Amenability-Like properties of C(X,A)

TL;DR

This paper explores how amenability-like properties transfer between a Banach algebra A and the algebra of continuous functions C(X, A) over a compact Hausdorff space, focusing on induced homomorphisms and their amenability implications.

Contribution

It introduces induced homomorphisms between A and C(X, A) and analyzes conditions under which (weak) amenability properties are preserved or transferred.

Findings

Conditions for $ au$-(weak) amenability transfer from $C(X, A)$ to $A$

Conditions for $ ilde{ au}$-(weak) amenability transfer from $A$ to $C(X, A)$

Characterization of when induced homomorphisms preserve amenability properties

Abstract

Let $A$ be a Banach algebra and $X$ be a compact Hausdorff space. Given homomorphisms $ \sigma \in Hom(A)$ and $\tau \in Hom(C(X, A))$, we introduce induced homomorphisms $\tilde{\sigma}\in Hom(C(X, A)) $ and $\tilde{\tau}\in Hom(A)$, respectively. We study when $\tau$-(weak) amenability of $C(X, A)$ implies $\tilde{\tau}$-(weak) amenability of $A$. We also investigate where $ \sigma$-weak amenability of $A$ yields $\tilde{\sigma}$-weak amenability of $C(X, A)$.

Equations

$$\sigma (a)\cdot u_{\alpha }-u_{\alpha }\cdot \sigma (a)⟶0,\text{and}\pi (u_{\alpha })\sigma (a)⟶\sigma (a)(a\in A),$$

$$\Gamma (u,\alpha )=\sum _{i,j}{u}_i{\alpha }_j\otimes v_i\beta j\in C(X,A)\hat{\otimes }C(X,A)$$

$$\Vert \tilde{\sigma }(a).U-U.\tilde{\sigma }(a)\Vert <\epsilon \text{and}\Vert \pi (U)\tilde{\sigma }(a)-\tilde{\sigma }(a)\Vert <\epsilon (a\in F)$$

$$\Vert \sigma (b).\alpha -\alpha .\sigma (b)\Vert <\frac{\epsilon }{4cNL},\Vert \pi (\alpha ).\sigma (b)-\sigma (b)\Vert <\frac{\epsilon }{NL}(b\in F_A).$$

$$\Vert f.u-u.f\Vert <\frac{\epsilon }{2\Vert \alpha \Vert LN}(f\in F_C).$$

$$\Vert \tilde{\sigma }(a).U-U.\tilde{\sigma }(a)\Vert$$

$$=\Vert \sum _k\Gamma (f_{k}u,\sigma (a_k)\alpha -\alpha \sigma (a_k))+\Gamma (f_{k}u-u{f}_k,\alpha \sigma (a_k))\Vert$$

$$\le \sum _k(L\Vert u\Vert \Vert \sigma (a_k)\alpha -\alpha \sigma (a_k)\Vert +L\Vert \alpha \Vert \Vert f_{k}u-u{f}_k\Vert )$$

$$\le NL(2c\frac{\epsilon }{4cNL}+\Vert \alpha \Vert \frac{\epsilon }{2\Vert \alpha {\Vert }_{P}LN})=\epsilon$$

$$\Vert \pi (U)\tilde{\sigma }(a)-\tilde{\sigma }(a)\Vert$$

$$\le L\sum _k\Vert \pi (\alpha )\sigma (a_k)-\sigma (a_k)\Vert <NL\frac{\epsilon }{NL}=\epsilon .$$

$$\Vert \tilde{\sigma }(a_{\epsilon }).U-U.\tilde{\sigma }(a_{\epsilon })\Vert <\frac{\epsilon }{2}\text{and}\Vert \pi (U)\tilde{\sigma }(a_{\epsilon })-\tilde{\sigma }(a_{\epsilon })\Vert <\frac{\epsilon }{2}(a_{\epsilon }\in F_{\epsilon }).$$

$$\Vert \tilde{\sigma }(a).U-U.\tilde{\sigma }(a)\Vert <\epsilon \text{and}\Vert \pi (U)\tilde{\sigma }(a)-\tilde{\sigma }(a)\Vert <\epsilon (a\in F)$$

$$\tau (f)(x):=\tau (1_{f(x)})(x):=\eta (1_{f(x)})(x)(f\in C(X,A),x\in X).$$

$$⟨\sigma (a_0),(R_{\lambda }\circ D)(a_0)⟩=⟨\sigma (a_0)\cdot D{a}_0,\lambda ⟩=1$$

$$D(fg\sigma (e_v))$$

$$=w{k}^{\ast }-\lim D(f\sigma (e_{\mu })).\tilde{\sigma }(g\sigma (e_v))+\tilde{\sigma }(f\sigma (e_{\mu })).D(g\sigma (e_v))$$

$$=\tilde{D}(f).\tilde{\sigma }(g\sigma (e_v))+f.D(g\sigma (e_v))$$

$$\tilde{D}(fg)=\tilde{D}(f).g+f.\tilde{D}(g)(f,g\in C(X)).$$

$$D(f\sigma (a))=\tilde{D}(f)\cdot \sigma (a)+f\cdot \tilde{D}(\sigma (a))=0(a\in A,f\in C(X)).$$

$$\tilde{D}:I\times A⟶E,\tilde{D}(a,b):=D(ab)-\sigma (b)\cdot D(a)$$

$$\tilde{D}(a_1{a}_2,b_1{b}_2)$$

$$=\sigma (a_1{b}_1)\cdot D(a_2{b}_2)+\sigma (a_2{b}_2)\cdot D(a_1{b}_1)-\sigma (b_1{b}_2)\cdot D(a_1{a}_2)$$

$$=\sigma (b_1)\cdot [\sigma (a_1)\cdot D(a_2{b}_2)-\sigma (b_2)\sigma (a_1)\cdot D(a_2)]$$

$$+\sigma (b_2)\cdot [\sigma (b_1{a}_1)\cdot D(a_2)+\sigma (a_2)\cdot D(a_1{b}_1)-\sigma (b_1)\cdot D(a_1{a}_2)]$$

$$=\sigma (b_1)\cdot [D(a_1{a}_2{b}_2)-\sigma (a_2{b}_2)\cdot D(a_1)-\sigma (b_2{a}_1)\cdot D(a_2)]$$

$$+\sigma (b_2)\cdot [D(a_1{a}_2{b}_1)-\sigma (b_1)\cdot D(a_1{a}_2)]$$

$$=\sigma (b_1)\cdot [D(a_1{a}_2{b}_2)-\sigma (b_2)\cdot D(a_1{a}_2)]$$

$$+\sigma (b_2)\cdot [D(a_1{a}_2{b}_1)-\sigma (b_1)\cdot D(a_1{a}_2)]$$

$$=\sigma (b_1)\cdot \tilde{D}(a_1{a}_2,b_2)+\sigma (b_2)\cdot \tilde{D}(a_1{a}_2,b_1)$$

$$\tilde{D}:I\times A\to F,(a,b)⟼(j\circ D)(a,b)-\sigma (b)\cdot (j\circ D)(a)$$

$$\sigma (a)c\cdot D(b)=(j\circ D)(b)(\sigma (a)c)=(\sigma (a)\cdot (j\circ D)(b))(c)=\tilde{D}(a,b)(c)=0.$$

$$\overline{\widetilde{{\sigma }^{♯}}(C(X,A))}=\overline{\tilde{\sigma }(C(X,A))}=C(X,A)$$

Full text

Keywords:

$\sigma$-amenability, $\sigma$-weak amenability, $\sigma$-bounded approximate diagonal.

MSC 2010: Primary: 46M10; Secondary: 46M18, 46H20.

1. Introduction

Let $X$ be a compact Hausdorff space and let $A$ be a Banach algebra. It is known that $\mathit{C(X,A)}$, the set of all $A$-valued continuous functions on $X$, is a Banach algebra with pointwise algebraic operations and the uniform norm $||f||_{\infty}:=\sup_{x\in X}||f(x)||$, $f\in C(X,A)$ [4]. In the nice papers [2, 8], Ghamarshoushtari and Zhang studied amenability and weak amenability of $\mathit{C(X,A)}$. They showed that $\mathit{C(X,A)}$ is amenable if and only if $A$ is amenable. Further, if $A$ is commutative, they proved that $C(X,A)$ is weakly amenable if and only if $A$ is weakly amenable.

Let $A$ be a Banach algebra. We denote by $Hom(A)$ the space of all continuous homomorphisms from $A$ into $A$. Let $A$ be a Banach algebra, $E$ be a Banach $A$-bimodule and let $\sigma\in Hom(A)$. A bounded linear map $D:A\longrightarrow E$ is $\sigma$-derivation if $D(ab)=D(a)\cdot\sigma(b)+\sigma(a)\cdot D(b)$, for $a,b\in A$. A $\sigma$-derivation $D:A\longrightarrow E$ is $\sigma$-inner derivation if there exists $x\in E$ such that $D(a)=\sigma(a)\cdot x-x\cdot\sigma(a)$, for all $a\in A$. A Banach algebra $A$ is $\sigma$-amenable if for every Banach $A$-bimodule $E$, every $\sigma$-derivation $D:A\longrightarrow E^{*}$ is $\sigma$-inner. Especially, a Banach algebra $A$ is $\sigma$-weakly amenable if every $\sigma$-derivation $D:A\longrightarrow A^{*}$ is $\sigma$-inner. It is known that the projective tensor product $A\hat{\otimes}A$ is a Banach $A$-bimodule in a natural way. A bounded net $(u_{\alpha})\subset A\hat{\otimes}A$ is a $\sigma$-bounded approximate diagonal for $A$ if

[TABLE]

where $\pi:A\hat{\otimes}A\longrightarrow A$ is the product map defined by $\pi(a\hat{\otimes}b)=ab$.

Before preceding further, we set up our notations. Let $X$ be a compact Hausdorff space and let $A$ be a Banach algebra. For each $x\in X$, we consider the continuous epimorphism $T_{x}:C(X,A)\longrightarrow A$ defined by $T_{x}(f):=f(x)$, for all $f\in C(X,A)$. For every $a\in A$, we define $1_{a}\in C(X,A)$ by the formula $1_{a}(x):=a$, for all $x\in X$. We notice that, every homomorphism $\sigma\in Hom(A)$ induces a homomorphism $\tilde{\sigma}\in Hom(C(X,A))$ defined by $\tilde{\sigma}(f):=\sigma f$, $f\in C(X,A)$. Conversely, for every homomorphism $\tau\in Hom(C(X,A))$ and $x\in X$, we introduce the map $\tilde{\tau}_{x}:A\longrightarrow A$ through $\tilde{\tau}_{x}(a):=T_{x}(\tau(1_{a}))$, $a\in A$. Since $1_{ab}=1_{a}1_{b}$, $\tilde{\tau}_{x}\in Hom(A)$. It is readily seen that $(T_{x}\tau)(1_{a})=(\tilde{\tau}_{x}T_{x})(1_{a})$, for each $a\in A$. Next, for $f\in C(X)$ and $a\in A$ we define $fa\in C(X,A)$ via $fa(x)=f(x)a$ for each $x\in X$. Throughout the paper, we keep the above definitions and notations.

In this paper, we deal with amenability-like properties of the Banach algebra $C(X,A)$. Suppose that $\sigma$ and $\tau$ are homomorphisms on $A$ and $C(X,A)$, respectively. For a given $\sigma$-bounded approximate diagonal for $A$, we construct a $\tilde{\sigma}$-bounded approximate diagonal for $C(X,A)$ (Theorem 2.1). We show that under some certain conditions, $\tau$-(weak) amenability of $C(X,A)$ implies $\tilde{\tau}_{x}$-(weak) amenability of $A$ (Theorems 2.4 and 3.4). For a commutative Banach algebra $A$, we prove that $\sigma$-weak amenability of $A$ yields $\tilde{\sigma}$-weak amenability of $C(X,A)$ in the presence of a bounded approximate identity for $A$ (Theorem 3.2). Finally, we show that Theorem 3.2 is still true without the existence of a bounded approximate identity for $A$ (Theorem 3.9).

2. $\sigma$-amenability

Suppose that $A$ is a Banach algebra and $X$ is a compact Hausdorff space. For $u=\sum_{i}u_{i}\otimes v_{i}\in C(X)\hat{\otimes}C(X)$ and $\alpha=\sum_{j}\alpha_{j}\otimes\beta_{j}\in A\hat{\otimes}A$, we consider

[TABLE]

so that $\|\Gamma(u,\alpha)\|\leq\|u\|\ \|\alpha\|$.

Theorem 2.1**.**

Let $X$ be a compact Hausdorff space, $A$ be a Banach algebra and let $\sigma\in Hom(A)$. If $A$ has a $\sigma$-bounded approximate diagonal, then $C(X,A)$ has a $\tilde{\sigma}$-bounded approximate diagonal.

Proof.

We follow the standard argument in [2]. Let $(\alpha_{v})\in A\hat{\otimes}A$ be a $\sigma$-bounded approximate diagonal for $A$ such that $\|\alpha_{v}\|_{p}\leq M$ for all $v$. We claim that for any $\varepsilon>0$ and any finite set $F\subset C(X,A)$, there is $U=U_{(F,\varepsilon)}\in C(X,A)\hat{\otimes}C(X,A)$ with $\|U\|\leq 2Mc$ such that

[TABLE]

where $c>0$ is the constant asserted in [2, Corollary 1.2]. Given $\varepsilon>0$ and a finite set $F\subset C(X,A)$. We first assume that each $a\in F$ is of the form of a finite sum $a=\sum_{k}f_{k}a_{k}$, with $f_{k}\in C(X)$ and $a_{k}\in A$. It is easy to check that $\tilde{\sigma}(a)=\sum_{k}f_{k}\sigma(a_{k})$. We denote by $F_{A}$ $(\subset A)$ the finite set of all elements $a_{k}$ associated to $a$ for all $a\in F$, and by $F_{C}$ $(\subset C(X))$ the finite set of all functions $f_{k}$ associated to $a$ for all $a\in F$. Let $N>0$ be an integer that is greater than the number of the terms of $a=\sum_{k}f_{k}a_{k}$ for all $a\in F$, and set $L:=\max\{\|\sigma(b)\|\ ,\ \|f\|\ :\ b\in F_{A},f\in F_{C}\}$. By the assumption, there is $\alpha\in(\alpha_{v})$ such that

[TABLE]

On the other hand, by the same argument as in the proof of [2, Theorem 2.1], we obtain an element $u\in C(X)\otimes C(X)$ with $\pi(u)=1$ and $\|u\|\leq 2c$ for which

[TABLE]

Putting $U:=\Gamma(u,\alpha)$, for $a=\sum_{k}f_{k}a_{k}$ we see that

[TABLE]

and

[TABLE]

Now, we assume that $F\subset C(X,A)$ is an arbitrary finite set. From the proof of [2, Theorem 2.1], we know that for each $a\in F$ there exists an element $a_{\varepsilon}=\sum_{k}f_{k}a_{k}$ where the right side of $a_{\varepsilon}$ is a finite sum, $f_{k}\in C(X)$ and $a_{k}\in A$ such that \|\tilde{\sigma}(a)-\tilde{\sigma}(a_{\varepsilon})\|<\min\big{\{}\dfrac{\varepsilon}{4},\dfrac{\varepsilon}{8Mc}\big{\}}. Applying the above argument for the finite set $F_{\varepsilon}:=\{a_{\varepsilon}:a\in F\}$, we get $U\in C(X,A)\hat{\otimes}C(X,A)$ such that $\|U\|\leq 2cM$ and

[TABLE]

Therefore

[TABLE]

so that the claim is proved. Finally, the net $(U_{F,\varepsilon})$ with the natural partial order $(F_{1},\varepsilon_{1})<(F_{2},\varepsilon_{2})$ if and only if $F_{1}\subset F_{2}$ and $\varepsilon_{1}\geq\varepsilon_{2}$ is the desired $\tilde{\sigma}$-approximate diagonal for $C(X,A)$. ∎

Remark 2.2**.**

To our knowledge, we do not know whether or not the existence of $\sigma$-bounded approximate diagonal is equivalent to $\sigma$-amenability. Hence, we can not prove or disprove if $\sigma$-amenability of $A$ implies $\tilde{\sigma}$-amenability of $C(X,A)$.

Proposition 2.3**.**

Suppose that $\sigma\in Hom(A)$ and $\tau\in Hom(B)$, where $A$ and $B$ are Banach algebras. Suppose that $\phi:A\longrightarrow B$ is a continuous homomorphism with a dense range and $\tau\phi=\phi\sigma$. If $A$ is $\sigma$-amenable, then $B$ is $\tau$-amenable.

Proof.

We may either prove it or else look at [5, Proposition 3.3]. ∎

Theorem 2.4**.**

Let $X$ be a compact Hausdorff space, let $A$ be a Banach algebra and let $\tau\in Hom(C(X,A))$ such that $T_{x_{0}}\tau=\tilde{\tau}_{x_{0}}T_{x_{0}}$, for some $x_{0}\in X$. If $C(X,A)$ is $\tau$-amenable, then $A$ is $\tilde{\tau}_{x_{0}}$-amenable.

Proof.

We have already seen that the map $T_{x_{0}}$ is surjective, so this is immediate by Proposition 2.3. ∎

We note that homomorphisms $\tau\in Hom(C(X,A))$ satisfying the condition of Theorem 2.4 exist in abundance. Take an arbitrary homomorphism $\eta\in Hom(C(X,A))$ and define the map $\tau:C(X,A)\longrightarrow C(X,A)$ by

[TABLE]

Then, we may check that $\tau\in Hom(C(X,A))$ and $T_{x}\tau=\tilde{\tau}_{x}T_{x}$ for all $x\in X$.

3. $\sigma$-weak amenability

For a Banach algebra $A$ and a homomorphism $\sigma$ on $A$, we write $Z^{1}_{\sigma}(A,E)$ for the set of all $\sigma$-derivations from $A$ into a Banach $A$-bimodule $E$.

Let $A$ be a $\sigma$-weakly amenable Banach algebra, and $\sigma\in Hom(A)$ such that $\sigma(A)$ is a dense subset of $A$. A more or less verbatim of the classic argument, shows that $\overline{A^{2}}=A$ (see [6, Theorem 6] for details).

Proposition 3.1**.**

Let $A$ be a commutative Banach algebra and let $\sigma\in Hom(A)$ with a dense range. If $A$ is $\sigma$-weakly amenable, then $Z^{1}_{\sigma}(A,E)={0}$ for each Banach $A$-module $E$.

Proof.

Assume towards a contradiction that there is a nonzero element $D\in Z^{1}_{\sigma}(A,E)$. Since $\overline{A^{2}}=A$, there exists $a_{0}\in A$ with $D(a_{0}^{2})\neq 0$. Hence $\sigma(a_{0})\cdot Da_{0}\neq 0$ and so there exists $\lambda\in E^{*}$ with $\langle\sigma(a_{0})\cdot Da_{0},\lambda\rangle=1$. Consider $R_{\lambda}\in{}_{A}{B(E,A^{*})}$ such that $\langle a,R_{\lambda}x\rangle=\langle a\cdot x,\lambda\rangle$, for $a\in A$ and $x\in E$ [1, Proposition 2.6.6]. It is not hard to see that $R_{\lambda}\circ D\in Z^{1}_{\sigma}(A,A^{*})$. Then we have

[TABLE]

so that $R_{\lambda}\circ D\neq 0$, a contradiction of the fact that $A$ is $\sigma$-weakly amenable. ∎

Proposition 3.2**.**

Let $X$ be a compact Hausdorff space and let $A$ be a commutative Banach algebra with a bounded net $(e_{v})$ for which $(\sigma(e_{v}))$ is a bounded approximate identity for $A$, where $\sigma$ belongs to $Hom(A)$ with a dense range. If $A$ is $\sigma$-weakly amenable, then $C(X,A)$ is $\tilde{\sigma}$-weakly amenable.

Proof.

Using the map $a\longmapsto 1_{a}$, we may consider $A$ as a closed subalgebra of the commutative Banach algebra $C(X,A)$. Suppose that $D:C(X,A)\longrightarrow C(X,A)^{*}$ is a $\tilde{\sigma}$-derivation. Notice that $C(X,A)$ is naturally a commutative $A$-bimodule with actions $a\ .\ f(x)=f\ .\ a(x):=af(x)$, ($a\in A,f\in C(X,A),x\in X$). We also note that $\tilde{\sigma}=\sigma$ on $A$. Therefore $D\mid_{A}:A\longrightarrow C(X,A)^{*}$ is a $\sigma$-derivation and then, by Proposition 3.1, $D\mid_{A}=0$. On the other hand, $(\sigma(e_{v}))$ is also a bounded approximate identity for $C(X,A)$ and $wk^{*}$-$\lim D(\sigma(e_{v}))=0$. An argument similar to that in the proof of [2, Proposition 4.1] shows that $wk^{*}$-$\lim D(f\sigma(e_{v}))$ exists for each $f\in C(X)$. So we may define $\tilde{D}:C(X)\longrightarrow C(X,A)^{*}$ via $\tilde{D}(f)=wk^{*}-\lim_{v}D(f\sigma(e_{v}))$. Clearly $C(X,A)$ is a commutative $C(X)$-bimodule. Then

[TABLE]

for $f,g\in C(X)$. Taking $wk^{*}$-limit in $v$, we get

[TABLE]

Hence $\tilde{D}$ is a derivation on amenable $C(X)$. Therefore $\tilde{D}=0$, and then

[TABLE]

Whence $D=0$ on the linear span of $\{f\sigma(a):f\in C(X),a\in A\}$ which is dense in the linear span of $\{fa:f\in C(X),a\in A\}$, by the density of range of $\sigma$. The latter is itself dense in $C(X,A)$ by [2], so that $D=0$ on the whole $C(X,A)$. ∎

In Proposition 3.2, as an special case, we may suppose that $(e_{v})$ is itself a bounded approximate identity for $A$. Indeed if $(e_{v})$ is a bounded approximate identity for $A$, then the density of the range of $\sigma$ shows that $(\sigma(e_{v}))$ is still a bounded approximate identity for $A$.

The following was proved in [6, Proposition 18], and so we omit its proof.

Proposition 3.3**.**

Suppose that $\sigma\in Hom(A)$ and $\tau\in Hom(B)$, where $A$ and $B$ are Banach algebras such that $A$ is commutative and $\sigma$ has a dense range. Suppose that $\phi:A\longrightarrow B$ is a continuous epimorphism for which $\tau\phi=\phi\sigma$. If $A$ is $\sigma$-weakly amenable, then $B$ is $\tau$-weakly amenable.

The following is the converse of Proposition 3.2.

Theorem 3.4**.**

Let $X$ be a compact Hausdorff space, let $A$ be a commutative Banach algebra and let $\tau\in Hom(C(X,A))$ with a dense range such that $T_{x_{0}}\tau=\tilde{\tau}_{x_{0}}T_{x_{0}}$, for some $x_{0}\in X$. If $C(X,A)$ is $\tau$-weakly amenable, then $A$ is $\tilde{\tau}_{x_{0}}$-weakly amenable.

Proof.

We use Proposition 3.3. ∎

We extend [1, Theorem 1.8.4] as follows, where the proof reads somehow the same lines.

Proposition 3.5**.**

Let $I$ be an ideal in a commutative algebra $A$, $E$ be an $A$-module and $D:I\longrightarrow E$ be a $\sigma$-derivation. Then the map

[TABLE]

is bilinear such that

(i) $\tilde{D}(a,b)=\sigma(a)\cdot D(b)\ \ \ (a,b\in I)$;

(ii) for each $a\in I^{2}$, the map $b\longmapsto\tilde{D}(a,b),$ $A\longrightarrow E$ is a $\sigma$-derivation.

Proof.

We only prove the clause (ii). For $a_{1},a_{2}\in I$, and $b_{1},b_{2}\in A$, we have

[TABLE]

as required. ∎

The following is an analogue of [1, Lemma 2.8.68].

Lemma 3.6**.**

Let $A$ be a $\sigma$-weakly amenable commutative Banach algebra, $I$ be a closed ideal in $A$, and $E$ be a Banach $A$-module. Take $\sigma\in Hom(A)$ with a dense range such that $\sigma(I)\subseteq I$. Then $D\mid_{I^{4}}=0$ for each $D\in Z^{1}_{\sigma}(I,E)$.

Proof.

We first observe that $F:={}_{A}{\mathcal{B}(I,E)}$ is a Banach $A$-module for the action $(a\cdot T)(b)=T(ab)$, $(a\in A,b\in I)$. Then we notice that the map $j:E\longrightarrow F$ with $j(x)(a)=a\cdot x$, $(a\in I,x\in E)$ belongs to ${}_{A}{\mathcal{B}(E,F)}$. Whence $j\circ D\in Z^{1}_{\sigma}(I,E)$. Clearly the map

[TABLE]

is bilinear. Therefore $\tilde{D}(a,b)=\sigma\ (a)\cdot(j\circ D)(b)$, by Proposition 3.5 (i), for $a,b\in I$. Take $a\in I^{2}$. By Proposition 3.5 (ii), the map $A\longrightarrow F$, $b\longmapsto\tilde{D}(a,b)$, is a $\sigma$-derivation. It follows from Proposition 3.1, that this map is zero and so $\tilde{D}(I^{2}\times A)=0$. Hence, for $a\in I^{2}$ and $b,c\in I$, we have

[TABLE]

In particular, choosing $c\in\sigma(I)$, we may see that $\sigma(I^{3})\cdot D(I)=0$, and whence $D\mid I^{4}=0$. ∎

Proposition 3.7**.**

Let $A$ be a $\sigma$-weakly amenable commutative Banach algebra, $I$ be a closed ideal in $A$, and $\sigma\in Hom(A)$ with a dense range such that $\overline{\sigma(I)}=I$. Then $I$ is $\sigma$-weakly amenable if and only if $\overline{I^{2}}=I$.

Proof.

If $I$ is $\sigma$-weakly amenable, then $\overline{I^{2}}=I$ by [6, Theorem 6].

Conversely if $\overline{I^{2}}=I$, then $\overline{I^{4}}=I$. Suppose that $D\in Z^{1}_{\sigma}(I,I^{*})$. Then by Lemma 3.6, $D\mid_{I^{4}}=0$ so that $D=0$. Hence $I$ is $\sigma$-weakly amenable. ∎

Lemma 3.8**.**

Let $X$ be a compact Hausdorff space, let $A$ be a Banach algebra and let $\sigma\in Hom(A)$. If $\sigma$ has a dense range, then $\tilde{\sigma}$ has a dense range as well.

Proof.

Take $f\in C(X)$ and $a\in A$ and put $h:=fa\in C(X,A)$. By the assumption, there exists a sequence $(a_{n})\subseteq A$ such that $\lim_{n}\sigma(a_{n})=a$. Define $h_{n}:=fa_{n}$, $(n=1,2,...)$. Then it is easy to verify that $\lim_{n}\tilde{\sigma}(h_{n})=h$. This completes the proof, since the set of all linear combinations of elements of $C(X,A)$ of the form $fa$ $(f\in C(X),a\in A)$ is dense in $C(X,A)$ [2]. ∎

We recall that a homomorphism $\sigma\in Hom(A)$ is extended to a homomorphism $\sigma^{\sharp}\in Hom(A^{\sharp})$ through $\sigma^{\sharp}(a+\lambda e)=\sigma(a)+\lambda e$ $(a\in A,\lambda\in\mathbb{C})$, where $e$ is the identity of $A^{\sharp}$, the unitization of $A$.

Now, we are ready to prove our last goal.

Theorem 3.9**.**

Let $X$ be a compact Hausdorff space, let $A$ be a commutative Banach algebra and let $\sigma\in Hom(A)$ with a dense range. If $A$ is ${\sigma}$-weakly amenable, then $C(X,A)$ is $\tilde{\sigma}$-weakly amenable.

Proof.

Suppose that $A$ is ${\sigma}$-weakly amenable. By [6, Theorem 12], $A^{\sharp}$ is $\sigma^{\sharp}$-weakly amenable. Applying Proposition 3.2, we see that $C(X,A^{\sharp})$ is $\tilde{\sigma^{\sharp}}$-weakly amenable. Our assumptions together with [6, Theorem 6], imply that $A^{2}$ is dense in $A$. We learn from the proof of [8, Theorem 1] that ${C(X,A)^{2}}$ is dense in $C(X,A)$, and also $C(X,A)$ is a closed ideal of $C(X,A^{\sharp})$. Next, it is easy to verify that $\tilde{\sigma^{\sharp}}(f)=\tilde{\sigma}(f)$, for all $f\in C(X,A)$. Hence

[TABLE]

by Lemma 3.8. Now, an application of Proposition 3.7 yields that $C(X,A)$ is $\tilde{\sigma}$-weakly amenable. ∎