystyle\leq$
$\displaystyle||a\cdot
m_{\alpha}-\varphi(a)m_{\alpha}||\cdot||y_{F}||+|m_{\alpha}(x)|.$

It follows that $||a\cdot n_{\alpha}-\varphi(a)n_{\alpha}||\rightarrow 0$ and
$n_{\alpha}(\varphi|_{J})=m_{\alpha}(\varphi)-m_{\alpha}(x_{\varphi})\rightarrow
1$ for all $a\in J$. Then, by Proposition 2.1, $J$ is approximately
$\varphi|_{J}$-amenable.

The proof in the case of uniform approximate $\varphi$-amenability is similar.
$\blacksquare$

######  Lemma 3.4.

Let $A$ be a Banach algebra and $J$ is an ideal of $A$, which has a right or
left approximate identity. Then every $\varphi\in\sigma(J)$ can be extended to
a $\widetilde{\varphi}$ in $\sigma(A)$.

Proof. Assume that $\\{e_{\alpha}\\}$ is right approximate identity for $J$
and $\varphi\in\sigma(J)$. Choose $u_{0}\in J$ such that $\varphi(u_{0})=1$,
then $J=\mathbb{C}u_{0}\oplus I_{\varphi}$, and $u_{0}a-u_{0}\in I_{\varphi}$
for all $a\in J$, where $I_{\varphi}=\ker\varphi$.

Set $\widetilde{\varphi}(a)=\varphi(u_{0}a)$ for all $a\in A$. Then
$\widetilde{\varphi}|_{J}=\varphi$, and
$\widetilde{\varphi}(a_{1}a_{2})=\varphi(u_{0}a_{1}a_{2})=\widetilde{\varphi}(a_{1})\widetilde{\varphi}(a_{2})$,
since
$u_{0}a_{1}a_{2}-u_{0}a_{1}u_{0}a_{2}=(u_{0}a_{1}-u_{0}a_{1}u_{0})a_{2}=\lim\limits_{\alpha}(u_{0}a_{1}-u_{0}a_{1}u_{0})e_{\alpha}a_{2}\in
I_{\varphi}$. It follows that $\widetilde{\varphi}\in\sigma(A)$.

The proof in the case of a left approximate identity is similar.
$\blacksquare$

######  Theorem 3.5.

Let $A$ be a Banach algebra and $J$ a weakly complemented ideal of $A$ with
(bounded) left and right approximate identities. Suppose that $A$ is
(uniformly) approximately character amenable. Then $J$ is (uniformly,
respectively)approximately character amenable.

Proof. Clearly $J$ is (uniformly) approximately $\varphi$-amenable, for any
$\varphi\in\sigma(J)\cup\\{0\\}$, by Proposition 2.7, 3.3 and Lemma 3.4. That
is to say, $J$ is (uniformly) approximately right character amenable. The
proof in the case of (uniform) approximate left character amenability is
similar. Thus $J$ is (uniformly) approximately character amenable.
$\blacksquare$

By [18, Theorem 2.6] a Banach algebra $A$ without a unit is character amenable
if and only if $A^{\sharp}$ is character amenable. We obtain a similar result
for approximate character amenability.

######  Lemma 3.6.

Let $A$ be a Banach algebra without identity and let $A^{\sharp}$ denote the
unitization of $A$ by adjoining an identity $e$. Let
$\varphi\in\sigma(A)\cup\\{0\\}$ and let $\varphi_{e}$ be the unique extension
of $\varphi$ to an element of $\sigma(A^{\sharp})$. Then $A^{\sharp}$ is
approximately $\varphi_{e}$-amenable if $A$ is approximately
$\varphi$-amenable.

Proof. Assume that $A$ is approximately $\varphi$-amenable, where
$\varphi\in\sigma(A)$. Then the standard argument of [16, Lemma 3.2] applies,
so that $A^{\sharp}$ is approximately $\varphi_{e}$-amenable.

If $\varphi=0$, $A^{\sharp}=\mathbb{C}e\oplus A$ and $\varphi_{e}(A)=0$,
$\varphi_{e}(e)=1$. Since $A$ is approximately $0$-amenable, it follows from
Proposition 2.7 that $A$ has a right approximate identity $(b_{\alpha})$.

Set $m_{\alpha}=e-b_{\alpha}$, then
$||am_{\alpha}-\varphi_{e}(a)m_{\alpha}||\rightarrow 0$ and
$\varphi_{e}(m_{\alpha})=1$, for all $a\in A^{\sharp}$. Thus $A^{\sharp}$ is
approximately $\varphi_{e}$-amenable, by Proposition 2.1/2.2. $\blacksquare$

######  Theorem 3.7.

Let $A$ be a Banach algebra without a unit. Then $A$ is approximately
character amenable if and only if $A^{\sharp}$ is approximately character
amenable.

Proof. Assume $A$ is approximately character amenable, the argument of Lemma
3.6 applies, $A^{\sharp}$ is approximately character amenable.

For the converse, assume that $A^{\sharp}$ is approximately character
amenable, then $A^{\sharp}$ is both approximately right and left character
amenable. We shall show that $A$ has both right and left approximately
identities.

Define $\varphi\in\sigma(A^{\sharp})$ by $\varphi(A)=0,\varphi(e)=1$. Since
$A^{\sharp}$ is right approximately character amenable, Proposition 2.1/2.2
gives a net $(m_{\alpha})$ in $A^{\sharp}$ such that
$\varphi(m_{\alpha})\rightarrow 1$ and $||bm_{\alpha}||\rightarrow 0$ for all
$b\in A$. Writing $m_{\alpha}=\lambda_{\alpha}e+b_{\alpha}$ where
$\lambda_{\alpha}\in\mathbb{C},b_{\alpha}\in A$, it follows that
$||bb_{\alpha}+b||\rightarrow 0$ and hence $A$ has a right approximately
identity. Similarly, $A$ has a left approximately identity. Then, by Theorem
3.5, $A$ is approximately character amenable. $\blacksquare$

##  4\. Characterization of (approximate) character contractibility

In this section, we first characterize $\varphi$-contractibility, approximate
$\varphi$-contractibility and uniform approximate $\varphi$-contractibility in
two different ways and then characterize character contractibility,
approximate character contractibility and uniform approximate character
contractibility in the same different ways.

######  Proposition 4.1.

Let $A$ be a Banach algebra and $\varphi\in\sigma(A)$. Then the following
conditions are equivalent:

$(i)$ $A$ is $\varphi$-contractible;

$(ii)$ There exists $m\in A$ such that $\varphi(m)=1$ and $am=\varphi(a)m$ for
all $a\in A$.

Proof. $(i)\Rightarrow(ii)$ Choose $u_{0}\in A$ such that $\varphi(u_{0})=1$.
Then $A=\mathbb{C}u_{0}\oplus I_{\varphi}$, where $I_{\varphi}=\ker\varphi$.
We define an action of $A$ on $X=A$ by $a\cdot x=ax,x\cdot a=\varphi(a)x,a\in
A,x\in X$. Then $X$ is a $(A,\varphi)$-bimodule.

Moreover, define a derivation $D:A\rightarrow A$ given by
$D(a)=au_{0}-\varphi(a)u_{0}$. Then $D(a)\in\ker\varphi$ and $\ker\varphi$ is
a submodule of $X$. Since $A$ is $\varphi$-contractible, it follows that there
exists $n\in\ker\varphi$ such that $D(a)=a\cdot n-n\cdot a$. Set $m=u_{0}-n$.
Then $\varphi(m)=1$ and $am=\varphi(a)m$ for all $a\in A$ .

$(ii)\Rightarrow(i)$ $m\in A$ is such that $\varphi(m)=1$ and $am=\varphi(a)m$
for all $a\in A$. Let $X$ be a $(A,\varphi)$-bimodule, and let $D:A\rightarrow
X$ be a derivation. Set $x=D(m)$, then

$\displaystyle a\cdot D(m)$ $\displaystyle=$ $\displaystyle D(am)-D(a)\cdot m$
$\displaystyle=$ $\displaystyle D(am)-\varphi(m)D(a)$ $\displaystyle=$
$\displaystyle\varphi(a)D(m)-D(a).$

It follows that $D(a)=\varphi(a)D(m)-a\cdot D(m)=D(m)\cdot a-a\cdot D(m)$ for
all $a\in A$. Since $D$ was arbitrary, it follows that $A$ is
$\varphi$-contractible. $\blacksquare$

######  Proposition 4.2.

For a Banach algebra $A$ the following are equivalent:

$(i)$ $A$ is approximately $\varphi$-contractible;

(ii) There exists a net $\\{m_{\alpha}\\}\subset A$ such that
$\varphi(m_{\alpha})=1$ and $||am_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow
0$ for all $a\in A$;

$(iii)$ There exists a net $\\{m_{\alpha}\\}\subset A$ such that
$\varphi(m_{\alpha})\rightarrow 1$ and
$||am_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ for all $a\in A$.

Proof. The proof is a minor modification of the proof of the analogous
statement in Proposition 4.1. $\blacksquare$

######  Lemma 4.3.

Let $A$ be a Banach algebra and $\varphi\in\sigma(A)$. If the ideal
$I_{\varphi}=\ker\varphi$ has a right identity, then $A$ is
$\varphi$-contractible.

Proof. Choose $u_{0}\in A$ such that $\varphi(u_{0})=1$. Then
$A=\mathbb{C}u_{0}\oplus I_{\varphi}$ and there exists $a_{0}\in I_{\varphi}$
such that $u_{0}^{2}-u_{0}=a_{0}$. Let $b_{0}$ be a right identity for
$I_{\varphi}$. Set $m=u_{0}-u_{0}b_{0}\in A$. Then, for any $b\in
I_{\varphi}$, $b(u_{0}-u_{0}b_{0})=bu_{0}-bu_{0}b_{0}=0.$ Furthermore,

$u_{0}(u_{0}-u_{0}b_{0})-(u_{0}-u_{0}b_{0})$
$=u_{0}^{2}-u_{0}^{2}b_{0}-u_{0}+u_{0}b_{0}$ $=a_{0}-a_{0}b_{0}=0.$

It follows that $am-\varphi(a)m=0$ for all $a\in A$ and $\varphi(m)=1$. Thus
$A$ is $\varphi$-contractible by Proposition 4.1. $\blacksquare$

######  Lemma 4.4.

Suppose that $A$ is $\varphi$-contractible for some $\varphi\in\sigma(A)$ and
that $A$ has a right identity. Then $I_{\varphi}=\ker\varphi$ has a right
identity.

Proof. Assume that $A=\mathbb{C}u_{0}\oplus I_{\varphi}$ for some $u_{0}\in
A$, and $n=u_{0}+b_{0}$ be a right identity for $A$, where $b_{0}\in
I_{\varphi}$. Since $A$ is $\varphi$-contractible, it follows from Proposition
4.1 that there exist an $m=u_{0}+b_{1}\in A$ such that $am-\varphi(a)m=0$,
where $b_{1}\in I_{\varphi}$. Set $e=b_{0}-b_{1}$. Then $be-b=0,$ for any
$b\in I_{\varphi}$. In fact, for any $b\in I_{\varphi}$, $bn-b=0$, and $bm=0$.
It follows that $bu_{0}+bb_{0}-b=0$ and $bu_{0}+bb_{1}=0$. Thus $be-b=0$ for
all $b\in I_{\varphi}$. $\blacksquare$

The following result follows immediately from Lemma 4.3/4.4 and we omit its
proof.

######  Proposition 4.5.

Let $A$ be a Banach algebra with a right identity and $\varphi\in\sigma(A)$.
Then $A$ is $\varphi$-contractible if and only if $I_{\varphi}$ has a right
identity.

For $0$-contractibility and (uniform) approximate $0$-contractibility we have
the following parallel results to approximate $0$-amenability, the proofs of
Proposition 4.6 and Proposition 4.7 are minor modifications of the proof of
the analogous statements in Proposition 2.7 and will be omitted.

######  Proposition 4.6.

Banach algebra $A$ is $0$-contractible if and only if $A$ has a right
identity.

######  Proposition 4.7.

Banach algebra $A$ is approximately $0$-contractible if and only if $A$ has a
right approximate identity.

######  Proposition 4.8.

Banach algebra $A$ is uniformly approximately $0$-contractible if and only if
$A$ has a right identity.

Proof. It suffices to show that if $A$ is uniformly approximately
$0$-contractible, then $A$ has a right identity. Define an action of $A$ on
$E=A$ by $a*x=ax,x*a=0,~{}a\in A,x\in E,$ then $E$ is $(A,0)$-bimodule. The
natural injection $a\mapsto a:A\rightarrow A$ is a derivation. Thus there is a
net $\\{e_{\alpha}\\}$ in $A$ such that $ae_{\alpha}\rightarrow a$ uniformly
for $||a||\leq 1$. Let $R_{b}$ denote right multiplication by $b\in A$. Then
there is $e_{\lambda}\in\\{e_{\alpha}\\}$ with $||R_{e_{\lambda}}a-a||<||a||$
for all $a\in A$. Thus $R_{e_{\lambda}}$ is invertible. It follows that, there
is $e\in A$ such that $ee_{\lambda}=e_{\lambda}$, whence $(ae-a)e_{\lambda}=0$
for all $a\in A$. Then $e$ is a right identity of $A$ by injectivity of
$R_{e_{\lambda}}$. $\blacksquare$

Finally, we characterize character contractibility and (uniform) approximate
character contractibility, whose proofs are minor modifications of the proofs
of the analogous statements in Theorem 2.8/2.9 and will be omitted.

######  Theorem 4.9.

For a Banach algebra $A$ the following are equivalent:

$(i)$ $A$ is character contractible;

$(ii)$ $A$ has an identity and for any $\varphi\in\sigma(A)$, there exists
$m_{1},m_{2}\in A$ such that $m_{i}(\varphi)=1,(i=1,2)$ and
$am_{1}-\varphi(a)m=0$, $m_{2}a-\varphi(a)m_{2}=0$ for all $a\in A$;

$(iii)$ $A$ has an identity and for any $\varphi\in\sigma(A)$, there exists
bounded nets
$\\{M_{\alpha}\\},\\{M_{\alpha}^{{}^{\prime}}\\}\subset(A\widehat{\otimes}A)$
such that $\varphi(\bigtriangleup M_{\alpha})=1$, $\varphi(\bigtriangleup
M_{\alpha}^{{}^{\prime}})=1(or~{}\varphi(\bigtriangleup M_{\alpha})\rightarrow
1,\varphi(\bigtriangleup M_{\alpha}^{{}^{\prime}})\rightarrow 1)$ and
$||a\cdot M_{\alpha}-\varphi(a)M_{\alpha}||+||M_{\alpha}^{{}^{\prime}}\cdot
a-\varphi(a)M_{\alpha}^{{}^{\prime}}||\rightarrow 0$ for all $a\in A$.

######  Theorem 4.10.

Let $A$ be a Banach algebra. Then $A$ is character contractible if and only if
$\ker\varphi$ has an identity for every $\varphi\in\sigma(A)\cup\\{0\\}$.

######  Theorem 4.11.

For a Banach algebra $A$ the following are equivalent:

$(i)$ $A$ is (uniformly) approximately character contractible;

$(ii)$ $A$ has (an identity, respectively) both right and left approximate
identities and for any $\varphi\in\sigma(A)$, there exist nets
$\\{m_{\alpha}\\},\\{m_{\alpha}^{{}^{\prime}}\\}\subset A$ such that
$m_{\alpha}(\varphi)=1,m_{\alpha}^{{}^{\prime}}(\varphi)=1(or~{}m_{\alpha}(\varphi)\rightarrow
1,m_{\alpha}^{{}^{\prime}}(\varphi)\rightarrow 1)$ and
$||am_{\alpha}-\varphi(a)m_{\alpha}||+||m_{\alpha}^{{}^{\prime}}a-\varphi(a)m_{\alpha}^{{}^{\prime}}||\rightarrow
0$ for all $a\in A$ (uniformly on the unit ball of $A$, respectively);

$(iii)$ $A$ has (an identity, respectively) both right and left approximate
identities and for any $\varphi\in\sigma(A)$, there exist nets
$\\{M_{\alpha}\\},\\{M_{\alpha}^{{}^{\prime}}\\}\subset(A\widehat{\otimes}A)$
such that $\varphi(\bigtriangleup M_{\alpha})=1$, $\varphi(\bigtriangleup
M_{\alpha}^{{}^{\prime}})=1(or~{}\varphi(\bigtriangleup M_{\alpha})\rightarrow
1,\varphi(\bigtriangleup M_{\alpha}^{{}^{\prime}})\rightarrow 1)$ and
$||a\cdot M_{\alpha}-\varphi(a)M_{\alpha}||+||M_{\alpha}^{{}^{\prime}}\cdot
a-\varphi(a)M_{\alpha}^{{}^{\prime}}||\rightarrow 0$ for all $a\in A$
(uniformly on the unit ball of $A$, respectively).

##  5\. Relations between generalized notions of character amenability

In this section, we are concerned with relations among generalized notions of
character amenability (contractibility). Firstly, we prove that
$w^{*}$-approximate $\varphi$-amenability, approximate $\varphi$-amenability
and approximate $\varphi$-contractibility are the same properties. We then
prove that approximate character amenability and approximate character
contractibility are the same properties, as are uniform approximate character
amenability (contractibility) and character amenability (contractibility,
respectively). Moreover, we obtain that character contractibility and
contractibility are the same properties for commutative Banach algebra.

######  Lemma 5.1.

For a Banach algebra $A$ and $\varphi\in\sigma(A)$, the following are
equivalent:

$(i)$ A is approximately $\varphi$-contractible;

$(ii)$ A is approximately $\varphi$-amenable;

$(iii)$ A is $w^{*}$-approximately $\varphi$-amenable.

Proof. It suffices to show that $(iii)\Rightarrow(i)$. Suppose that $(iii)$
holds. By the standard argument of Proposition 2.1, it follows that there is a
net $\\{m_{\alpha}\\}\subset A^{**}$ such that $(a\cdot
m_{\alpha}-\varphi(a)m_{\alpha})(f)\rightarrow
0,m_{\alpha}(\varphi)\rightarrow 1$ for all $a\in A,f\in A^{*}$. It follows
from the proof of Proposition 2.2 that there exists a net
$\\{n_{\beta}\\}\subset A$ such that $\varphi(n_{\beta})\rightarrow 1$ and
$||an_{\beta}-\varphi(a)n_{\beta}||\rightarrow 0$ for all $a\in A$. Thus A is
approximately $\varphi$-contractible by Proposition 4.2. $\blacksquare$

######  Theorem 5.2.

A Banach algebra $A$ is approximately character contractible if and only if
$A$ is approximately character amenable.

Proof. The standard argument of Proposition 2.7/4.7 and Lemma 5.1 applies.
$\blacksquare$

Recall that a Banach algebra $A$ is uniformly approximately
$\varphi$-amenable, if for every $(\varphi,A)$-bimodule $E$, every derivation
$D$ from $A$ into the dual $A$-bimodule $E^{*}$ may be approximated uniformly
on the unit ball of A by inner derivations. Clearly any $\varphi$-amenable
($\varphi$-contractible) Banach algebra is uniformly approximately
$\varphi$-amenable ($\varphi$-contractible, respectively). In the following
theorems we prove that the converse is also true. That is to say, uniform
approximate character amenability is equivalent to character amenability, and
uniform approximate character contractibility is equivalent to character
contractibility.

######  Lemma 5.3.

[18, Corollary 2.7] Let $A$ be a Banach algebra, if $\ker\varphi$ has both
bounded left and right approximate identities for every
$\varphi\in\sigma(A)\cup\\{0\\}$. Then $A$ is character amenable.

######  Proposition 5.4.

Let $A$ be unital Banach algebra, then $A$ is uniformly approximately
character amenable if and only if it is character amenable.

Proof. It suffices to show that if $A$ is uniformly approximately character
amenable then $A$ is character amenable.

Take $\varphi\in\sigma(A)$. Since $A$ is uniformly approximately right
character amenable it follows from Proposition 2.1 that there is a net
$\\{m_{\alpha}\\}\subset A^{**}$, such that $m_{\alpha}(\varphi)=1$ and
$||a\cdot m_{\alpha}-\varphi(a)m_{\alpha}||\rightarrow 0$ uniformly for
$||a||\leq 1$. Note that $A=\mathbb{C}e\oplus I_{\varphi}$, where
$I_{\varphi}=\ker\varphi$, and let $J(\varphi)=\\{n\in
A^{**},n(\varphi)=0\\}$. $J(\varphi)$ is a $w^{*}$-closed ideal of $A^{**}$,
and $J(\varphi)$ can be canonically identified with the second dual
$(I_{\varphi})^{**}$. We have $||a\cdot m_{\alpha}||\rightarrow 0$ uniformly
on the unit ball of $I_{\varphi}$.

Set $n_{\alpha}=e-m_{\alpha}$. Then $n_{\alpha}(\varphi)=0$, that is,
$n_{\alpha}\in J(\varphi)$. Moreover, for any $a\in I_{\varphi}$,
$|(an_{\alpha}-a)(f)|=|(a\cdot n_{\alpha}-a)(f)|=|(a\cdot
m_{\alpha})(f)|\leq||a\cdot m_{\alpha}||\cdot||f||$. It follows that
$||an_{\alpha}-a||\rightarrow 0$ uniformly on the unit ball of $I_{\varphi}$.
Now take $s\in(I_{\varphi})^{**}$, then, there is a net $(s_{i})\subset
I_{\varphi}$ such that $||s_{i}||\leq||s||$ and $s_{i}\rightarrow
s(w^{*}~{}in~{}i)$. Thus $s_{i}n_{\alpha}-s_{i}\rightarrow
sn_{\alpha}-s(w^{*}~{}in~{}i)$ and
$||sn_{\alpha}-s||\leq\sup_{i}||s_{i}n_{\alpha}-s_{i}||$. It follows that
$||an_{\alpha}-a||\rightarrow 0$ uniformly for $a\in(I_{\varphi})^{**}$ and
$||a||\leq 1$.

Thus there is a sequence $(n_{k})\subset(I_{\varphi})^{**}$ and
$\varepsilon_{k}\rightarrow 0$ such that

$||an_{k}-a||\leq\varepsilon_{k}||a||,(a\in(I_{\varphi})^{**}).$

Thus, the multiplication operator
$R_{n_{k}}:(I_{\varphi})^{**}\rightarrow(I_{\varphi})^{**}$ defined by
$R_{n_{k}}(s)=sn_{k}$ satisfies $||R_{n_{k}}-id_{(I_{\varphi})^{**}}||<1$ for
$k$ sufficiently large. Take such $k$, so that $R_{n_{k}}$ is invertible. By
surjectivity, there is $x\in(I_{\varphi})^{**}$ such that $xn_{k}=n_{k}$. Then
for each $y\in(I_{\varphi})^{**}$ we have $(yx-y)n_{k}=0$. From the
injectivity of $R_{n_{k}}$ this implies $yx=y$ for all
$y\in(I_{\varphi})^{**}$. So $(I_{\varphi})^{**}$ has a right identity, then
$I_{\varphi}$ has a bounded right approximately identity.

On the other hand, since $A$ is uniformly approximately left character
amenable. We can deduce that $I_{\varphi}$ has a bounded left approximately
identity. Since $\varphi$ is arbitrary, it follows from Lemma 5.3 that $A$ is
character amenable. $\blacksquare$

######  Corollary 5.5.

If $A^{\sharp}$ is uniformly approximately character amenable, then $A$ has
bounded right and left approximate identity.

Proof. Choose $\varphi\in\sigma(A^{\sharp})$ such that
$\varphi(A)=0,\varphi(e)=1$, the standard argument of Proposition 5.4 applies.
$\blacksquare$

######  Lemma 5.6.

Let $A$ be a Banach algebra without a unit. Then $A$ is uniformly
approximately character amenable if and only if $A^{\sharp}$ is uniformly
approximately character amenable.

Proof. Assume $A$ is uniformly approximately character amenable, the standard
argument of Lemma 3.6 applies, $A^{\sharp}$ is uniformly approximately
character amenable.

For the inverse, assume $A^{\sharp}$ is uniformly approximately character
amenable. It follows from Corollary 5.5 that $A$ has bounded left and right
approximately identity. Thus $A$ is uniformly approximately character amenable
by Theorem 3.5. $\blacksquare$

Note that $A$ is (uniformly approximately) character amenable if and only if
its unitization $A^{\sharp}$ is (uniformly approximately, respectively)
character amenable [18, Theorem 2.6] and Lemma 5.6, then, by Proposition 5.4,
we have the following theorem.

######  Theorem 5.7.

A Banach algebra $A$ is uniformly approximately character amenable if and only
if it is character amenable.

######  Corollary 5.8.

If a finite-dimensional Banach algebra is uniformly approximately character
amenable, then it is character amenable.

For uniform approximate character contractibility and character
contractibility , we have the following parallel result whose proof is a minor
modification of the proof of the analogous statements in Proposition 5.4 and
will be omitted.

######  Theorem 5.9.

A Banach algebra $A$ is uniformly approximately character contractible if and
only if it is character contractible.

######  Corollary 5.10.

If a finite-dimensional Banach algebra is approximately character
contractible, then it is character contractible.

Now, we shall conclude this section by proving the following result.

######  Theorem 5.11.

Let $A$ be a commutative Banach algebra. Then $A$ is character contractible if
and only if $A$ is isomorphic to $\mathbb{C}^{n}$.

Proof. It suffices to show that if $A$ is character contractible then $A$ is
contractible.

Let $\mathfrak{M}$ be the maximal ideal space of $A$. It follows from Theorem
4.10 that, given $\varphi\in\mathfrak{M}\cup\\{0\\}$, there exists
$E_{\varphi}\in\ker\varphi$ such that $E_{\varphi}$ is an identity for
$\ker\varphi$. Arbitrarily choose $\varphi_{1},\varphi_{2}\in\mathfrak{M}$. If
$\varphi_{1}\neq\varphi_{2}$, then $\varphi_{1}(E\varphi_{2})=1$ and
$\varphi_{2}(E\varphi_{1})=1$. It follows that each point of $\mathfrak{M}$ is
isolated, so that $\mathfrak{M}$ is finite since $\mathfrak{M}$ is compact.
Hence $\mathfrak{M}=\\{\varphi_{1},\varphi_{2},\cdots,\varphi_{n}\\}$ and
$E_{\varphi_{i}}$ is the identity for $\ker\varphi_{i}$ for all $1\leq i\leq
n$. Hence, $E_{0}=E_{\varphi_{1}}\cdot E_{\varphi_{2}}\cdot\cdots\cdot
E_{\varphi_{n}}$ is an identity for
Rad$(A)=\ker\varphi_{1}\cap\cdots\cap\ker\varphi_{n}$. This is possible only
if Rad$(A)=\\{0\\}$, so that $A$ is semisimple. It follows that
$A\cong\mathbb{C}^{n}$, then, by [20], $A$ is contractible. $\blacksquare$

##  6\. Examples

In this section, we give three examples. The first example shows that there
exists a Banach algebra which is approximately character amenable but not
uniformly approximately character amenable. The second example shows that
there exists a Banach algebra which is character amenable but not character
contractible. The last example shows that there exists a Banach algebra which
is character contractible but not amenable.

Given a family $(A_{\alpha})_{\alpha\in\Lambda}$ of Banach algebras defined
their $l^{\infty}$ direct sum as

$l^{\infty}(A_{\alpha})=\\{(a_{\alpha}):a_{\alpha}\in
A_{\alpha},||(a_{\alpha})||=\sup||a_{\alpha}||<\infty\\}.$

Further, set

$c_{0}(A_{\alpha})=\\{(a_{\alpha}):(a_{\alpha})\in
l^{\infty}(A_{\alpha}),||a_{\alpha}||\rightarrow 0\\}.$

Example 1. There exists an approximately character amenable Banach algebra
which is not uniformly approximately character amenable.

For each $n\in\mathbb{N}$, take $A_{n}=\mathbb{C}^{n}$, each with the
corresponding $l^{1}$ norm. Then $A_{n}$ has an identity
$e_{n}=(1,1,\cdots,1)$ of norm $n$; clearly each $A_{n}$ is amenable, so
$c_{0}(A_{n}^{\sharp})$ is approximately amenable by [9, Example 6.1], then it
is also approximately character amenable. If $f_{\alpha}$ is a left
approximate identity for $c_{0}(A_{n}^{\sharp})$ then, given $n$, there is
$\alpha$ such that $||f_{\alpha}e_{n}-e_{n}||\leq 1$. Hence we have
$||f_{\alpha}||\geq||f_{\alpha}e_{n}||\geq n-1$ and then $\\{f_{\alpha}\\}$ is
unbounded.

But $c_{0}(A_{n}^{\sharp})$ is not uniformly approximately character amenable.
Indeed, if it were, Theorem 2.8 would imply that $c_{0}(A_{n}^{\sharp})$ had
bounded left and right approximate identity.

Example 2. There exists a character amenable Banach algebra which is neither
amenable nor character contractible.

Let $V$ be the Volterra operator, $A_{V}$ be the Banach algebra generated by
$V$ and $A$ be the Banach algebra generated by $V$ and $e$. Then
$A=\mathbb{C}e\oplus A_{V}$, and $A_{V}$ has a bounded two sided approximate
identity [6, Theorem 5.10]. And $\sigma(A)=\\{\varphi\\},$ here
$\varphi(A_{V})=0,\varphi(e)=1$. Thus $A$ is a character amenable by Lemma
5.3. However, $A$ is not amenable by [8].

Moreover, $A$ is not character contractible. Indeed, if it were, Proposition
4.1 implies that there exists an $m=e+b\in A$ such that
$a(e+b)=\varphi(a)(e+b)$ for all $a\in A$. If, in addition, $a\in A_{V}$ and
$a\neq 0$, then $a(e+b)=0$; since $e+b$ is invertible, we obtain $a=0$, a
contradiction.

Example 3. Let $A=B(H)$, where $H$ is an infinite-dimensional Hilbert space.
Then $A$ has an identity and $A$ is character contractible by Theorem 4.10.
But $A$ is not amenable by [20].

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