Approximate character amenability of Banach algebras

Abstract

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.

Full text

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
ββ
−→ 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