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Abstract and Applied Analysis
Volume 2012, Article ID 653140, 20 pages
doi:10.1155/2012/653140
Research Article
σ-Approximately Contractible Banach Algebras
M. Momeni,1T. Yazdanpanah,2and M. R. Mardanbeigi1
1Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU),
Tehran 1477893855, Iran
2Department of Mathematics, Persian Gulf University, Boushehr 75169, Iran
Correspondence should be addressed to M. Momeni, srb.maryam@gmail.com
Received 9 March 2012; Accepted 25 May 2012
Academic Editor: Qiji J. Zhu
Copyright q2012 M. Momeni et al. 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.
We investigate σ-approximate contractibility and σ-approximate amenability of Banach algebras,
which are extensions of usual notions of contractibility and amenability, respectively, where σis a
dense range or an idempotent bounded endomorphism of the corresponding Banach algebra.
1. Introduction
For a Banach algebra A,anA-bimodule will always refer to a Banach A-bimodule X,thatis,
a Banach space which is algebraically an A-bimodule, and for which there is a constant c≥0
such that for a∈A,x ∈X, we have
a·x≤cax,x·a≤cax.1.1
A derivation D:A→Xis a linear map, always taken to be continuous, satisfying
DabDa·ba·Db
a, b ∈A
.1.2
A Banach algebra Ais amenable if for any A-bimodule X, any derivation D:A→X
∗is
inner, that is, there exists x∗∈X
∗,with
Daa·x∗−x∗·aδx∗a
a∈A
.1.3
2 Abstract and Applied Analysis
Let Abe a Banach algebra and σa bounded endomorphism of A, that is, a bounded Banach
algebra homomorphism from Ainto A.Aσ-derivation from Ainto a Banach A-bimodule X
is a bounded linear map D:A→Xsatisfying
Dabσa·DbDa·σb
a, b ∈A
.1.4
For each x∈X, the mapping
δσ
x:A−→X 1.5
defined by δσ
xaσa·x−x·σa, for all a∈A,isaσ-derivation called an inner σ-
derivation.
Remark 1.1. Throughout this paper, we will assume that Ais a Banach algebra, and σis
a bounded endomorphism of Aunless otherwise specified. Also, we write σ-a.ifor σ-
approximately inner, σ-a.afor σ-approximately amenable, and σ-a.cfor σ-approximately
contractible.
The basic definition for the present paper is as follows.
Definition 1.2. Aσ-derivation D:A→Xis σ-a.i, if there exists a net xα⊆Xsuch that for
every a∈A,Dalimασa·xα−xα·σa, the limit being in norm and we write Dlim δσ
xα.
Note that we do not suppose xαto be bounded.
Definition 1.3. A Banach algebra Ais called σ-a.c if for any A-bimodule X, every σ-derivation
D:A→Xis σ-a.i.
Definition 1.4. A Banach algebra Ais called σ-a.a if for any A-bimodule X, every σ-derivation
D:A→X
∗is σ-a.i.
Definition 1.5. Let Abe a Banach algebra, and let Xand Ybe Banach A-bimodules. The linear
map T:X→Yis called a σ-A-bimodule homomorphism if
Ta·xσa·Tx,T
x·aTx·σa
a∈A,x ∈X
.1.6
2. Basic Properties
Proposition 2.1. Let Abe a σ-a.c Banach algebra. Then σAhas a left and right approximate
identity.
Proof. Consider XAas a Banach A-bimodule with the trivial right action, that is,
a·xax, x ·a0a∈A,x ∈X
.2.1
Abstract and Applied Analysis 3
Then D:A→Xdefined by Daσais a σ-derivation, and so there is a net {uα}⊆X
Asuch that Dlimαδσ
uα. Hence for each a∈A,
σaDalim
αδσ
uαalim
ασa·uα−uα·σalim
ασauα,2.2
which shows that {uα}is a right approximate identity for σA. Similarly, one can find a left
approximate identity for σA.
Corollary 2.2. Let Abe a σ-a.c Banach algebra and σa continuous epimorphism of A.ThenAhas
a left and right approximate identity.
Proposition 2.3. Let ϕbe a bounded endomorphism of Banach algebra A.IfAis σ-a.c, then Ais
ϕoσ-a.c.
Proof. Let Xbe a Banach A-bimodule and let D:A→Xbe a ϕoσ-derivation. Then X,∗
is an A-bimodule with the following module actions:
a∗xϕa·x, x ∗ax·ϕa
a∈A,x ∈X
.2.3
For each a, b ∈A, we have
Dabϕoσa·DbDa·ϕoσbσa∗DbDa∗σb.2.4
Thus D:A→X,∗is a continuous σ-derivation. Since Ais σ-a.c, there exists a net {xα}⊆X
such that Dlim δσ
xα. In fact,
Dalim
ασa∗xα−xα∗σa
lim
αϕoσa·xα−xα·ϕoσ a
lim
αδϕoσ
xαa
a∈A
.
2.5
Therefore, Dis a ϕoσ-a.i and so Ais ϕoσ-a.c.
Corollary 2.4. Let Abe an a.c Banach algebra. Then Ais σ-a.c for each bounded endomorphism σof
A.
Proposition 2.5. Let Abe a σ-a.c Banach algebra, where σis a bounded epimorphism of A.ThenA
is a.c.
Proof. Let Xbe a Banach A-bimodule and let d:A→Xbe a continuous derivation. It is
easy to see that doσ is a σ-derivation. Since Ais σ-a.c, there exists a net {xα}⊆Xsuch that
4 Abstract and Applied Analysis
doσalimασaxα−xασa.Nowforb∈Athere exists a∈Asuch that bσa,and,
therefore,
dbdσa lim
αxασa−σaxα
lim
αxαb−bxα,
2.6
which shows that dis approximately inner and so Ais a.c.
Corollary 2.6. Let ϕbe a bounded endomorphism of Banach algebra A.IfAis σ-a.a then it is ϕoσ-
a.a too.
Corollary 2.7. Let Abe an a.a Banach algebra. For each bounded endomorphism σ,Ais σ-a.a.
Corollary 2.8. Let Abe a σ-a.a Banach algebra, where σis a bounded epimorphism of A.ThenAis
a.a.
Proposition 2.9. Suppose that Bis a Banach algebra and ϕ:A→Bis a continuous epimorphism.
If Ais a.c, then Bis σ-a.c for each bounded endomorphism σof B.
Proof. Let σ:B→Bbe a bounded endomorphism of Band Xa Banach B-bimodule, then
X,∗is an A-bimodule with the following module actions:
a∗xσϕa·x, x ∗ax·σϕaa∈A,x ∈X
.2.7
Now let D:B→Xbe a continuous σ-derivation. It is easy to check that Doϕ :A→X,∗
is a derivation. Since Ais approximately contractible, there exists a net {xα}⊆Xsuch that
Doϕalimαδxαa. We have
DϕaDoϕalim
αδxαalim
αa∗xα−xα∗a
lim
ασϕaxα−xασϕaa∈A
.
2.8
Since ϕis an epimorphism, so for each b∈Bthere exists a∈Asuch that bϕa, and we
have
Dblim
ασbxα−xασb,2.9
which shows that Dis σ-a.i and so Bis σ-a.c.
Proposition 2.10. Suppose that Aand Bare Banach algebras, and let σand τbe bounded
endomorphism of Aand B, respectively. Let ϕ:A→Bbe a bounded epimorphism such that
ϕoσ τoϕ.IfAis σ-a.c, then Bis τ-a.c.
Abstract and Applied Analysis 5
Proof. Let Xbe a Banach B-bimodule and D:B→Xa continuous τ-derivation. Then X,∗
is an A-bimodule with the following actions:
a∗xϕa·x, x ∗ax·ϕa
a∈A,x ∈X
.2.10
It is easy to check that Doϕ :A→X,∗is a σ-derivation. Since Ais σ-a.c, there exists a net
{xα}⊆Xsuch that Doϕalimαδσ
xαa, so we have
Dϕalim
ασa∗xα−xα∗σa
lim
αϕσa ·xα−xα·ϕσa
lim
ατϕa·xα−xα·τϕaa∈A
.
2.11
Since ϕis epimorphism, so Dblimατbxα−xατbfor all b∈B, and hence Bis τ-a.c.
3. σ-Approximate Contractibility for Unital Banach Algebras
In this section we state some properties of σ-approximate contractibility when Ahas an
identity. First we express the following proposition that one can see its proof in 1,Proposition
3.3, and bring some corollaries when σAis dense in A.
Proposition 3.1. Let Abe a unital Banach algebra with unit e, σAdense in A,Xa Banach A-
bimodule, and D:A→Xaσ-derivation. Then, there is a σ-derivation D1:A→e·X·eand
η∈X, such that DD1δη.
The following definition extends the definition of the unital Banach A-module in the
classical sense.
Definition 3.2. Let Abe a unital Banach algebra with identity e. Banach A-bimodule Xis
called σ-unital if Xσe·X·σe.
Corollary 3.3. Let Abe a unital Banach algebra and σAdense in A. Then, Ais σ-a.c (resp.,σ-a.a)
if and only if for all σ-unital Banach A-bimodule X, every σ-derivation D:A→Xresp., D :
A→X
∗is σ-a.i.
Proof. Since σeis a unit for σA,andσAis dense in A,weseethatσee,sothat
e·X·eis a σ-unital Banach A-bimodule. Now by Proposition 3.1, the proof is complete.
Corollary 3.4. Suppose that Ais a unital Banach algebra and σAis dense in A.LetXbe a Banach
A-bimodule and D:A→X
∗aσ-derivation. If Ais σ-a.a, then there exists a net ηα⊆e·X
∗·e,
and η∈X
∗, such that Dlimαδσ
ηαδη.
Proof. By Proposition 3.1,DD1δηsuch that η∈X
∗and D1:A→e·X
∗·eis a σ-
derivation. Since e·X
∗·e∼
e·X·e∗and Ais σ-a.a, D1:A→e·X·e∗is σ-a.i. Hence
D1limαδσ
ηαfor some net ηα⊆e·X
∗·e.
6 Abstract and Applied Analysis
In the following proposition we consider σ-approximate contractibility when σis an
idempotent endomorphism of A. We can see the proof of the following proposition in 1,
Proposition 4.1.
Proposition 3.5. Assume that Ahas an element ewhich is a unit for σAand Xis a Banach A-
bimodule. If σis a bounded idempotent endomorphism of A, then for each σ-derivation D:A→X
there exists a σ-derivation D1:A→σe·X·σeand η∈X, such that DD1δη.
Corollary 3.6. Assume that Ahas an element ewhich is a unit for σAand σis a bounded
idempotent endomorphism of A,thenAis σ-a.c (resp., σ-a.a) if and only if for all σ-unital Banach
A-bimodule, X, every σ-derivation D:A→X(resp., D:A→X
∗is σ-a.i.
Lemma 3.7. Assume that Ais a unital Banach algebra with the identity e, and X,∗is a σ-unital
Banach A-bimodule with the following module actions:
a∗xσax, x ∗axσa
a∈A,x ∈X
.3.1
If D:A→X
∗is a σ-derivation, then De0.
Proof. We have DeDeeσeDeDeσeand
e∗x, Deσex, Deσe∗ex, Deσeσe
x, Deσee∗x, Dex∈X
.
3.2
Hence DeσeDeand so σeDe0. Hence De0.
Proposition 3.8. Let σbe a bounded idempotent endomorphism of Banach algebra A.IfAis σ-a.a,
then A#is σ-a.a, where σis the endomorphism of A#induced by σ, that is, σaασaα.
Proof. Let Xbe a Banach A#-bimodule and D:A#→X
∗a continuous σ-derivation. By
Proposition 3.5, there exits η∈X
∗and D1:A#→σe·X
∗·σesuch that DD1δη.Set
d:D1|A:A→σe·X∗·σe. It is easy to check that dis a σ-derivation. Since Ais σ-a.a, there
exists a net x∗
γ⊆X
∗such that dlimγδσ
x∗
γ. Hence D1alimγσax∗
γ−x∗
γσa,a∈A.
Since σe·X
∗·σeis σ-unital, by Lemma 3.7,D1e0 and for each aα∈A
#we have
D1aαD1aαD1eD1alim
γσax∗
γ−x∗
γσa
lim
γσaα−αx∗
γ−x∗
γσaα−α
lim
γϕaαx∗
γ−x∗
γϕaα.
3.3
This shows that D1is σ-a.i, and so A#is σ-a.a.
Proposition 3.9. Let σbe a bounded endomorphism of Banach algebra A.IfA#is σ-a.a, then Ais
σ-a.a.
Abstract and Applied Analysis 7
Proof. Let Xbe a Banach A-bimodule and D:A→X
∗a continuous σ-derivation. Xis a
Banach A#-bimodule with the following module actions:
aα·xa·xαx, x ·aαx·aαx, 3.4
for all a∈A,x ∈X,α ∈C. Define D#:A#→X
∗with D#aαDa. Clearly, D#is a
continuous σ-derivation. Hence, there is a net x∗
γ⊆X
∗such that D#limσ
γδx∗
γ. Hence, for
each a∈Awe have
DaD#aαlim
γσaαx∗
γ−x∗
γσaαlim
γσax∗
γ−x∗
γσa3.5
which shows that Dis σ-a.i and so Ais σ-a.a.
4. σ-Approximate Amenability When AHas
a Bounded Approximate Identity
Lemma 4.1. Let Abe a Banach algebra with bounded approximate identity and Xa Banach A-
bimodule with trivial left or right action, then every σ-derivation D:A→X
∗is σ-inner.
Proof. Let Xbe a Banach A-bimodule with trivial left action. Hence, X∗is a Banach A-
bimodule with trivial right action, that is,
x∗·a0,a·x∗ax∗x∗∈X
∗,a∈A
.4.1
Let D:A→X
∗be a continuous σ-derivation and eαa bounded approximate identity of A.
By Banach Alaoglu’s Theorem, Deα has a subnet Deβ such that Deβw∗
→x∗
0, for some
x∗
0∈X
∗. Since a·eβ
·
→aand Dis continuous, Da·eβ·
→Da. Hence, Da·eβw∗
→Da.
On the other hand, Da·eβσaDeβw∗
→σax∗
0and so Daσax∗
0. Hence,
Daσax∗
0−x∗
0σaand Dis σ-inner.
The following definitions extends the definition of the neo-unital and essential Banach
A-bimodule in the classical sense.
Definition 4.2. Let Xbe a Banach A-bimodule. Then Xis called σ-neo-unital σ-pseudo-
unital,ifXσA·X·σA. Similarly, one defines σ-neo-unital left and right Banach
modules.
Definition 4.3. Let Xbe a Banach A-bimodule. Then Xis called σ-essential if X
σAXσAspan σA·X·σA. Similarly, one defines σ-essential left and right Banach
modules.
We recall that a bounded approximate identity in Banach algebra Afor Banach
A-bimodule Xis a bounded net eαin Asuch that for each x∈X,eαx→xand xeα→x.
Proposition 4.4. Assume that Ahas a left bounded approximate identity, σis a bounded idempotent
endomorphism of A, and Xis a left Banach A-module. Then Xis σ-neo-unital if and only if Xis
σ-essential.
8 Abstract and Applied Analysis
Proof. Let Xbe a σ-essential Banach A-bimodule. Since σis idempotent, σAis Banach
subalgebra of A.Leteα⊆Abe left approximate identity with bound m. First suppose that
z∈span σA·X, so there exist a1,...,a
n∈A,x1,...,x
n∈Xsuch that zn
i1σaixi. For
1≤i≤n,eαai→aiand, therefore, σeαz→z.
Now suppose that z∈σAX. There exists {zn}⊆span σA·Xsuch that zn→z.
Thus,
∃n0∈Ns.t.∀nn≥n0;zn−z<ε
2σm14.2
On the other hand, for each n∈Nwe have σeαzn
α
→znand so σeαzn0
α
→zn0.
Therefore,
∃α0;∀αα≥α0;σeαzn0−zn0<ε
2.4.3
Now we have
σeαz−z≤σeαz−σeαzn0σeαzn0−zn0zn0−z
≤σeαz−zn0σeαzn0−zn0zn0−z
<σeα1z−zn0ε
2
<σm1ε
σm12ε
2ε,
4.4
which shows that σeα ⊆σAis a left bounded approximate identity for X.Nowby
Cohen factorization Theorem, XσA·X.SoXis σ-neo-unital. The other side is trivial.
Corollary 4.5. Every σ-neo-unital left Banach A-module is essential.
Proof. Let Xbe a σ-neo-unital left Banach A-module. We have XσA·X ⊆ A·X ⊆ AX ⊆ X
so XAX.
Proposition 4.6. Let Abe a Banach algebra with a left bounded approximate identity, σbe a bounded
idempotent endomorphism of A, and Xa left Banach A-module. Then σA·Xis closed weakly
complemented submodule of X.
Proof. Set YσAX,sinceAhas a left bounded approximate identity, by Cohen
factorization Theorem A2A, and we have σAYσAσAXσA2XσAXY,
which shows that Yis σ-essential by Proposition 4.4,Yis σ-neo unital that is, YσA·Y.
Hence, σAXYσA·Y⊆σA·Xand so σAXσA·X.ThusσA·Xis closed
submodule of X.
Now we prove that σA·Xis weakly complemented in X.Leteαbe a left
approximate identity in Awith bound m, and define a net Tαin BX∗by setting Tαx∗
x∗·σeαx∗∈X
∗. We have Tα≤σm.ThusTαis a bounded net in BX∗since
BX∗X∗⊗X∗and ball BX∗is w∗-compact, so there exists T∈BX∗such that we
Abstract and Applied Analysis 9
may suppose that w∗−limαTαTand T≤σm. For each a∈A,x∈X,andx∗∈X
∗,we
have
σa·x, T x∗lim
ασa·x, x∗·σeα
lim
ασeασa·x, x∗
σa·x, x∗,
4.5
and so x∗−Tx∗∈σA·X⊥. On other hand, for each x∗∈X
∗,
T2x∗TTx∗lim
αTx∗σeαlim
αx∗σeαTx∗.4.6
Thus Tis projection, and IX∗−T:X∗→σA·X⊥is projection. So σA·Xis weakly
complemented in Xand, we have X∗σA·X⊥⊕σA·X∗.
Corollary 4.7. Let Ahave a bounded approximate identity, and let Xbe a Banach A-bimodule and
σa bounded idempotent endomorphism of A.Then
iσA·X·σAis a closed weakly complemented submodule of X,
iiAis σ-a.a if and only if for every σ-neo-unital Banach A-bimodule X, every σ-derivation
D:A→X
∗is σ-approximately inner.
Proof. Set YσA·X.ByProposition 4.6,Yis a closed and weakly complemented
submodule of X,andT:X∗→Y
∗and I−T:X∗→Y
⊥are projection maps. Let D:A→X
∗
be a σ-derivation, so ToD and I−ToD are σ-derivations and DToDI−ToD.
Since A·X/Y{0}by Lemma 4.1,I−ToD is σ-inner. So there exists J0∈Y
⊥such that
I−ToD δσ
J0.ThusDToD δσ
J0and so Dis σ-a.i if and only if ToD :A→Y
∗is σ-a.i.
Now let ZY·σA.ByProposition 4.6,Zis a closed weakly complemented in Y,
and T:Y∗→Z
∗and I−T:Y∗→Z
⊥are projection maps. Assume that D1:A→Y
∗is a σ-
derivation, thus ToD and I−ToD are σ-derivations, and we have D1ToD1I−T·D1.
Since Y/Z·A {0},byLemma 4.1,I−T·D1is σ-inner and so there exists z0∈Z⊥
such that I−ToD1δσ
Z0. Therefore, D1ToD1δσ
Z0.Thus,D1is σ.a.i if and only if
ToD1is σ.a.i. Set DoT D1.Thus,DToD1δσ
Z0δσ
J0. Therefore, Dis σ-a.i, if and only if
ToD1:A→Z
∗σA·X·σA∗is σ.a.i. Recall that Zis σ-neo-unital. Thus, Ais σ-a.a if
and only if for every σ-neo-unital Banach A-bimodul, X, every σ-derivation D:A→X∗is
σ-a.i.
Corollary 4.8. Let Ahave a bounded approximate identity, and let Xbe a Banach A-bimodule and σ
a bounded idempotent endomorphism of A.ThenAis σ-a.a if and only if for every σ-essential Banach
A-bimodule X, every σ-derivation D:A→X
∗is σ-approximately inner.
Proposition 4.9. Suppose that σis a bounded idempotent endomorphism of Aand define σ:A#→
A#with σaασaα. The following statements are equivalent.
1Ais σ-a.a.
2Thereisanetμα⊆A#
⊗A#∗∗ such that for each a∈A
#,σa·μα−μα·σa→0and
π∗∗μα→e.
10 Abstract and Applied Analysis
3Thereisanetμ
α⊆A#
⊗A#∗∗ such that for each a∈A
#,σa·μα−μα·σa→0and
for every α,π∗∗μ
αe.
Proof. 1⇒3Suppose that Ais σ-a.a, by Proposition 3.8,A#is σ-a.a. Let ue⊗e∈A
#
⊗A#.
A#
⊗A#is a Banach A#-bimodule with the following module actions:
a·b⊗cσab⊗c,b⊗c·ab⊗cσaa, b, c ∈A
#.4.7
Set δu:A#→ker π∗∗ with definition δuaσa·u−u·σaa∈A
#.δuis σ-derivation.
Recall that ker π∗∗ ker π∗∗. Since A#is σ-a.a, thus there exists eα⊆ker π∗∗ such that
δualim
ασaeα−eασaa∈A
#.4.8
Set μ
αu−eα∈A#
⊗A#∗∗. We have
σaμ
α−μ
ασaσau−uσa−σaeα−eασa −→ 0,4.9
and for each α,
π∗∗μ
απ∗∗u−eαπ∗∗ u−π∗∗ eαπue. 4.10
3⇒2is clear.
2⇒1By Proposition 3.9,itissufficient to show that A#is σ-a.a.
Let D:A#→X
∗be a derivation. By Corollary 4.7, we may take Xto be σ-neo-unital.
We run the standard argument, so for each α∈I,setfαxμαψx, where for a, b ∈A
#,
x∈X, we have ψxa⊗bx, σaDb. Then, mγ
α⊂A#
⊗A#converging ω∗to μαα∈I
and noting that for m∈A
#
⊗A#,a ∈A
#,x∈X, then
ψσax−xσamσaψx−ψxσam−x, σπmDa.4.11
Since Xis σ-neo-unital, so XXσA#. So for each a∈Aand x∈X, we have
σax−xσa,f
αψσax−xσa,μ
α
lim
γmγ
α,ψσax−xσa
σaψx−ψxσa,μ
α−lim
γx, σπmγ
αDa
ψx,μ
ασa−σaμα−x, π∗∗ μαDa.
4.12
Abstract and Applied Analysis 11
Thus,
x, σafα−fασa−x, Da
≤
ψx,σaμα−μασa
x
π∗∗μα−e
Da
≤D·x
σaμα−μασa
x
π∗μα−e
Da,
4.13
and, therefore, Dlimαδσ
fα. It follows that A#is σ-a.a and so Ais σ-a.a.
Proposition 4.10. Suppose that Ais σ-a.a, and let
Σ:0−→ X ∗f
−→ Y g
−→ Z −→ 0,4.14
be an admissible short exact sequence of left A-module and left σ-A-module homomorphism. Then
Σ,σ-approximately split, that is, there is a net Gα:Z→Yof right inverse maps to gsuch that
limασaGα−Gασa 0for a∈A, and a net Fα:Y→X
∗of left inverse maps to fsuch that
limασafα−fασa 0for a∈A.
Proof. Following the proof of 2, Theorem 2.3, for a right inverse Gfor g,σ-approximate
amenability gives a net ϕα⊆BZ,X∗such that
σa·G−G·σalim
ασa·fGα−fGα·σaa∈A
.4.15
Setting GαG−fϕαgives the required net. Applying the same argument as 2,
Proposition 1.1provides Fα.
We recall that if Ais a Banach algebra with a weak left rightapproximate identity,
then Ahas a left rightapproximate identity 1, Lemma 2.2.
Corollary 4.11. Suppose that Banach algebra Ais σ-a.a, then σAhas left and right approximate
identities.
Corollary 4.12. Suppose that Banach algebra Ais σ-a.a and σis a bounded epimorphism of A,then
Ahas left and right approximate identities.
Lemma 4.13. Let σbe a bounded idempotent endomorphism of Banach algebra Aand Xaσ-neo-
unital Banach A-module. If eααis a bounded approximate identity in A,thenσeααis a bounded
approximate identity for X.
Proof. For every a∈Awe have eασa→σa. Since σis idempotent, σeασa→σa.
For each x∈X, there exists a∈Aand y∈Xsuch that xσa·y. Therefore,
σeα·xσeασa·y−→ σa·yx, 4.16
which shows that σeα is a bounded approximate identity for X.
12 Abstract and Applied Analysis
It is often convenient to extend a derivation to a large algebra. If a Banach algebra Iis
contained as a closed ideal in another Banach algebra A, then the strict topology on Awith
respect to Iis defined through the family of seminorms Pii∈I, where
Pia:aiiaa∈A
.4.17
Note that the strict topology is Hausdorffonly if {a∈A:a·II·a{0}} {0}3.
Proposition 4.14. Let Abe a Banach algebra and Ia closed ideal in A.letσbe a bounded idempotent
endomorphism of Aand Ihas a bounded approximate identity. Let Xbe a σ-neo-unital Banach I-
module and D:I→X
∗aσ-derivation. Then, Xis a Banach A-bimodule in a canonical fashion, and
there is a unique σ-derivation
D:A→X
∗such that
i
D|ID,
ii
Dis continuous with respect to the strict topology on Aand the ω∗-topology on X∗.
Proof. Since Xis a σ-neo-unital Banach I-module, so for each x∈X, there exists i∈Iand
y∈Xsuch that xσi·y. Define a·xσai·ya∈A.
We claim that a·xis well defined, that is, independent of the choices of iand y.Let
i∈Iand y∈Xbe such that xσi·y,andleteααbe a bounded approximate identity
for I. For each a∈Aand x∈Xwe have
a·xσai·ylim
ασaeαi·y
lim
ασaeασi·ylim
ασaeαx
lim
ασaeασi·ylim
ασaeαi·y
σai·y.
4.18
It is obvious that this operation of Aon Xturns Xinto a left Banach A-module. Similarly,
one defines a right Banach A-module structure on X. So that, eventually, Xbecomes a Banach
A-bimodule. To extend D,let
D:A−→X
∗,a−→ ω∗−lim
αDaeα−σa·Deα.4.19
We claim that
Dis well-defined, that is, the limit in 4.19does exist. Let x∈X,andleti∈I
and y∈Xsuch that xy·σi.ByLemma 4.13,σeαis bounded approximate identity for
X, and we have
x, Daeα−σa·Deαy·σi,D
aeα−σa·Deα
y, σiDaeα−σia·Deα
Abstract and Applied Analysis 13
y, Diaeα−Diσaeα−DiaeαDiaσeα
σeα·y, Dia−σaeα·y, Di
α
−→ y, Dia−σa·y, Dia∈A
.
4.20
So the limit in 4.19exists. Furthermore, for i∈I,
Diω∗−lim
αDieα−σi·Deα
ω∗−lim
αDieα−DieαDiσeαDi,
4.21
so
Dis an extension of D.Alsofora∈Aand i∈Iwe have
Da·σiω∗−lim
αDaeα·σi−σa·Deα·σi
ω∗−lim
αDaeαi−σaeα·Di−σa·Deαiσaσeα·Di
ω∗−lim
αDaeαi−σa·Deαi Dai−σa·Di.
4.22
We claim that
Dis continuous with respect to the strict topology on Aand the ω∗-topology
an X∗.
Let an
strict
→ain A.
∀i∈I, aniian−→ aiia.4.23
For each x∈X,
x,
Dan−x,
Da
lim
α|x, Daneα−σan·Deα−x, Daeα−σa·Deα|
lim
α|x, Daneα−Daeα−σanDeασaDeα|
≤lim
αxDaneα−Daeα −σa0·Deα−σanDeα
≤lim
αxDaneα−aeασan−σaDeα
≤lim
αxDan−aeασan−aDeα−→ 0,
4.24
so
Dis continuous.
14 Abstract and Applied Analysis
It remains to show that
Dis a σ-derivation. From the definition of the strict topology,
we have aeα→ain the strict topology for all a∈Abecause aeαiiaeαα
→aiiai∈
Iand so
Daeαw∗
→
Da. Therefore,
Dabω∗−lim
αlim
β
Daeαbeβ
ω∗−lim
αlim
βDaeαbeβ
ω∗−lim
αlim
βσaeαDbeβDaeα·σbeβ
ω∗−lim
αlim
βσaeα
Dbeβ
Daeα·σbeβ
σa
Db
Daσb,
4.25
that is,
Dis σ-derivation.
Corollary 4.15. Suppose that Ais σ-a.a, where σis bounded idempotent endomorphism of A,I is a
closed ideal in A.IfIhas a bounded approximate identity, then Iis σ-a.a.
Proof. Suppose that Ihas a bounded approximate identity, Xis a σ-neo-unital Banach I-
bimodule, and D:I→X
∗is a σ-derivation. By Proposition 4.14,Xbecomes to a Banach
A-bimodule and Dhas a unique extension
D:A→X
∗which is a σ-derivation. Since Ais
σ-a.a,
∃{x∗
α}⊆X
∗s.t.
Dalim
ασa·x∗
α−x∗
α·σa
a∈A
.4.26
So we have Di
Dilimασi·x∗
α−x∗
α·σi, which shows that Dlimαδσ
x∗
αis σ-a.i, and
Iis σ-a.a.
Corollary 4.16. Let Abe an a.a Banach algebra and Ia closed ideal of A.ThenA/I is σ-a.a for each
bounded endomorphism σof A/I.
Proposition 4.17. Let Ibe a closed ideal of Asuch that σI⊆I.IfAis σ-a.a, then A/I is σ-a.c,
where σis an endomorphism of A/I induced by σ(i.e., σaIσaIfor a∈A.
Proof. Let Xbe a Banach A/I-bimodule and D:A/I →Xaσ-derivation. Then Xbecomes
an A-bimodule with the following module actions:
a·xπa·x, x ·ax·πa
a∈A,x ∈X
,4.27
Abstract and Applied Analysis 15
where πis the canonical homomorphism π:A→A/I. It is easy to see that Doπ :A→X
becomes a σ-derivation. Since Ais σ-a.c, there exists a net {xα}⊆Xsuch that Doπa
limασa·xα−xα·σaa∈A. Therefore, for each a∈A,
DaIDoπalim
ασa·xα−xα·σa
lim
απσa ·xα−xα·πσa
lim
ασaI·xα−xα·σaI
lim
ασaIxα−xασaI.
4.28
Thus, A/I is σ-a.c.
Proposition 4.18. Suppose that Iis a closed ideal in A.IfIis σ-amenable and A/I is a.a, then Ais
σ-a.a.
Proof. Let Xbe a Banach A-bimodule and D:A→X
∗aσ-derivation. Xis a Banach I-
bimodule too.
Clearly, dD|I:I→X
∗is a σ-derivation, and by σ-amenability of Ithere exists
x∗
0∈X
∗such that Dδσ
x∗
0, and, therefore, for each i∈Iwe have diσi·x∗
0−x∗
0·σi.Set
D1D−δσ
x∗
0. Clearly, D1is σ-derivation and D1|I0. Now let X0spanX·σI∪σI·X·
X/X0is a Banach A/I-bimodule via the following module actions:
aIxX0σaxX0,xX0aIxσaX0x∈X,a∈A
.4.29
Now we define
D:A
I−→ X
X0∗
;xX0,
DaIx, D1aa∈A,x ∈X
.4.30
Let aIaIand xX0xX0for some a, a∈Aand x, x∈X.Soa−a∈I, and we
have D1a−a0. Thus, D1aDa. Now we have
xX0,
DaIxX0,
DaI.4.31
Thus, x, D1ax,D
1ax,D
1a, and, therefore,
x−x,D
1a0.4.32
It is enough to show that D1ais zero on X0. Suppose that σix∈X
0, we have
σix, D1ax, D1aσix, D1ai−σaD1i0,
xσi,D
1ax, σiD1ax, D1ia−D1iσa0.
4.33
16 Abstract and Applied Analysis
So for all a∈A,D1a0onσI·X∪X·σIand so for all a∈A,D1a0onX0. Since
x−x∈X
0, therefore x−x,D
1a0 which shows that D1is well defined. We claim that
Dis a derivation;
xX0,
DaIbIx, D1ab
x, σaD1bD1aσb
xσa,D
1bσbx, D1a
xσaX0,
DbI
σbxX0,
DaI
xX0aI,
DbI
bIxX0,
DaI.
4.34
So there exists a net ϕα⊆X/X0∗such that
Dlimαδϕα.Letq:X→X/X0be the
quotient map. For every α,ϕαoq∈X
∗.Setx∗
αϕαoq⊆X
∗. We have
x, D1axX0,
DaI
xX0,lim
αaIϕα−ϕαaI
lim
αxσaX0,ϕ
α−σaxX0,ϕ
α
lim
αqxσa,ϕ
α−qσax,ϕ
α
lim
αϕαoqxσa−σaxxσa−σax, x∗
α
lim
αx, σax∗
α−x∗
ασa
x, lim
αδσ
x∗
αa.
4.35
So D1D−δσ
x∗limαδσ
x∗
α, and, therefore, Dlimαδσ
x∗
α−x∗
0. Which shows that Dis σ-a.i and
so Ais σ-a.a.
Example 4.19. Let Abe a Banach algebra and let 0 /
ϕ∈BallA∗. Then Awith the product
a·aϕaabecomes a Banach algebra. We denote this algebra with Aϕ.Itiseasytoseethat
Aϕhas a left identity e, while it has not right approximate identity, so Aϕis not contractible
and is not approximately contractible. Also Aϕis biprojective. Now suppose that σ:Aϕ→
Aϕbe defined by σaϕae. We have
σ2aσϕaeϕaσeϕaϕeeϕaeσa.4.36
Abstract and Applied Analysis 17
Thus σis idempotent. It is easy to see that eis identity for σAϕ, and since Ais biprojective
by 1, Corollary 5.3,Aϕis σ-biprojective. Thus by 1, Theorem 4.3,Aϕis σ-contractible and
so Aϕis σ-a.c.
It is easy to see that ker ϕand all subspaces of ker ϕare all ideals of Aϕand σker ϕ⊆
ker ϕso σI⊆Ifor each ideal of A. Therefore, by Proposition 4.17,Aϕ/I is σ-a.c for each
ideal Iof Aϕ, where σaIσaIϕaeI.
Corollary 4.20. Suppose that σis a bounded idempotent endomorphism of Banach algebra A.Then
Ais σ-a.a if and only if there are nets μ
αin A
⊗A∗∗ and Fα,Gα⊆A
∗∗, such that for each
a∈A,
1σa·μ
α−μ
α·σaFα⊗σa−σa⊗Gα→0,
2σa·Fα→σa,G
α·σa→σa,
3π∗∗μ
α·σa−Fα·σa−Gα·σa→0.
Proof. Suppose that Ais σ-a.a, take the net μαgiven in Proposition 4.9 and write
μαμ
α−Fα⊗e−e⊗Gαcαe⊗e, 4.37
where μ
α⊆A
⊗A∗∗,Fα,Gα⊆A∗∗ ,andcα⊆C. Applying π∗∗ ,π∗∗μ
α−Fα−Gαcαe→
e, hence cα→1, then
π∗∗μ
α·σa−Fα·σa−Gα·σae·σa−→ e·σa
a∈A
.4.38
So we have iiifurther, by Proposition 4.9,fora∈A
#,
σa·μ
α−σa·Fα⊗e−σa⊗Gασa⊗e
μ
α·σaFα⊗σae⊗Gα·σa−e⊗σa−→ 0.
4.39
Thus σa·μ
α−μ
α·σaFα⊗σa−σa⊗Gα→0, and σa·Fα→σa,G
α·σa→σa.
So for a∈A,
σa·μ
α−μ
α·σaFα⊗σa−σa⊗Gα−→ 0,
σa·Fα−→ σa,G
α·σa−→ σa.
4.40
18 Abstract and Applied Analysis
Conversely, set cα1andμαμ
α−Fα⊗e−e⊗Gαe⊗e. We have
σaα·μα−μα·σaασaα·μα−μα·σaα
σa·μα−μα·σaaμα−αμα
σa·μα−μα·σa
σa·μ
α−σaFα⊗e−σa⊗Gα
σa⊗e−μ
α·σa
Fα⊗σae⊗Gασa−e⊗σa
σa·μ
α−μ
α·σa
Fα⊗σa−σa⊗Gα→0a∈A
.
4.41
So σa·μα−μα·σa→0a∈A
#.Also
π∗∗μα·σaπ∗∗ μ
α−Fα⊗e−e⊗Gαe⊗eσa
π∗∗μ
ασa−Fα·σa
−Gα·σaσa−→ σa
a∈A
,
4.42
and so π∗∗μα→e.Now,byProposition 4.9,Ais σ-a.a.
For σ-approximate contractibility we have the following parallel result.
Proposition 4.21. Ais σ-a.c if and only if any of the following equivalent conditions hold:
1there is a net μα⊂A
#
⊗A#such that for each a∈A
#,σa·μα−μα·σa→0and
πμα→e;
2there is a net μ
α⊂A
#
⊗A#such that for each a∈A
#,σa·μ
α−μ
α·σa→0and
πμ
αe;
3there are nets μ
α⊂A
⊗A,Fα,Gα⊂A, such that for each a∈A,
iσa·μ
α−μ
α·σaFα⊗σa−σa⊗Gα→0;
iiσi·Fα→σa,G
α·σa→σa;
iiiπμ
α·σa−Fα·σa−Gα·σa→0.
We know Banach algebra Ais amenable if and only if Ahas bounded approximate
diagonal 3.
Proposition 4.22. Banach algebra Ais σ-amenable if and only if Ahas bounded approximate σ-
diagonal, that is, there is a bounded net μα⊆A
⊗A such that for each a∈A,σa·μα−μα·σa→
0and πμα·σa→σa.
Proposition 4.23. If Banach algebra Ais σ-amenable, then Ais σ-a.c.
Abstract and Applied Analysis 19
Proof. Suppose that Ais σ-amenable. Then there exists a bounded net μαin A⊗Asuch that
for each a∈A,
σa·μα−μα·σa−→ 0,π
μα·σa−→ σa.4.43
Set fαπμα. It is easy to see that fαis a bounded approximate identity. Then μ
α
μαfα⊗fαand FαGαfαsatisfy i–iiiof Proposition 4.21, because
iσa·μ
α−μ
α·σafα⊗σa−σa⊗fασa·μα−μα·σaσafα⊗fα−fα⊗
σafα⊗σa−σa⊗fα→0a∈A,
iiσa·fασa·πμα→σa,fα·σaπμα·σa→σa,
iiiπμ
α·σaπμαfα⊗fα·σafα·σaf2
α·σa.
So
πμ
α·σa−Fα·σa−Gα·σafα·σaf2
α·σa−fα·σa−fα·σa−→ 0.4.44
Note that f2
αis a bounded approximate identity too, thus, by Proposition 4.21,Ais σ-a.c.
Corollary 4.24. Suppose that Ais a σ-a.a Banach algebra where σis an idempotent endomorphism of
Aand Iis a closed two-sided ideal of Awhich σIhas a bounded approximate identity and σI⊆I.
Then, Iis σ-a.a.
Proof. Let {eα}be a bounded approximate identity in σI,so{eα}is bounded net in σI∗∗,
and so by Banach-Alaoglu theorem there exists a subnet {eβ}⊆{eα}and E∈σI∗∗ such that
eβ
w∗
→E.Eis a right identity in σI∗∗ because for each F∈σI∗∗ and f∈σI∗,
f, FEf·F, Elim
βeβ,fFlim
βeβf, Ff, F.4.45
Also EactsasanidentityonσIitself. Let μα,Fα,Gαbe the nets given by
Corollary 4.20 for A. Define μ
αE·μα·E∈I
⊗I∗∗,F
αE·Fα∈I∗∗,andG
αGα·E∈I∗∗.
Then, for i∈I,
iwe consider
σi·μ
α−μ
α·σiF
α⊗σi−σi⊗G
α
σi·E·μα·E−E·μα·E·σiE·Fα⊗σi−σi⊗Gα·E
σi·μα·E−E·μασiE·Fα⊗σi−σi⊗Gα·E
E·σi·μα·E−E·μα·σi·E
E·Fα⊗σi·E−E·σi⊗Gα·E
Eσi·μα−μα·σiFα⊗σi−σi⊗Gα·E−→ 0,
4.46
20 Abstract and Applied Analysis
iiwe consider
σi·F
ασi·E·Fασi·Fα−→ σi,
G
α·σiGα·E·σiGα·σi−→ σi
4.47
iiiwe consider
π∗∗μ
α·
σa−F
α·
σa−G
α−
σa
π∗∗E·μα·E·σa−E·Fα·σa−Gα·E·σa
E·π∗∗μα·E·σa−E·Fα·σa−Gα·σa
E·π∗∗μα·σa−E·Fα·σa−Gα·σa−E·GασaE·Gασa
E·π∗∗μα·σa−Fα.σa−GασaE−eGασa−→ 0.
4.48
An alternative proof would be to follow the standard argument stated in Corollary 4.15.
References
1P. C. Curtis Jr. and R. J. Loy, “The structure of amenable Banach algebras,” Journal of the London
Mathematical Society, vol. 40, no. 1, pp. 89–104, 1989.
2M. Eshaghi Gordji, “Point derivations on second duals and unitization of Banach algebras,” Nonlinear
Functional Analysis and Applications, vol. 13, no. 2, pp. 271–275, 2008.
3M. Eshaghi Gordji, “Homomorphisms, amenability and weak amenability of Banach algebras,”
Vietnam Journal of Mathematics, vol. 36, no. 3, pp. 253–260, 2008.
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