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# Approximate character amenability of Banach algebras

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In this paper we consider some property of approximate character 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.
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
2013 M. Momeni and T. Yazdanpanah. This is an open access article
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 Xis 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·xαxα·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.IfϕΦA∪{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 Xis given by
x·a=ϕ(a)x(x∈X
,a∈A).
Deﬁnition 1. Let Abe a Banach algebra and ϕΦA∪{0}.Ais called ap-
proximately ϕ-amenable if there exists a net {mα}⊂A
∗∗ such that mα(ϕ)−→
1 and ||a·mαϕ(a)mα|| −0 for all a∈A. Banach algebra Ais called
approximately character amenable if Ais approximately ϕ-amenable for every
ϕΦA∪{0}.
2. SOME PROPERTY OF APPROXIMATELY ϕAMENABLE
BANACH ALGEBRAS
Proposition 2. Let Abe a Banach algebra and ϕΦA∪{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)αI⊆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−Xbe a continuous
derivation. Set D=D|X:X−A
and gα=(D)(mα)∈X
,for each
αI. 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 αIwe 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).
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)gαmα(ϕ)D(a).Since mα(ϕ)α
−→ 1, so we have
D(a) = lim
α(ϕ(a)gαa·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 ϕΦA, the following are equiv-
alent:
i) There exists a net {mα}⊂A
∗∗ such that mα(ϕ)−→ 1and for each a∈A,
||a·mαϕ(a)mα|| −0.
ii) There exists a net {nβ}⊂Asuch that ϕ(nβ)−→ 1and for each a∈A,
||a·nβϕ(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 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·mαϕ(a)mα|| −0(a∈A).
Clearly the dual space Ais linear, closed, and A-invariant. By this deﬁni-
tion, Banach algebra Ais called approximately ϕ-amenable if Ais approxi-
mately ϕ-left amenable.
Proposition 6. Let Abe a commutative Banach algebra and Xa linear,
closed, A-left invariant subspace of Asuch that ΦA⊆X. Suppose P:A−→
Xis a continuous linear map such that PA)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}αI⊆X
such that mα(ϕ)α
−→ 1 and ||a.mαϕ(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 αI, the functional m
α:A−→
C, deﬁned by m
α=mαoP belong to A∗∗ and m
α(ϕ)=mα()=mα(ϕ)α
−→
1. Also we have,
||a·m
αϕ(a)m
α|| ≤ ||a·mαϕ(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 ϕΦA.If
Ais approximately ϕ-amenable, Then for every f∈A
there exists a net
(λα)αICsuch that for each αI,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}αI⊂A
∗∗ such that mα(ϕ)α
−→ 1 and ||a·mαϕ(a)mα|| −0(a
A).For ﬁxed α0I, 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, mα0C. 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
ϕΦT(A), the Banach algebra T(A)is approximately ϕ-amenable.
Proof. For each ϕΦT(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(aϕ(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 ˜ϕΦA.
According to proposition 3, by approximate ˜ϕ-amenability of A, there exists
a net (nα)αIin Asuch that ˜ϕ(nα)−→ 1 and ||anα˜ϕ(a)nα|| −0 for each
a∈A.
Now for each αI, set mα:= T(x)nα. So we have
ϕ(mα)=ϕ(T(x)nα)= ˜ϕ(nα)−→ 1.
Also, for each a∈A,
T(a)mαϕ(T(a))mα=T(a)T(x)nαϕ(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 ϕΦT(A)∪{0}the Banach algebra T(A)is approximately ϕ-
amenable.
(ii)For each ϕΦT(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 ϕΦT(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 ϕΦT(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 ϕΦT(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 ϕΦT(A),T(A)is
approximately ϕ- amenable if and only if Iϕhas a right approximate identity.
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