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Int. Journal of Math. Analysis, Vol. 7, 2013, no. 19, 899 - 906

HIKARI Ltd, www.m-hikari.com

Approximate Character Amenability

of Banach Algebras

M. Momeni

Department of Mathematics, Science and Research Branch

Islamic Azad University (IAU), Tehran Iran

srb.maryam@gmail.com

T. Yazdanpanah

Department of Mathematics, Persian Gulf University

Boushehr, 75168, Iran

yazdanpanah@pgu.ac.ir

Copyright c

2013 M. Momeni and T. Yazdanpanah. This is an open access article

distributed under the Creative Commons Attribution License, which permits unrestricted

use, distribution, and reproduction in any medium, provided the original work is properly

cited.

Abstract. In this paper we consider some property of approximate charac-

ter amenable Banach algebras. Also we express the concept of approximately

ϕ-left amenability of subspaces in A∗. Then we investigate the multipliers on

such algebras.

Mathematics Subject Classiﬁcation: Primary: 43A20; Secondary 46H20,

46H25

Keywords: Approximate Character Amenability, Character, Derivation,

Multiplier

900 M. Momeni and T. Yazdanpanah

1. Introduction

Suppose that Ais a Banach algebra and Xis a Banach A-bimodule. then

X∗, the dual of X, has a natural Banach A-bimodule structure deﬁned by

<x,a·x∗>=<x·a, x∗>, <x,x

∗·a>=<a·x, x∗>(a∈A,x∈X ,x

∗∈X

∗).

Such a Banach A- bimodule X∗is called a dual A-bimodule. Also For x∗∈X

∗

and x∈X deﬁne x∗·x∈A

∗by

<a,x

∗·x>=<x·a, x∗>(a∈A).

Let Abe a Banach algebra and Xbe a Banach A-bimodule. A derivation

D:A−→X is a linear map, always taken to be continuous, satisfying

D(ab)=D(a)·b+a·D(b)(a, b ∈A).

Given x∈X,the map δx(a)=ax −xa is a derivation on Awhich is called an

inner derivation.Aderivation D:A−→X is called approximately inner, if

there exists a net {xα}⊂X such that

D(a) = lim

α(a·xα−xα·a)(a∈A).

The limit being in norm. Note {xα}in the above is not necessarily bounded.

A Banach algebra Ais approximately amenable if for any A-bimodule X,

any derivation D:A−→X

∗is approximately inner. A Banach algebra A

is approximately contractible if every derivation from Ainto every Banach

A-bimodule Xis approximately inner. [4].

Let Abe a commutative Banach algebra. A map T:A−→Ais said to be

a multiplier if it satisﬁes

aT (b)=T(a)b(a, b ∈A).

We denote the set of all multiplier on Aby M(A). If Ahas a bounded

approximate identity, then M(A) is a closed subalgebra of B(A). (The Banach

algebra of all bounded operators on A), and in this case for T∈M(A)we

have

T(ab)=aT (b)=T(a)b(a, b ∈A).

Now Given a commutative complex Banach algebra Awith or without iden-

tity. Let ΦAstand for the spectrum of A, i.e., the set of all nontrivial multi-

plicative linear functional on A.Ifϕ∈ΦA∪{0}, then a Banach A-module

X, can be considered as a Banach left and right A-module, which denoted

Approximate character amenability of Banach algebras 901

by (ϕ, A)- bimodule and (ϕ, A)- bimodule Respectively, with the following

module actions

a·x=ϕ(a)X,x·a=ϕ(a)X(a∈A,x∈X).

Note that for left module action in the above it is easily veriﬁed that the right

action of Aon the dual A-module X∗is given by

x∗·a=ϕ(a)x∗(x∗∈X

∗,a∈A).

Deﬁnition 1. Let Abe a Banach algebra and ϕ∈ΦA∪{0}.Ais called ap-

proximately ϕ-amenable if there exists a net {mα}⊂A

∗∗ such that mα(ϕ)−→

1 and ||a·mα−ϕ(a)mα|| −→ 0 for all a∈A. Banach algebra Ais called

approximately character amenable if Ais approximately ϕ-amenable for every

ϕ∈ΦA∪{0}.

2. SOME PROPERTY OF APPROXIMATELY ϕ−AMENABLE

BANACH ALGEBRAS

Proposition 2. Let Abe a Banach algebra and ϕ∈ΦA∪{0}. Then the

following conditions are equivalent.

i) Ais approximately ϕ-amenable.

ii)For each (ϕ, A)-bimodule X, every continuous derivation D:A−→X

∗

is approximately inner.

Proof. (i) ⇒(ii). Suppose that Ais approximately ϕ-amenable, so there exists

a net (mα)α∈I⊆A

∗∗ such that mα(ϕ)−→ 1 and for each a∈A,||a·mα−

ϕ(a)mα|| −→ 0.

Let Xbe a Banach (ϕ, A)-bimodule and D:A−→X∗be a continuous

derivation. Set D=D∗|X:X−→A

∗and gα=(D)∗(mα)∈X

∗,for each

α∈I. Then for all a, b ∈A, and x∈X, we have;

b, D(a·x)=a·x, D(b)

=ϕ(a)x, D(b)

=ϕ(a)b, D(x).

Hence D(a·x)=ϕ(a)D(x). This implies that

x, gα·a=a·x, gα=a·x, (D)∗(mα)

=D(a·x),m

α=ϕ(a)D(x),m

α

=ϕ(a)x, (D)∗(mα)=ϕ(a)x, gα.

902 M. Momeni and T. Yazdanpanah

Therefore for all a∈Aand α∈Iwe have gα·a=ϕ(a)gα. Since Dis a

derivation we have,

b, D(x·a)=x·a, D(b)

=x, a ·D(b)=x, D(ab)−D(a)·b

=x, D(ab)−x, D(a)·b

=ab, D(x)−ϕ(b)x, D(a)

=b, D(x)·a−x, D(a)ϕ(b).

Thus for each a∈Aand x∈X we have D(x·a)=D(x)·a−<x,D(a)>ϕ.

It follows that

x, a ·gα=x·a, gα=x·a, (D)∗(mα)

=D(x.a),m

α

=D(x).a, mα−x, D(a)ϕ, mα

=ϕ(a)D(x),m

α−x, D(a)ϕ, mα

=ϕ(a)x, gα−x, D(a)ϕ, mα

Therefore a·gα=ϕ(a)gα−mα(ϕ)D(a).Since mα(ϕ)α

−→ 1, so we have

D(a) = lim

α(ϕ(a)gα−a·gα).

Combining this with the equation gα·a=ϕ(a)gα, we obtain,

D(a) = lim

αa·(−gα)−(−gα)·a

= lim

αδ(−gα)(a).

So Dis approximately inner. For (ii) ⇒(i) see [10,11].

One can see the proof of following proposition in [11], which we apply it in

the Next section.

Proposition 3. For a Banach algebra Aand ϕ∈ΦA, the following are equiv-

alent:

i) There exists a net {mα}⊂A

∗∗ such that mα(ϕ)−→ 1and for each a∈A,

||a·mα−ϕ(a)mα|| −→ 0.

ii) There exists a net {nβ}⊂Asuch that ϕ(nβ)−→ 1and for each a∈A,

||a·nβ−ϕ(a)nβ|| −→ 0.

iii) Ais approximately ϕ-amenable.

Deﬁnition 4. Let Abe a Banach algebra and Xbe a subspace of A∗.Xis

called A-left (resp. right) invariant if X·A⊆X (resp. A·X ⊆ X). It is

called A-invariant if it is A-left and A-right invariant.

Approximate character amenability of Banach algebras 903

Now we deﬁne approximately ϕ-left amenability of subspaces of A∗.

Deﬁnition 5. Let Abe a commutative Banach algebra and Xa linear, closed,

A-left invariant subspace of A∗, such that ΦA⊆X.Xis called approximately

ϕ-left amenable if there exists a net {mα}⊆X

∗with the following properties:

i) mα(ϕ)α

−→ 1

ii) ||a·mα−ϕ(a)mα|| −→ 0(a∈A).

Clearly the dual space A∗is linear, closed, and A-invariant. By this deﬁni-

tion, Banach algebra Ais called approximately ϕ-amenable if A∗is approxi-

mately ϕ-left amenable.

Proposition 6. Let Abe a commutative Banach algebra and Xa linear,

closed, A-left invariant subspace of A∗such that ΦA⊆X. Suppose P:A∗−→

Xis a continuous linear map such that P(ΦA)⊆ΦAand P(a∗·a)=P(a∗)·a,

for each a∗∈A

∗and a∈A. Then Xis approximately ϕ-left amenable if and

only if Ais approximately ϕ-amenable.

Proof. Let X⊆A

∗be approximately ϕ-left amenable. Then there exists a net

{mα}α∈I⊆X

∗such that mα(ϕ)α

−→ 1 and ||a.mα−ϕ(a)mα|| −→ 0(a∈A).

It is easy to see that ϕ·a=a·ϕ=ϕ(a)ϕand P(ϕ)·a=ϕ(a)P(ϕ),(a∈A).

So P(ϕ)=ϕ.

The continuity of Pimplies that for each α∈I, the functional m

α:A∗−→

C, deﬁned by m

α=mαoP belong to A∗∗ and m

α(ϕ)=mα(Pϕ)=mα(ϕ)α

−→

1. Also we have,

||a·m

α−ϕ(a)m

α|| ≤ ||a·mα−ϕ(a)mα|| ||P|| −→ 0

Hence Ais approximately ϕ-amenable. The converse of the proposition is

trivial by deﬁnition.

example 1. i) Let Abe a Banach algebra with a bounded approximate iden-

tity and X=AA∗.SoXis a (proper) closed submodule of A∗[12], and

Hom(A,C)⊂AA

∗. For a ﬁxed b∈A, the map Pb:A∗−→ X deﬁned by

Pb(f)=b·fis a continuous linear map and Pb(f·a)=Pb(f)·a, for every

a∈A. Then by the previous theorem Xis left approximate ϕ-amenable if and

only if Ais approximately ϕ-amenable.

Note by the same argument we can prove the previous proposition in case

that Ais not commutative.

Proposition 7. Let Abe a commutative Banach algebra and ϕ∈ΦA.If

Ais approximately ϕ-amenable, Then for every f∈A

∗there exists a net

(λα)α∈I⊂Csuch that for each α∈I,λαϕ∈{f·a;a∈A}

−w∗.

904 M. Momeni and T. Yazdanpanah

Proof. Let Abe an approximately ϕ-amenable Banach algebra. So there exists

a net {mα}α∈I⊂A

∗∗ such that mα(ϕ)α

−→ 1 and ||a·mα−ϕ(a)mα|| −→ 0(a∈

A).For ﬁxed α0∈I, by Goldstein’s theorem, there exists a net {aα0

β}β⊆A

such that

aα0

β

β

−→ mα0.Hence a·

aα0

β

β

−→ a·mα0,(a∈A).Thus

f, a ·

aα0β=f·a,

aα0

ββ

−→ f·a, mα0=f, a ·mα0.

Let λα0=f, mα0∈C. We ﬁnd λα0ϕ∈{f·a;a∈A}

−w∗.By the same

argumen for each α>α

0we ﬁnd net (λα) such that λαϕ∈{f·a, a ∈A}

−w∗.

The converse is clear by [13, theorem 3.3].

3. MULTIPLIERS WITH CLOSED RANGE

In this section we consider multiplier Ton a commutative approximately

ϕ-amenable Banach algebras and express some interesting results that show

the relation between approximate character amenability of Aand existence of

approximate identity in Iϕ=kerϕ.We start this section with the important

following lemma.

Lemma 8. If Ais an approximately character amenable commutative Banach

algebra and T:A−→Abe a multiplier with closed range, then for each

ϕ∈ΦT(A), the Banach algebra T(A)is approximately ϕ-amenable.

Proof. For each ϕ∈ΦT(A), we can choose x∈Afor which ϕ(T(x)) = 1. If

now deﬁne the linear functional ˜ϕon Aby ˜ϕ(a):=ϕ(T(xa)) for a∈A, then

˜ϕis non-zero and multiplicative because,

˜ϕ(a)˜ϕ(b)=ϕ(T(xa))ϕ(T(xb)) = ϕ(T(x)T(x)ab)

=ϕ(T(x))ϕ(T(xab)) = ˜ϕ(ab)

Also the deﬁnition of ˜ϕis independent of the choice of x. Therefore ˜ϕ∈ΦA.

According to proposition 3, by approximate ˜ϕ-amenability of A, there exists

a net (nα)α∈Iin Asuch that ˜ϕ(nα)−→ 1 and ||anα−˜ϕ(a)nα|| −→ 0 for each

a∈A.

Now for each α∈I, set mα:= T(x)nα. So we have

ϕ(mα)=ϕ(T(x)nα)= ˜ϕ(nα)−→ 1.

Also, for each a∈A,

T(a)mα−ϕ(T(a))mα=T(a)T(x)nα−ϕ(T(a))T(x)nα

≤T(x)·T(a)nα−˜ϕ(T(a))nα−→0.

Note that ˜ϕ(T(a)) = ϕ(T(x)T(a)) = ϕ(T(x))ϕ(T(a)) = ϕ(T(a)).Thus T(A)

is approximately ϕ- amenable.

Approximate character amenability of Banach algebras 905

Corollary 9. Let Abe a commutative Banach algebra and T:A−→Abe a

multiplier with closed range. Then the following assertion are equivalent.

(i)For each ϕ∈ΦT(A)∪{0}the Banach algebra T(A)is approximately ϕ-

amenable.

(ii)For each ϕ∈ΦT(A)∪{0}, every continuous derivation D:T(A)−→ X ∗

is approximately inner for each Banach T(A)-bimodule X, such that T(a)·x=

ϕ(T(a)) ·x.

The following theorem is a direct result of [11,proposition 2.7].

Theorem 10. Let Abe a commutative Banach algebra and T:A−→Abe a

multiplier with closed range. Then for each ϕ∈ΦT(A)∪{0}the Banach algebra

T(A)is approximately ϕ-amenable if and only if T(A)has a right approximate

identity.

By combination of lemma 8 and theorem 10 we have the following important

result.

Proposition 11. Suppose that Ais a commutative Banach algebra. If Ais an

approximately character amenable and T:A−→Ais a multiplier with closed

rang, then the Banach algebra T(A)has a bounded approximate identity.

Proposition 12. Let Abe a commutative Banach algebra and T:A−→A

a multiplier with closed rang. If for ϕ∈ΦT(A)the ideal Iϕ=kerϕhas a right

approximate identity, Then T(A)is approximately ϕ-amenable.

Proposition 13. Suppose that Ais a commutative approximately character

amenable Banach algebra and T:A−→A is a multiplier with closed rang.

Then for each ϕ∈ΦT(A),Iϕ=kerϕhas a right approximate identity.

Proof. By proposition 11, since Ais approximately character amenable, So

T(A) has a bounded approximate identity. Now by lemma 13 the proof is

completed.

The following result follows immediately from two above propositions.

Proposition 14. Suppose that Ais a commutative Banach algebra and T:

A−→Aa multiplier with closed rang. Then for each ϕ∈ΦT(A),T(A)is

approximately ϕ- amenable if and only if Iϕhas a right approximate identity.

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Received: December, 2012