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Weak amenability of Banach algebras with respect to characters.

1 Introduction

An important result by Johnson [10] is that a locally compact group G is amenable if and only if the group algebra [L.sup.1](G) is amenable as a Banach algebra; that is, the first cohomology groups [H.sup,1]([L.sup.1](G),X*) vanishes for all Banach [L.sup.1](G)-bimodules X. This equivalence does not remain true for the convolution semigroup algebra [l.sup.1](S) of a discrete semigroup S.

Lau [16] introduced and investigated a weaker property on F-algebras, called left amenability, by relativizing amenability to the identity element of the dual W*-algebra. Recall that an F-algebras is a Banach algebra L which is the unique predual of a W*-algebra M and the identity element u of M is a character on L; later, in Pier [27], F-algebras were called Lau algebras. In fact, the Lau algebra L is said to be left amenable if [H.sup,1](L;X*) = {0} for all Banach L-bimodules X with the left action defined by l x x = u(l)x for all [lambda] [member of] L and x [member of] X; see also [6], [14], [15], [17], [19], [20] and [21]. Lau [16] proved that [l.sup.1](S) is left amenable if and only if S is left amenable.

The class of Lau algebras includes the Fourier algebra A(G) and the group algebra [L.sup.1](G) of a locally compact group G, as well as the measure algebra M(S) of a locally compact topological semigroup S; see Lau [16]. It also includes the Fourier-Stieltjes algebra B(G) of any topological group G; see Lau and Ludwig [19].

A considerable generalization of the concept of left amenability was introduced by Kaniuth, Lau and Pym [12] (see also Monfared [22]) for a Banach algebra A with respect to an arbitrary character f on A; in fact, the Banach algebra A is called [phi]-amenable if [H.sup,1](A;X*) vanishes for all Banach A-bimodules X for which the left module action of A on X is defined by a x x = [phi](a)x for a [member of] A and x [member of] X. For some related works on amenability of Banach algebras with respect to a character, see [1, 2, 5, 13, 23, 24, 25, 26].

Moreover, the notion of weak amenability for an arbitrary Banach algebra was introduced and studied by Johnson [11]; the Banach algebra A is called weakly amenable if [H.sup,1](A,A*) = {0}, where the dual space A* of A is equipped with the Arens module structure; see also the recent works [7, 18].

Our goal in this work is to introduce and to study weak character amenability of a Banach algebra A as a newnotion of amenability. In Section 2,we showthat if A has a bounded left approximate identity, then A is weakly character amenable; however the converse is not true. In Section 3, we will focus on a special class of module extension Banach algebras to characterize their derivations; we also use this result to give a class of Banach algebras that are not weakly character amenable. We finally compare the notion of weak character amenability with some different versions of amenability.

2 Weak character amenability

Let A be a Banach algebra and let X be a Banach A-bimodule. Then the dual X* of X, has an Arens A-bimodule structure defined by

(a x [LAMBDA])(x) = [LAMBDA] (x x a),

([LAMBDA] x a)(x) = [LAMBDA] (a x x)

for all a [member of] A, x [member of] X and [LAMBDA] [member of] X*.

A derivation of A into X* is a linear mapping D : A -[right arrow] X* such that

D(ab) = D(a) x b + a x D(b)

for all a, b [member of] A. For each [LAMBDA] [member of] X*, the mapping [ad.sub.[LAMBDA]] : A [right arrow] X* defined by

[ad.sub.[LAMBDA]](a) = a x [LAMBDA] - [LAMBDA] x a

is a derivation, called the inner derivation associated with [LAMBDA].

Let Y be a Banach right A-module and [phi] [member of] [DELTA](A) [union] {0}, where [DELTA](A) is the spectrum of A consisting of all characters from A into the complex numbers. We denote by [sub.[phi]]Y the Banach A-bimodule Y such that the left action satisfies

a x y = [phi](a)y

for all a [member of] A and y [member of] Y. Then ([sub.[phi]]Y)* is also a Banach A-bimodule with the natural dual actions.

We always consider A as a Banach right A-module with the actions obtained from the multiplication of A. Then [sub.[phi]]A is a Banach A-bimodule with the left and right actions

a x x = [phi](a)x and x x a = xa

for all a, x [member of] A, respectively. Moreover, ([sub.[phi]]A)* is a Banach A-bimodule with the left and right actions defined by

a x f = a f and f x a = [phi](a) f

for all a [member of] A and [phi] [member of] A*, respectively; here a [phi] [member of] A* is defined by (a f)(b) = f (ba) for all b [member of] A.

Definition 2.1. Let A be a Banach algebra and let [phi] [member of] [DELTA](A) [union] {0}. We say that A is weakly [phi]-amenable if

[H.sup,1](A, ([sub.[phi]]A)*) = {0}.

We also say that A is weakly character amenable whenever A is weakly [phi]-amenable for all [phi] [member of] [DELTA](A) [union] {0}.

In the following, we introduce a large class of weakly character amenable Banach algebras.

Theorem 2.2. Let A be a Banach algebra. If A has a bounded left approximate identity, then A is weakly character amenable. In particular, the unitization [A.sup.#] of A is weakly character amenable.

Proof. For [phi] [member of] [DELTA](A) [union] {0}, suppose that D : A [right arrow] ([sub.[phi]]A)* is a derivation and let [([e.sub.[gamma]]).sub.[gamma][member of][GAMMA]] be a bounded left approximate identity for A. For [gamma] [member of] [GAMMA], define the functional [f.sub.[gamma]] [member of] A* by

[f.sub.[gamma]](a) := <D(a), [e.sub.[gamma]]>

for all a [member of] A. The bounded net [([f.sub.[gamma]]).sub.[gamma][member of][GAMMA]] [subset or equal to] A* has a subnet [([f.sub.[beta]]).sub.[beta][member of][GAMMA]] converging to a point f in the weak*- topology of A*. Then for each a, b [member of] A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This means that D = [ad.sub.f] which completes the proof.

The following example shows that Theorem 2.2 does not remain valid for all Banach algebras with approximate identity (not necessarily bounded).

Example 2.3. Let A be a Banach algebra without right identity such that A is isometrically isomorphic to the dual space A*. Then A is not weakly 0-amenable; indeed, the identity map

I : A -[right arrow] ([sub.0]A)*

is a derivation which is not inner. In particular, the discrete Banach algebra [l.sup.2](N) of all sequences of complex numbers a := (a(n)) with

[parallel]a[[parallel].sub.2] := [([[infinity].summation over (n=1)][[absolute value of a(n)].sup.2]).sup.1/2] < [infinity]

endowed with pointwise product is not weakly 0-amenable. Note that [l.sup.2](N) has a left approximate identity.

In the sequel, we investigate the relationship between weak [phi]-amenability of A and the 2-nd Hochschild cohomology group of A with coefficients in C. We first state some preliminary results and notations.

Let A be a Banach algebra and let X be a Banach A-bimodule. Then the space

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

becomes a Banach A-bimodule, through

(a x T)([a.sub.1], ..., [a.sub.k]) := a x T([a.sub.1], ..., [a.sub.k])

and

(T x a)([a.sub.1], ..., [a.sub.k]) : = T(a[a.sub.1], ..., [a.sub.k]) + [k-1.summation over (j=1)(-1)]T(a, [a.sub.1], ..., [a.sub.j][a.sub.j+1], ..., [a.sub.k]) + [(-1).sup.k]T(a, [a.sub.1], ..., [a.sub.k-1]) x [a.sub.k].

for all T [member of] [L.sup.k](A,X) and a, [a.sub.1], ..., [a.sub.k] [member of] A; for more details see [28]. Now, consider the operators

[[delta].sup.1] : [L.sup.1](A,X) [right arrow] [L.sup.2](A,X)

and

[[delta].sup.2] : [L.sup.2](A,X) [right arrow] [L.sup.3](A,X)

with formulates

([[delta].sup.1]S)([a.sub.1], [a.sub.2]) := [a.sub.1] x S[a.sub.2] - S([a.sub.1][a.sub.2]) + S([a.sub.1]) x [a.sub.2], ([[delta].sup.2]T)([a.sub.1], [a.sub.2], [a.sub.3]) := [a.sub.1] x T([a.sub.2], [a.sub.3]) - T([a.sub.1][a.sub.2], [a.sub.3]) + T([a.sub.1], [a.sub.2][a.sub.3]) - T([a.sub.1], [a.sub.2]) x [a.sub.3]

for all T [member of] [L.sup.2](A,X), S [member of] [L.sup.1](A,X) and [a.sub.1], [a.sub.2], [a.sub.3] [member of] A. Then the 2-th Hochschild cohomology group of A with coefficients in X is defined by

[H.sup.2](A,X) := [Z.sup.2](A,X)/[N.sup.2](A,X),

where

[N.sup.2](A,X) := ran([[delta].sup.1])

and

[Z.sup.2](A,X) := ker([[delta].sup.2]).

Let C be the Banach space of complex numbers. We denote by [sub.0][C.sub.[phi]] the Banach A-bimodule endowed with the actions defined by

a x [lambda] = 0 and [lambda] x a = [phi](a) [lambda]

for all a [member of] A, [lambda] [member of] C and [phi] [member of] [DELTA](A) [union] {0}. In Theorem 2.2, we show that A is weakly [phi]-amenable for all Banach algebras A with a bounded left approximate identity. In the following result, we extend that result to a considerably larger family of Banach algebras; that is, Banach algebras A for which [H.sup.2](A,[sub.0][C.sub.[phi]]) = {0} for all [phi] [member of] [DELTA](A) [union] {0}.

Theorem 2.4. Let A be a Banach algebra and let [phi] [member of] [DELTA](A) [union] {0}. Consider the following statements

(a) A has a bounded left approximate identity.

(b) [H.sup.2](A, [sub.0][C.sub.[phi]]) = {0}.

(c) A is weakly [phi]-amenable.

Then we have (a) [??] (b) [??] (c).

Proof. That (a) implies (b) follows from the fact that [H.sup.2](A,X*) = {0} for all Banach A-bimodules X with the right module action x x a = 0 (a [member of] A, x [member of] X); see [10], Proposition 1.5.

To show that (b) implies (c), consider the short exact sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of Banach A-bimodules, where i is the natural embedding and [[phi].sub.[infinity]](a, [lambda]) = [lambda] for all a [member of] A and [lambda] [member of] C. The sequence [SIGMA] is admissible; that is, [[phi].sub.[infinity]] has a bounded right inverse. Hence its dual [SIGMA] is admissible; see [3], Theorem 2.8.31. We therefore have a long exact sequence which contains the subsequence

... [right arrow] [H.sup,1](A, ([sub.[phi]][A.sup.#])*) [right arrow] [H.sup,1](A, ([sub.[phi]]A)*) [right arrow] [H.sup.2](A, [sub.0][C.sub.[phi]]) [right arrow] ...

by Theorem 2.8.25 of [3]. Let [[phi].sup.#] [member of] D([A.sup.#]) denote the character defined by

[[phi].sup.#](a, [lambda]) = [phi](a) + [lambda]

for all a [member of] A and [lambda] [member of] C. Suppose that D : A -[right arrow] ([sub.[phi]][A.sup.#])* is a derivation and extend derivation D to a derivation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] given by

[D.sup.#](a, [lambda]) := D(a)

for all a [member of] A and [lambda] [member of] C. Since [A.sup.#] is weakly character amenable, D is inner and

[H.sup,1](A, ([sub.[phi]][A.sup.#])*) = {0}.

But [H.sup.2](A, [sub.0][C.sub.[phi]]) = {0} and so

[H.sup,1](A, ([sub.[phi]]A)*) = {0};

that is, A is weakly [phi]-amenable.

Let [OMEGA] be a set and let [l.sup.1]([OMEGA]) = {a : [OMEGA] -[right arrow] C | [parallel]a[[parallel].sub.1] := [[SIGMA].sub.s[member of][OMEGA]] [absolute value of a(s)] < [infinity]}. Then [l.sup.1]([OMEGA]) with respect to the pointwise operations and the norm [parallel].[[parallel].sub.1] is a Banach algebra. We have

[DELTA]([l.sup.1]([OMEGA])) = {[[phi].sub.s] : s [member of] [OMEGA]},

where [[phi].sub.s] [member of] [l.sup.1]([OMEGA])* is the functional defined by

[[phi].sub.s](a) = a(s)

for all a [member of] [l.sup.1]([OMEGA]). Let us recall that [l.sup.1]([OMEGA]) has no bounded left approximate identity if [OMEGA] is infinite.

We continue this section by the following result which shows that the statements (a) and (b) in Theorem 2.4 are not equivalent.

Proposition 2.5. Let [OMEGA] be a set. Then [H.sup.2]([l.sup.1]([OMEGA]), [sub.0][C.sub.[phi]]) = {0} for all [phi] [member of] D([l.sup.1]([OMEGA])) [union] {0}.

Proof. Fix s [member of] [OMEGA] and consider [[phi].sub.s] [member of] [DELTA]([l.sup.1]([OMEGA])). We know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by Theorem 2.4.6 of [28]. Also, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [l.sup.1]([OMEGA])* is a Banach [l.sup.1]([OMEGA])-bimodule with the following module actions

a x f = 0, f x a = a f - f (a)[[phi].sub.s]

for all a [member of] [l.sup.1]([OMEGA]) and [phi] [member of] [l.sup.1]([OMEGA])*. Therefore, it is sufficient to prove that

[H.sup,1]([l.sup.1]([OMEGA]), [l.sup.1]([OMEGA])*) = {0}.

For this end, let D : [l.sup.1]([OMEGA]) [right arrow] [l.sup.1]([OMEGA])*be a derivation and define

g(t) := -<D([e.sub.t]), [e.sub.t]>

for t [member of] [OMEGA], where et is defined on [OMEGA] by [e.sub.t](u) = 1 for u = t and [e.sub.t](u) = 0 for all u [member of] [OMEGA] with u [not equal to] t. Then g defines an element of [l.sup.1]([OMEGA])* and we have

0 = D([e.sub.t][e.sub.u]) = [e.sub.t] x D([e.sub.u]) + D([e.sub.t]) x [e.sub.u] = 0 + [e.sub.u]D([e.sub.t]) - <D([e.sub.t]), [e.sub.u]>[[phi].sub.s]

for all t, u [member of] [OMEGA] with t [not equal to] u. Now, choose t, u [member of] [OMEGA] \ {s} with t 6 = u, and note that

<D([e.sub.t]), [e.sub.u]> = 0. (1)

A similar argument gives

<D([e.sub.s]), [e.sub.t]> = 0 t [member of] [OMEGA] (2)

<D([e.sub.t]), [e.sub.s]> = -<D([e.sub.t]), [e.sub.t]> t [member of] [OMEGA] \ {s}. (3)

For each a [member of] [l.sup.1]([OMEGA]), note that a can be written as

a = summation over (t[member of]c(a))]a(t)[e.sub.t]

where

c(a) = {t [member of] [OMEGA]|a(t) [not equal to] 0}.

On the one hand, it follows from (1), (2) and (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all a, b [member of] [l.sup.1]([OMEGA]). On the other hand, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all a, b [member of] [l.sup.1]([OMEGA]). Thus D(a) = a x g - g x a; that is, D is inner. It follows that

[H.sup,1]([l.sup.1]([OMEGA]), [l.sup.1]([OMEGA])*) = {0},

and therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A similar argument shows that [H.sup.2]([l.sup.1]([OMEGA]), [sub.0][C.sub.0]) = {0}.

As a consequence of this result together with Theorem 2.4, we have the following result which shows that the converse of Theorem 2.2 is not valid.

Corollary 2.6. For each set W, the Banach algebra [l.sup.1]([OMEGA]) is weakly character amenable.

It is still an open question whether the converse of the implication (b) [??] (c) in Theorem 2.4 holds.

Question 1. Let A be a Banach algebra and let [phi] [member of] [DELTA](A)[union]{0}. Does [H.sup.2](A, [sub.0][C.sub.[phi]]) = {0} when A is weakly [phi]-amenable?

3 A class of non-weakly character amenable Banach algebras

In this section, we introduce a class of Banach algebras which are not weakly character amenable. For this purpose, let A be a Banach algebra, let [phi] [member of] [DELTA](A) [union] {0} and let Y be a Banach right A-module. Denote by A [direct sum] [sub.[phi]]Y the module extension of A corresponding to [sub.[phi]]Y; that is, the [l.sup.1]-direct sum of A and [sub.[phi]]Y as a Banach algebra endowed with the product defined by

(a, y)(b, y') = (ab, [phi](a)y' + y x b)

for all (a, y), (b, y') [member of] A[direct sum] [sub.[phi]]Y. Then the dual of A[direct sum] [sub.[phi]]Y can be identified with the [.sup.[infinity]]-direct sum A* x Y* in the natural way

<(f ,[LAMBDA]), (a, y)> = f (a) + [LAMBDA] (y)

for all (f , [LAMBDA]) [member of] (A [direct sum] [sub.[phi]]Y)* and (a, y) [member of] A [direct sum] [sub.[phi]]Y. Furthermore,

[DELTA](A [direct sum] [sub.[phi]]Y) = {([psi], 0) : [psi] [member of] [DELTA](A)}.

Moreover, the (A [direct sum] [sub.[phi]]Y)-bimodule actions of (([sub.[phi],0])(A [direct sum] [sub.[phi]]Y))* are formulated as follows

(a, y)(f , [LAMBDA]) = (a f + [LAMBDA] (y)[phi], a x [LAMBDA])

and

(f , [LAMBDA]) x (a, y) = ([phi](a) f , [phi](a) [LAMBDA])

where (a, y) [member of] A [direct sum] [sub.[phi]]Y and (f , [LAMBDA]) [member of] A* x [sub.[phi]]Y*; see [3] and [29] for more details.

The following result characterizes derivations on A [direct sum] [sub.[phi]]Y.

Theorem 3.1. Let A be a Banach algebra, let [phi] [member of] [DELTA](A), and let Y be a Banach right A-module. Then D : A [direct sum] [sub.[phi]]Y -[right arrow] ([sub.([phi],0])](A [direct sum] [sub.[phi]]Y))* is a derivation if and only if

D(a, y) = ([D.sub.A*] (a) + T(y), [D.sub.Y*] (a))

for all (a, y) [member of] A [direct sum] [sub.[phi]]Y such that

(a) [D.sub.A*] : A -[right arrow] ([sub.[phi]]A)* is a derivation.

(b) [D.sub.Y*] : A -[right arrow] ([sub.[phi]]Y)* is a derivation.

(c) T : [sub.[phi]]Y -[right arrow] ([sub.[phi]]A)* is a bounded linear map with

T(y x a) = [D.sub.Y*](a)(y)[phi] + [phi] (a)T(y), [phi] (a)T(y) = aT(y).

for all a [member of] A and y [member of] Y. In this case,

(i) D is inner if and only if [D.sub.A*] is inner and there is [LAMBDA] [member of] Y* such that T(y) = [LAMBDA] (y)[phi] for all y [member of] Y.

(ii) If there is [LAMBDA] [member of] Y* such that T(y) = [LAMBDA](y)[phi], then [D.sub.Y*] = [ad.sub.[LAMBDA]].

Proof. Suppose that D : A [direct sum] [sub.[phi]]Y -[right arrow] ([sub.([phi],0)]A [direct sum] [sub.[phi]]Y)* is a derivation. Then

D(a, y) = D([alpha], 0) + D(0, y)

for all a [member of] A and y [member of] Y. Since D([alpha], 0), D(0, y) [member of] A* xY*, we can define themaps

[D.sub.A*] : A -[right arrow] ([sub.[phi]]A)*,

[D.sub.Y*] : A -[right arrow] ([sub.[phi]]Y)*,

T : [sub.[phi]]Y -[right arrow] ([sub.[phi]]A)*,

and

S : [sub.[phi]]Y -[right arrow] ([sub.[phi]]Y)*

such that

D([alpha], 0) = ([D.sub.A*] (a), [D.sub.Y*] (a)),

and

D(0, y) = (T(y), S(y)).

Then

D(a, y) = ([D.sub.A*] (a) + T(y), [D.sub.Y*] (a) + S(y)).

For every (a, y), (a' , y') [member of] A [direct sum] [sub.[phi]]Y, on the one hand,

D((a, y)(a' , y'))= D((aa' , [phi](a)y' + y x a')) =([D.sub.A*] (aa') + T([phi](a)y' + y x a'), [D.sub.Y*](aa') + S([phi](a)y' + y x a')),

and on the other hand

(a, y)([D.sub.A*] (a')+T(y'), [D.sub.Y*](a')+S(y'))+ [phi](a')(([D.sub.A*] (a)+T(y), [D.sub.Y*] (a)+S(y)) = (a[D.sub.A*] (a') + aT(y') + [D.sub.Y*](a')(y)[phi] + S(y')(y)[phi], a x [D.sub.Y*](a') + a x S(y')) + ([phi](a')[D.sub.A*](a) + [phi](a')T(y), [phi](a')[D.sub.Y*](a) + [phi](a')S(y'))

Since D is a derivation,

[D.sub.A*](aa') + T([phi](a)y' + y x a')= a[D.sub.A*] (a') + aT(y') + [D.sub.Y*](a')(y)[phi] + S(y')(y)[phi] + [phi](a')[D.sub.A*](a) + [phi](a')T(y),

and

[D.sub.Y*](aa') + S([phi](a)y' + y x a') = a x [D.sub.Y*](a') + a x S(y') + [phi](a')[D.sub.Y*] (a) + [phi](a')S(y')

If we take y = y' = 0, it follows that

[D.sub.A*](aa') = a[D.sub.A*] (a') + [phi](a')[D.sub.A*](a),

and

[D.sub.Y*](aa') = a x [D.sub.Y*](a') + [phi](a')[D.sub.Y*] (a).

If we take a = a' = 0, it follows that S(y')(y) = 0 for all y, y' [member of] Y, and hence S = 0. If we take a = y' = 0, it follows that

T(y x a') = [D.sub.X*](a')(y)[phi] + [phi](a')T(y),

and if we take a' = y = 0, it follows that

T(a x y') = T([phi](a)y') = aT(y').

The converse is an easy computation.

To prove (i), in this case suppose that D is an inner derivation. Then there is (f ,[LAMBDA]) [member of] (A [direct sum] [sub.[phi]]Y)* such that

D(a, y) = (a, y)(f, [LAMBDA]) - [phi](a)(f, [LAMBDA]).

Therefore,

[D.sub.A*](a) + T(y) = a f + [LAMBDA](y)[phi] - [phi](a) f

and

[D.sub.Y*](a) = a x [LAMBDA] - [phi](a) [LAMBDA].

That is, [D.sub.Y*] = [ad.sub.[LAMBDA]], T(y) = [LAMBDA](y)[phi] and

[D.sub.A*](a) = a f - [phi](a) f .

The converse of (i) and the statement (ii) are obvious.

As a consequence of Theorem 3.1, we have the following result. First, for a Banach space E, we denote by [sub.[phi]][E.sub.[phi]] the Banach A-bimodule obtained from E by putting

a x [xi] := [phi](a) [xi] := [xi] x a

for all a [member of] A and [xi] [member of] E.

Corollary 3.2. Let A be a Banach algebra, let [phi] [member of] [DELTA](A), and let E be a Banach space. Then D : A [direct sum] [sub.[phi]][E.sub.[phi]] -[right arrow] ([sub.([phi],0)](A [direct sum] [sub.[phi]][E.sub.[phi]]))* is a derivation if and only if

D(a, [xi]) = ([D.sub.A*](a) + T([xi]), 0)

for all a [member of] A and [xi] [member of] E such that

(a) [D.sub.A*] : A -[right arrow] ([sub.[phi]]A)* is a derivation.

(b) T : [sub.[phi]][E.sub.[phi]] -[right arrow] ([sub.[phi]]A)* is a bounded linear map such that for each a [member of] A and [xi] [member of] E,

[phi](a)T([xi]) = aT([xi]).

In this case, D is inner if and only if [D.sub.A*] is inner and there is [LAMBDA] [member of] E* such that T([xi]) = [LAMBDA]([xi])[phi] for all [xi] [member of] E.

Let E be a Banach space and let [alpha] [member of] E*\{0}. Then E with the multiplication [xi] [xi]' = [alpha]([xi]') [xi] for all [xi], [xi]' [member of] E is a Banach algebra which is denoted by [Alg.sub.[alpha]](E). It is easy to see that

[DELTA]([Alg.sub.[alpha]](E)) = {[alpha]}.

Proposition 3.3. Suppose that E is a Banach space with dim(E) [greater than or equal to] 2. Then [Alg.sub.[alpha]](E) [direct sum] [sub.[alpha]][([Alg.sub.[alpha]](E)*).sub.[alpha]] is not weakly ([alpha], 0)-amenable.

Proof. Suppose to the contrary that [Alg.sub.[alpha]](E) [direct sum] [alpha]([Alg.sub.[alpha]](E)*)[alpha] is weakly ([alpha], 0)-amenable. Let I : [Alg.sub.[alpha]](E)* -[right arrow] [Alg.sub.[alpha]](E)* be the identity mapping and define

D : [Alg.sub.[alpha]](E) [direct sum] [alpha]([Alg.sub.[alpha]](E)*)[alpha] -[right arrow] ([sub.([alpha],0)]([Alg.sub.[alpha]](E) [direct sum] [alpha]([Alg.sub.[alpha]](E)*)[alpha]))*

given by

D([xi], f) := (I(f), 0)

for all [xi] [member of] [Alg.sub.[alpha]](E) and [phi] [member of] [Alg.sub.[alpha]](E)*. Then D is a derivation. By Corollary 3.2, there is [m.sub.0] [member of] [Alg.sub.[alpha]](E)** such that f = [m.sub.0](f)[alpha] for all [phi] [member of] [Alg.sub.[alpha]](E)*. It follows that E* is generated by {[alpha]} which is a contradiction.

Theorem 3.4. Let A be a Banach algebra and let Y be a Banach right A-module. Then D : A [direct sum] [sub.0]Y -[right arrow] ([sub.(0,0)](A [direct sum] [sub.0]Y))* is a derivation if and only if

D(a, y) = ([D.sub.A*](a) + T(y), [D.sub.Y*] (a) + S(y))

for all a [member of] A and y [member of] Y such that

(a) [D.sub.A*] : A -[right arrow] ([sub.0]A)* is a derivation.

(b) [D.sub.Y*] : A -[right arrow] ([sub.0]Y)* is a derivation.

(c) T : [sub.0]Y -[right arrow] ([sub.0]A)* is an A-bimodule homomorphism.

(d) S : [sub.0]Y -[right arrow] ([sub.0]Y)* is an A-bimodule homomorphism.

In this case, D is inner if and only if [D.sub.A*] and [D.sub.Y*] are inner, T = 0 and S = 0.

Proof. The proof is similar to the proof of Theorem 3.1.

At the end of this section, we start making a comparison between some different versions of amenability related to weak character amenability.

Remark 3.5. (a) Character amenability implies automatically weak character amenability. The converse of this statement is false. For an example, let G be a locally compact group and let M(G) be the measure algebra of G. Since M(G) has an identity, it follows that M(G) is weakly character amenable by Theorem 2.2. Moreover, it is known that M(G) is weakly amenable if and only if G is discrete; see Theorem 1.2 of [4]. Also, M(G) is character amenable if and only if G is discrete and amenable; see Corollary 2.5 of [22]. So, in the case where G is non-discrete, M(G) is weakly character amenable, but neither weakly amenable nor character amenable.

(b) Let us remark that amenability implies character amenability and weak amenability. But the converse is not true. For example, consider the C*-algebra B(H) of all bounded operators on an infinite-dimensional Hilbert space H. In fact, every C*-algebra is both weakly amenable and character amenable; see [9] and [12]. Moreover, a C*-algebra is amenable if and only if it is nuclear. So, we only need to recall from [8] that B(H) is not nuclear.

(c) Let N denote the natural number and consider the Banach algebra [l.sup.2](N) of all sequences a := (a(n)) of complex numbers with

[parallel]a[[parallel].sub.2] := [[infinity].summation over (n=1)] [[absolute value of a(n)].sup.2] < [infinity]

endowed with the pointwise product. Then [l.sup.2](N) is weakly amenable by Example 4.1.42 (iii) [3], whereas in Example 2.3, we have shown that [l.sup.2](N) is not weakly character amenable.

We complete the following schematic diagram that illustrates the implications between weak character amenability and some related notions of amenability. Here "-" indicates the fact that there is a counter-example to the relevant implication.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Acknowledgements. The authors would like to sincerely thank the referee for invaluable his/her suggestions and comments on the paper. They acknowledge that the research was supported by a grant from INSF, Iran National Science Foundation, under the name Fourier Analysis and Applications.

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Received by the editors in October 2015--In revised form in June 2016.

Communicated by F. Bastin.

2010 Mathematics Subject Classification : Primary 46H05, 46H25, 47B47. Secondary 43A10.

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

emails : isfahani@cc.iut.ac.ir, shahmoradi@math.iut.ac.ir, simasoltani@cc.iut.ac.ir
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