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Abstract

We investigate Οƒ -approximate contractibility and Οƒ -approximate amenability of Banach algebras, which are extensions of usual notions of contractibility and amenability, respectively, where Οƒ is a dense range or an idempotent bounded endomorphism of the corresponding Banach algebra.
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 653140, 20 pages
doi:10.1155/2012/653140
Research Article
Οƒ-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 Οƒ-approximate contractibility and Οƒ-approximate amenability of Banach algebras,
which are extensions of usual notions of contractibility and amenability, respectively, where Οƒis 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 cβ‰₯0
such that for a∈A,x ∈X, we have
aΒ·x≀cax,xΒ·a≀cax.1.1
A derivation D:A→Xis a linear map, always taken to be continuous, satisfying
DabDa·bξ€…aΒ·Db
a, b ∈A
.1.2
A Banach algebra Ais amenable if for any A-bimodule X, any derivation D:A→X
βˆ—is
inner, that is, there exists xβˆ—βˆˆX
βˆ—,with
DaaΒ·xβˆ—βˆ’xβˆ—Β·aδxβˆ—ξ€‚a
a∈A
.1.3
2 Abstract and Applied Analysis
Let Abe a Banach algebra and Οƒa 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
Dabσa·DbDa·σb
a, b ∈A
.1.4
For each x∈X, the mapping
δσ
x:Aβˆ’β†’X 1.5
defined by δσ
xaσa·xβˆ’x·σa, for all a∈A,isaΟƒ-derivation called an inner Οƒ-
derivation.
Remark 1.1. Throughout this paper, we will assume that Ais a Banach algebra, and Οƒis
a bounded endomorphism of Aunless otherwise specified. Also, we write σ-a.ifor Οƒ-
approximately inner, σ-a.afor Οƒ-approximately amenable, and σ-a.cfor Οƒ-approximately
contractible.
The basic definition for the present paper is as follows.
Definition 1.2. AΟƒ-derivation D:Aβ†’Xis Οƒ-a.i, if there exists a net xΞ±ξ€ƒβŠ†Xsuch that for
every a∈A,Dalimασa·xΞ±βˆ’xα·σa, the limit being in norm and we write Dlim δσ
xΞ±.
Note that we do not suppose xαto be bounded.
Definition 1.3. A Banach algebra Ais called Οƒ-a.c if for any A-bimodule X, every Οƒ-derivation
D:A→Xis σ-a.i.
Definition 1.4. A Banach algebra Ais called Οƒ-a.a if for any A-bimodule X, every Οƒ-derivation
D:A→X
βˆ—is Οƒ-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
TaΒ·xσa·Tx,T
xΒ·aTx·σa
a∈A,x ∈X
.1.6
2. Basic Properties
Proposition 2.1. Let Abe a Οƒ-a.c Banach algebra. Then σAhas a left and right approximate
identity.
Proof. Consider XAas a Banach A-bimodule with the trivial right action, that is,
aΒ·xax, x Β·a0a∈A,x ∈X
.2.1
Abstract and Applied Analysis 3
Then D:Aβ†’Xdefined by Daσais a Οƒ-derivation, and so there is a net {uΞ±}βŠ†X
Asuch that Dlimαδσ
uα. Hence for each a∈A,
σaDalim
αδσ
uαalim
ασa·uΞ±βˆ’uα·σalim
ασauΞ±,2.2
which shows that {uΞ±}is a right approximate identity for σA. Similarly, one can find a left
approximate identity for σA.
Corollary 2.2. Let Abe a Οƒ-a.c Banach algebra and Οƒa continuous epimorphism of A.ThenAhas
a left and right approximate identity.
Proposition 2.3. Let Ο•be a bounded endomorphism of Banach algebra A.IfAis Οƒ-a.c, then Ais
ϕoσ-a.c.
Proof. Let Xbe a Banach A-bimodule and let D:Aβ†’Xbe a ϕoσ-derivation. Then X,βˆ—ξ€ƒ
is an A-bimodule with the following module actions:
aβˆ—xϕa·x, x βˆ—ax·ϕa
a∈A,x ∈X
.2.3
For each a, b ∈A, we have
Dabϕoσa·DbDa·ϕoσbσaξ€ƒβˆ—DbDaξ€ƒβˆ—Οƒξ€‚b.2.4
Thus D:A→X,βˆ—ξ€ƒis a continuous Οƒ-derivation. Since Ais Οƒ-a.c, there exists a net {xΞ±}βŠ†X
such that Dlim δσ
xΞ±. In fact,
Dalim
ασaξ€ƒβˆ—xΞ±βˆ’xΞ±βˆ—Οƒξ€‚a
lim
αϕoσa·xΞ±βˆ’xΞ±Β·Ο•oΟƒ a
lim
αδϕoΟƒ
xαa
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 Οƒ-a.c for each bounded endomorphism Οƒof
A.
Proposition 2.5. Let Abe a Οƒ-a.c Banach algebra, where Οƒis 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 doΟƒ is a Οƒ-derivation. Since Ais Οƒ-a.c, there exists a net {xΞ±}βŠ†Xsuch that
4 Abstract and Applied Analysis
doσalimασaxΞ±βˆ’xασa.Nowforb∈Athere exists a∈Asuch that bσa,and,
therefore,
dbdσa lim
Ξ±xασaξ€ƒβˆ’Οƒξ€‚axΞ±
lim
Ξ±xΞ±bβˆ’bxΞ±,
2.6
which shows that dis approximately inner and so Ais a.c.
Corollary 2.6. Let Ο•be a bounded endomorphism of Banach algebra A.IfAis Οƒ-a.a then it is ϕoσ-
a.a too.
Corollary 2.7. Let Abe an a.a Banach algebra. For each bounded endomorphism Οƒ,Ais Οƒ-a.a.
Corollary 2.8. Let Abe a Οƒ-a.a Banach algebra, where Οƒis 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 Οƒ-a.c for each bounded endomorphism Οƒof B.
Proof. Let σ:B→Bbe a bounded endomorphism of Band Xa Banach B-bimodule, then
X,βˆ—ξ€ƒis an A-bimodule with the following module actions:
aβˆ—xσϕa·x, x βˆ—ax·σϕaa∈A,x ∈X
.2.7
Now let D:Bβ†’Xbe a continuous Οƒ-derivation. It is easy to check that DoΟ• :A→X,βˆ—ξ€ƒ
is a derivation. Since Ais approximately contractible, there exists a net {xΞ±}βŠ†Xsuch that
DoϕalimΞ±Ξ΄xαa. We have
DϕaDoϕalim
Ξ±Ξ΄xαalim
αaβˆ—xΞ±βˆ’xΞ±βˆ—a
lim
ασϕaxΞ±βˆ’xασϕaa∈A
.
2.8
Since Ο•is an epimorphism, so for each b∈Bthere exists a∈Asuch that bϕa, and we
have
Dblim
ασbxΞ±βˆ’xασb,2.9
which shows that Dis Οƒ-a.i and so Bis Οƒ-a.c.
Proposition 2.10. Suppose that Aand Bare Banach algebras, and let Οƒand Ο„be bounded
endomorphism of Aand B, respectively. Let ϕ:A→Bbe a bounded epimorphism such that
Ο•oΟƒ τoΟ•.IfAis Οƒ-a.c, then Bis Ο„-a.c.
Abstract and Applied Analysis 5
Proof. Let Xbe a Banach B-bimodule and D:Bβ†’Xa continuous Ο„-derivation. Then X,βˆ—ξ€ƒ
is an A-bimodule with the following actions:
aβˆ—xϕa·x, x βˆ—ax·ϕa
a∈A,x ∈X
.2.10
It is easy to check that DoΟ• :A→X,βˆ—ξ€ƒis a Οƒ-derivation. Since Ais Οƒ-a.c, there exists a net
{xΞ±}βŠ†Xsuch that Doϕalimαδσ
xαa, so we have
Dϕalim
ασaξ€ƒβˆ—xΞ±βˆ’xΞ±βˆ—Οƒξ€‚a
lim
αϕσa Β·xΞ±βˆ’xα·ϕσa
lim
ατϕa·xΞ±βˆ’xα·τϕaa∈A
.
2.11
Since Ο•is epimorphism, so DblimατbxΞ±βˆ’xατbfor all b∈B, and hence Bis Ο„-a.c.
3. Οƒ-Approximate Contractibility for Unital Banach Algebras
In this section we state some properties of Οƒ-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 σAis dense in A.
Proposition 3.1. Let Abe a unital Banach algebra with unit e, σAdense 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 DD1δη.
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 Οƒ-unital if Xσe·X·σe.
Corollary 3.3. Let Abe a unital Banach algebra and σAdense in A. Then, Ais Οƒ-a.c (resp.,Οƒ-a.a)
if and only if for all Οƒ-unital Banach A-bimodule X, every Οƒ-derivation D:Aβ†’Xresp., D :
A→X
βˆ—ξ€ƒis Οƒ-a.i.
Proof. Since σeis a unit for σA,andσAis dense in A,weseethatσee,sothat
eΒ·XΒ·eis a Οƒ-unital Banach A-bimodule. Now by Proposition 3.1, the proof is complete.
Corollary 3.4. Suppose that Ais a unital Banach algebra and σAis dense in A.LetXbe a Banach
A-bimodule and D:A→X
βˆ—aΟƒ-derivation. If Ais Οƒ-a.a, then there exists a net ξ€‚Ξ·Ξ±ξ€ƒβŠ†eΒ·X
βˆ—Β·e,
and η∈X
βˆ—, such that Dlimαδσ
ηαδη.
Proof. By Proposition 3.1,DD1δηsuch that η∈X
βˆ—and D1:Aβ†’eΒ·X
βˆ—Β·eis a Οƒ-
derivation. Since eΒ·X
βˆ—Β·e∼
eΒ·XΒ·eξ€ƒβˆ—and Ais Οƒ-a.a, D1:A→eΒ·XΒ·eξ€ƒβˆ—is Οƒ-a.i. Hence
D1limαδσ
Ξ·Ξ±for some net ξ€‚Ξ·Ξ±ξ€ƒβŠ†eΒ·X
βˆ—Β·e.
6 Abstract and Applied Analysis
In the following proposition we consider Οƒ-approximate contractibility when Οƒis 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 σAand Xis a Banach A-
bimodule. If σis a bounded idempotent endomorphism of A, then for each σ-derivation D:A→X
there exists a Οƒ-derivation D1:A→σe·X·σeand η∈X, such that DD1δη.
Corollary 3.6. Assume that Ahas an element ewhich is a unit for σAand Οƒis a bounded
idempotent endomorphism of A,thenAis Οƒ-a.c (resp., Οƒ-a.a) if and only if for all Οƒ-unital Banach
A-bimodule, X, every σ-derivation D:A→X(resp., D:A→X
βˆ—ξ€ƒis Οƒ-a.i.
Lemma 3.7. Assume that Ais a unital Banach algebra with the identity e, and X,βˆ—ξ€ƒis a Οƒ-unital
Banach A-bimodule with the following module actions:
aβˆ—xσax, x βˆ—axσa
a∈A,x ∈X
.3.1
If D:A→X
βˆ—is a Οƒ-derivation, then De0.
Proof. We have DeDeeσeDeDeσeand
ξ€Šeβˆ—x, Deσeξ€ƒξ€‹ξ€„ξ€Šx, Deσeξ€ƒβˆ—eξ€‹ξ€„ξ€Šx, Deσeσe
ξ€„ξ€Šx, Deσeξ€ƒξ€‹ξ€„ξ€Šeβˆ—x, Dex∈X
.
3.2
Hence DeσeDeand so σeDe0. Hence De0.
Proposition 3.8. Let Οƒbe a bounded idempotent endomorphism of Banach algebra A.IfAis Οƒ-a.a,
then A#is σ-a.a, where σis the endomorphism of A#induced by Οƒ, that is, σaασaα.
Proof. Let Xbe a Banach A#-bimodule and D:A#β†’X
βˆ—a continuous σ-derivation. By
Proposition 3.5, there exits η∈X
βˆ—and D1:A#→σe·X
βˆ—Β·ξ€„Οƒξ€‚esuch that DD1δη.Set
d:D1|A:A→σe·Xβˆ—Β·ξ€„Οƒξ€‚e. It is easy to check that dis a Οƒ-derivation. Since Ais Οƒ-a.a, there
exists a net xβˆ—
Ξ³ξ€ƒβŠ†X
βˆ—such that dlimγδσ
xβˆ—
Ξ³. Hence D1alimγσaxβˆ—
Ξ³βˆ’xβˆ—
γσa,a∈A.
Since σe·X
βˆ—Β·ξ€„Οƒξ€‚eis σ-unital, by Lemma 3.7,D1e0 and for each aξ€…Ξ±βˆˆA
#we have
D1aαD1aαD1eD1alim
γσaxβˆ—
Ξ³βˆ’xβˆ—
γσa
lim
γσaξ€…Ξ±ξ€ƒβˆ’Ξ±ξ€ƒxβˆ—
Ξ³βˆ’xβˆ—
γσaξ€…Ξ±ξ€ƒβˆ’Ξ±ξ€ƒ
lim
γϕaαxβˆ—
Ξ³βˆ’xβˆ—
γϕaα.
3.3
This shows that D1is σ-a.i, and so A#is σ-a.a.
Proposition 3.9. Let Οƒbe a bounded endomorphism of Banach algebra A.IfA#is σ-a.a, then Ais
Οƒ-a.a.
Abstract and Applied Analysis 7
Proof. Let Xbe a Banach A-bimodule and D:A→X
βˆ—a continuous Οƒ-derivation. Xis a
Banach A#-bimodule with the following module actions:
aα·xaΒ·xξ€…Ξ±x, x ·aαxΒ·aξ€…Ξ±x, 3.4
for all a∈A,x ∈X,Ξ± ∈C. Define D#:A#β†’X
βˆ—with D#aαDa. Clearly, D#is a
continuous σ-derivation. Hence, there is a net xβˆ—
Ξ³ξ€ƒβŠ†X
βˆ—such that D#limσ
Ξ³Ξ΄xβˆ—
Ξ³. Hence, for
each a∈Awe have
DaD#aαlim
γσaαxβˆ—
Ξ³βˆ’xβˆ—
γσaαlim
γσaxβˆ—
Ξ³βˆ’xβˆ—
γσa3.5
which shows that Dis Οƒ-a.i and so Ais Οƒ-a.a.
4. Οƒ-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
βˆ—is Οƒ-inner.
Proof. Let Xbe a Banach A-bimodule with trivial left action. Hence, Xβˆ—is a Banach A-
bimodule with trivial right action, that is,
xβˆ—Β·a0,aΒ·xβˆ—ξ€„axβˆ—ξ€‚xβˆ—βˆˆX
βˆ—,a∈A
.4.1
Let D:A→X
βˆ—be a continuous Οƒ-derivation and eαa bounded approximate identity of A.
By Banach Alaoglu’s Theorem, Deα has a subnet Deβ such that Deβwβˆ—
β†’xβˆ—
0, for some
xβˆ—
0∈X
βˆ—. Since aΒ·eΞ²
·
β†’aand Dis continuous, DaΒ·eβ·
β†’Da. Hence, DaΒ·eβwβˆ—
β†’Da.
On the other hand, DaΒ·eβσaDeβwβˆ—
→σaxβˆ—
0and so Daσaxβˆ—
0. Hence,
Daσaxβˆ—
0βˆ’xβˆ—
0σaand Dis Οƒ-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 Οƒ-neo-unital σ-pseudo-
unital,ifXσA·X·σA. Similarly, one defines Οƒ-neo-unital left and right Banach
modules.
Definition 4.3. Let Xbe a Banach A-bimodule. Then Xis called Οƒ-essential if X
σAXσAspan σA·X·σA. Similarly, one defines Οƒ-essential left and right Banach
modules.
We recall that a bounded approximate identity in Banach algebra Afor Banach
A-bimodule Xis a bounded net eαin Asuch that for each x∈X,eΞ±xβ†’xand xeΞ±β†’x.
Proposition 4.4. Assume that Ahas a left bounded approximate identity, Οƒis a bounded idempotent
endomorphism of A, and Xis a left Banach A-module. Then Xis Οƒ-neo-unital if and only if Xis
Οƒ-essential.
8 Abstract and Applied Analysis
Proof. Let Xbe a Οƒ-essential Banach A-bimodule. Since Οƒis idempotent, σAis Banach
subalgebra of A.LeteΞ±ξ€ƒβŠ†Abe left approximate identity with bound m. First suppose that
z∈span σA·X, so there exist a1,...,a
n∈A,x1,...,x
n∈Xsuch that zn
i1σaixi. For
1≀i≀n,eΞ±aiβ†’aiand, therefore, σeαzβ†’z.
Now suppose that zβˆˆΟƒξ€‚AX. There exists {zn}βŠ†span σA·Xsuch that znβ†’z.
Thus,
βˆƒn0∈Ns.t.βˆ€nnβ‰₯n0;znβˆ’z<Ξ΅
2σmξ€…14.2
On the other hand, for each n∈Nwe have σeαzn
Ξ±
β†’znand so σeαzn0
Ξ±
β†’zn0.
Therefore,
βˆƒΞ±0;βˆ€Ξ±ξ€ˆΞ±β‰₯Ξ±0;σeαzn0βˆ’zn0<Ξ΅
2.4.3
Now we have
σeαzβˆ’z≀σeαzβˆ’Οƒξ€‚eαzn0σeαzn0βˆ’zn0ξ€…zn0βˆ’z
≀σeαzβˆ’zn0σeαzn0βˆ’zn0zn0βˆ’z
<σeα1zβˆ’zn0Ρ
2
<σmξ€…1Ρ
σmξ€…12ξ€…Ξ΅
2Ρ,
4.4
which shows that σeα βŠ†Οƒξ€‚Ais a left bounded approximate identity for X.Nowby
Cohen factorization Theorem, XσA·X.SoXis Οƒ-neo-unital. The other side is trivial.
Corollary 4.5. Every Οƒ-neo-unital left Banach A-module is essential.
Proof. Let Xbe a Οƒ-neo-unital left Banach A-module. We have XσA·X βŠ† AΒ·X βŠ† AX βŠ† X
so XAX.
Proposition 4.6. Let Abe a Banach algebra with a left bounded approximate identity, Οƒbe a bounded
idempotent endomorphism of A, and Xa left Banach A-module. Then σA·Xis closed weakly
complemented submodule of X.
Proof. Set YσAX,sinceAhas a left bounded approximate identity, by Cohen
factorization Theorem A2A, and we have σAYσAσAXσA2XσAXY,
which shows that Yis Οƒ-essential by Proposition 4.4,Yis Οƒ-neo unital that is, YσA·Y.
Hence, σAXYσA·YβŠ†Οƒξ€‚A·Xand so σAXσA·X.ThusσA·Xis closed
submodule of X.
Now we prove that σA·Xis weakly complemented in X.Leteαbe a left
approximate identity in Awith bound m, and define a net Tαin BXβˆ—ξ€ƒby setting Tαxβˆ—ξ€ƒξ€„
xβˆ—Β·Οƒξ€‚eαxβˆ—βˆˆX
βˆ—ξ€ƒ. We have Tα≀σm.ThusTαis a bounded net in BXβˆ—ξ€ƒsince
BXβˆ—ξ€ƒξ€„ξ€‚Xβˆ—βŠ—Xξ€ƒβˆ—and ball BXβˆ—ξ€ƒis wβˆ—-compact, so there exists T∈BXβˆ—ξ€ƒsuch that we
Abstract and Applied Analysis 9
may suppose that wβˆ—βˆ’limΞ±TαTand T≀σm. For each a∈A,x∈X,andxβˆ—βˆˆX
βˆ—,we
have
ξ€ŠΟƒξ€‚a·x, T xβˆ—ξ€ƒξ€‹ξ€„lim
Ξ±ξ€ŠΟƒξ€‚a·x, xβˆ—Β·Οƒξ€‚eα
lim
Ξ±ξ€ŠΟƒξ€‚eασa·x, xβˆ—ξ€‹
ξ€„ξ€ŠΟƒξ€‚a·x, xβˆ—ξ€‹,
4.5
and so xβˆ—βˆ’Txβˆ—βˆˆξ€‚Οƒξ€‚A·XβŠ₯. On other hand, for each xβˆ—βˆˆX
βˆ—,
T2xβˆ—ξ€„TTxβˆ—ξ€ƒξ€„lim
Ξ±Txβˆ—ξ€ƒΟƒξ€‚eαlim
Ξ±xβˆ—Οƒξ€‚eαTxβˆ—ξ€ƒ.4.6
Thus Tis projection, and IXβˆ—βˆ’T:Xβˆ—β†’ξ€‚Οƒξ€‚A·XβŠ₯is projection. So σA·Xis weakly
complemented in Xand, we have Xβˆ—ξ€„ξ€‚Οƒξ€‚A·XβŠ₯βŠ•ξ€‚Οƒξ€‚A·Xξ€ƒβˆ—.
Corollary 4.7. Let Ahave a bounded approximate identity, and let Xbe a Banach A-bimodule and
Οƒa bounded idempotent endomorphism of A.Then
iσA·X·σAis a closed weakly complemented submodule of X,
iiAis Οƒ-a.a if and only if for every Οƒ-neo-unital Banach A-bimodule X, every Οƒ-derivation
D:A→X
βˆ—is Οƒ-approximately inner.
Proof. Set YσA·X.ByProposition 4.6,Yis a closed and weakly complemented
submodule of X,andT:Xβˆ—β†’Y
βˆ—and Iβˆ’T:Xβˆ—β†’Y
βŠ₯are projection maps. Let D:Aβ†’X
βˆ—
be a Οƒ-derivation, so ToD and Iβˆ’ToD are Οƒ-derivations and DToDIβˆ’ToD.
Since A·X/Y{0}by Lemma 4.1,Iβˆ’ToD is Οƒ-inner. So there exists J0∈Y
βŠ₯such that
Iβˆ’ToD δσ
J0.ThusDToD δσ
J0and so Dis σ-a.i if and only if ToD :A→Y
βˆ—is Οƒ-a.i.
Now let ZY·σA.ByProposition 4.6,Zis a closed weakly complemented in Y,
and T:Yβˆ—β†’Z
βˆ—and Iβˆ’T:Yβˆ—β†’Z
βŠ₯are projection maps. Assume that D1:Aβ†’Y
βˆ—is a Οƒ-
derivation, thus ToD and Iβˆ’ToD are Οƒ-derivations, and we have D1ToD1Iβˆ’T·D1.
Since Y/Z·A {0},byLemma 4.1,Iβˆ’T·D1is Οƒ-inner and so there exists z0∈ZβŠ₯
such that Iβˆ’ToD1δσ
Z0. Therefore, D1ToD1δσ
Z0.Thus,D1is Οƒ.a.i if and only if
ToD1is Οƒ.a.i. Set DoT D1.Thus,DToD1δσ
Z0δσ
J0. Therefore, Dis Οƒ-a.i, if and only if
ToD1:Aβ†’Z
βˆ—ξ€„ξ€‚Οƒξ€‚A·X·σAξ€ƒξ€ƒβˆ—is Οƒ.a.i. Recall that Zis Οƒ-neo-unital. Thus, Ais Οƒ-a.a if
and only if for every Οƒ-neo-unital Banach A-bimodul, X, every Οƒ-derivation D:Aβ†’Xβˆ—is
Οƒ-a.i.
Corollary 4.8. Let Ahave a bounded approximate identity, and let Xbe a Banach A-bimodule and Οƒ
a bounded idempotent endomorphism of A.ThenAis Οƒ-a.a if and only if for every Οƒ-essential Banach
A-bimodule X, every σ-derivation D:A→X
βˆ—is Οƒ-approximately inner.
Proposition 4.9. Suppose that Οƒis a bounded idempotent endomorphism of Aand define σ:A#β†’
A#with σaασaα. The following statements are equivalent.
1Ais Οƒ-a.a.
2Thereisanetξ€‚ΞΌΞ±ξ€ƒβŠ†ξ€‚A#
βŠ—A#ξ€ƒβˆ—βˆ— such that for each a∈A
#,σaξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·ξ€„Οƒξ€‚a→0and
Ο€βˆ—βˆ—ξ€‚ΞΌΞ±ξ€ƒβ†’ξ€„e.
10 Abstract and Applied Analysis
3Thereisanetμ
Ξ±ξ€ƒβŠ†ξ€‚A#
βŠ—A#ξ€ƒβˆ—βˆ— such that for each a∈A
#,σaξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·ξ€„Οƒξ€‚a→0and
for every Ξ±,Ο€βˆ—βˆ—ξ€‚ΞΌξ€‘
αe.
Proof. 1β‡’3Suppose that Ais Οƒ-a.a, by Proposition 3.8,A#is σ-a.a. Let ueβŠ—e∈A
#
βŠ—A#.
A#
βŠ—A#is a Banach A#-bimodule with the following module actions:
a·bβŠ—cσabβŠ—c,bβŠ—c·abβŠ—cσaξ€ƒξ€ˆa, b, c ∈A
#.4.7
Set δu:A#β†’ker Ο€βˆ—βˆ— with definition δuaσa·uβˆ’ξ€„u·σaa∈A
#.δuis σ-derivation.
Recall that ker Ο€βˆ—βˆ— ker Ο€ξ€ƒβˆ—βˆ—. Since A#is σ-a.a, thus there exists eΞ±ξ€ƒβŠ†ker Ο€βˆ—βˆ— such that
δualim
ασaeΞ±βˆ’eασaξ€ƒξ€ˆa∈A
#.4.8
Set μ
αuβˆ’eΞ±βˆˆξ€‚A#
βŠ—A#ξ€ƒβˆ—βˆ—. We have
σaμ
Ξ±βˆ’ΞΌξ€‘
ασaσauβˆ’ξ€„uσaξ€ƒβˆ’ξ€‚ξ€„Οƒξ€‚aeΞ±βˆ’eασa βˆ’β†’ 0,4.9
and for each Ξ±,
Ο€βˆ—βˆ—ξ€‚ΞΌξ€‘
Ξ±ξ€ƒξ€„Ο€βˆ—βˆ—ξ€‚ξ€„uβˆ’eΞ±ξ€ƒξ€„Ο€βˆ—βˆ— uξ€ƒβˆ’Ο€βˆ—βˆ— eαπue. 4.10
3β‡’2is clear.
2β‡’1By Proposition 3.9,itissufficient to show that A#is σ-a.a.
Let D:A#β†’X
βˆ—be a derivation. By Corollary 4.7, we may take Xto be Οƒ-neo-unital.
We run the standard argument, so for each α∈I,setfαxξ€ƒξ€„ΞΌΞ±ξ€‚Οˆx, where for a, b ∈A
#,
x∈X, we have ψxaβŠ—bξ€ƒξ€„ξ€Šx, σaDb. Then, mΞ³
Ξ±ξ€ƒβŠ‚A#
βŠ—A#converging Ο‰βˆ—to ΞΌΞ±ξ€‚Ξ±βˆˆI
and noting that for m∈A
#
βŠ—A#,a ∈A
#,x∈X, then
Οˆξ€„Οƒξ€‚axβˆ’xσamσaξ€ƒΟˆxβˆ’Οˆxσamξ€ƒβˆ’ξ€Šx, σπmDa.4.11
Since Xis σ-neo-unital, so XXσA#. So for each a∈Aand x∈X, we have
ξ€Šξ€„Οƒξ€‚axβˆ’xσa,f
Ξ±ξ€‹ξ€„ξ€ŠΟˆξ€„Οƒξ€‚axβˆ’xσa,ΞΌ
α
lim
Ξ³ξ€ŒmΞ³
Ξ±,Οˆξ€„Οƒξ€‚axβˆ’xσa
ξ€„ξ€Šξ€„Οƒξ€‚aξ€ƒΟˆxβˆ’Οˆxσa,ΞΌ
Ξ±ξ€‹βˆ’lim
Ξ³ξ€Œx, ξ€„Οƒξ€ˆΟ€ξ€ˆmΞ³
αDa
ξ€„ξ€ŠΟˆx,ΞΌ
ασaξ€ƒβˆ’ξ€„Οƒξ€‚aξ€ƒΞΌΞ±ξ€‹βˆ’ξ€Šx, Ο€βˆ—βˆ— μαDa.
4.12
Abstract and Applied Analysis 11
Thus,
ξ€Ž
ξ€Žξ€Šx, σafΞ±βˆ’fασaξ€ƒξ€‹βˆ’ξ€Šx, Daξ€ƒξ€‹ξ€Ž
ξ€Ž
β‰€ξ€Ž
ξ€Žξ€ŠΟˆx,σaξ€ƒΞΌΞ±βˆ’ΞΌΞ±ξ€„Οƒξ€‚aξ€ƒξ€‹ξ€Ž
ξ€Žξ€…ξ€„xξ€„ξ€Ž
ξ€ŽΟ€βˆ—βˆ—ξ€‚ΞΌΞ±ξ€ƒβˆ’ξ€„eξ€Ž
ξ€Žξ€„Da
≀D·xξ€„ξ€Ž
ξ€Žξ€„Οƒξ€‚aξ€ƒΞΌΞ±βˆ’ΞΌΞ±ξ€„Οƒξ€‚aξ€ƒξ€Ž
ξ€Žξ€…ξ€„xξ€„ξ€Ž
ξ€ŽΟ€βˆ—ξ€‚ΞΌΞ±ξ€ƒβˆ’ξ€„eξ€Ž
ξ€Žξ€„Da,
4.13
and, therefore, Dlimαδσ
fΞ±. It follows that A#is σ-a.a and so Ais Οƒ-a.a.
Proposition 4.10. Suppose that Ais Οƒ-a.a, and let
Ξ£:0βˆ’β†’ X βˆ—f
βˆ’β†’ Y g
βˆ’β†’ Z βˆ’β†’ 0,4.14
be an admissible short exact sequence of left A-module and left Οƒ-A-module homomorphism. Then
Σ,σ-approximately split, that is, there is a net Gα:Z→Yof right inverse maps to gsuch that
limασaGΞ±βˆ’Gασa  0for a∈A, and a net FΞ±:Yβ†’X
βˆ—of left inverse maps to fsuch that
limασafΞ±βˆ’fασa  0for a∈A.
Proof. Following the proof of 2, Theorem 2.3, for a right inverse Gfor g,Οƒ-approximate
amenability gives a net ξ€‚Ο•Ξ±ξ€ƒβŠ†BZ,Xβˆ—ξ€ƒsuch that
σa·Gβˆ’G·σalim
ασa·fGΞ±βˆ’fGα·σaa∈A
.4.15
Setting GαGβˆ’fϕαgives the required net. Applying the same argument as 2,
Proposition 1.1provides Fα.
We recall that if Ais a Banach algebra with a weak left rightapproximate identity,
then Ahas a left rightapproximate identity 1, Lemma 2.2.
Corollary 4.11. Suppose that Banach algebra Ais Οƒ-a.a, then σAhas left and right approximate
identities.
Corollary 4.12. Suppose that Banach algebra Ais Οƒ-a.a and Οƒis 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 eααis a bounded approximate identity in A,thenσeααis a bounded
approximate identity for X.
Proof. For every a∈Awe have eασa→σa. Since Οƒis idempotent, σeασa→σa.
For each x∈X, there exists a∈Aand y∈Xsuch that xσa·y. Therefore,
σeα·xσeασa·yβˆ’β†’ σa·yx, 4.16
which shows that σeα 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 Pii∈I, where
Pia:aiiaa∈A
.4.17
Note that the strict topology is Hausdorffonly if {a∈A:aΒ·IIΒ·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 Οƒ-neo-unital Banach I-
module and D:I→X
βˆ—aΟƒ-derivation. Then, Xis a Banach A-bimodule in a canonical fashion, and
there is a unique Οƒ-derivation 
D:A→X
βˆ—such that
i
D|ID,
ii
Dis continuous with respect to the strict topology on Aand the Ο‰βˆ—-topology on Xβˆ—.
Proof. Since Xis a Οƒ-neo-unital Banach I-module, so for each x∈X, there exists i∈Iand
y∈Xsuch that xσi·y. Define aΒ·xσai·ya∈A.
We claim that a·xis well defined, that is, independent of the choices of iand y.Let
iξ€‘βˆˆIand yξ€‘βˆˆXbe such that xσi·y,andleteααbe a bounded approximate identity
for I. For each a∈Aand x∈Xwe have
aΒ·xσai·ylim
ασaeΞ±i·y
lim
ασaeασi·ylim
ασaeαx
lim
ασaeασi·ylim
ασaeΞ±i·y
σai·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:Aβˆ’β†’X
βˆ—,aβˆ’β†’ Ο‰βˆ—βˆ’lim
αDaeΞ±ξ€ƒβˆ’Οƒξ€‚a·Deα.4.19
We claim that 
Dis well-defined, that is, the limit in 4.19does exist. Let x∈X,andleti∈I
and y∈Xsuch that xy·σi.ByLemma 4.13,σeαis bounded approximate identity for
X, and we have
ξ€Šx, DaeΞ±ξ€ƒβˆ’Οƒξ€‚a·DeΞ±ξ€ƒξ€‹ξ€„ξ€Šy·σi,D
aeΞ±ξ€ƒβˆ’Οƒξ€‚a·Deα
ξ€„ξ€Šy, σiDaeΞ±ξ€ƒβˆ’Οƒξ€‚ia·Deα
Abstract and Applied Analysis 13
ξ€„ξ€Šy, DiaeΞ±ξ€ƒβˆ’DiσaeΞ±ξ€ƒβˆ’DiaeαDiaσeα
ξ€„ξ€ŠΟƒξ€‚eα·y, Diaξ€ƒξ€‹βˆ’ξ€ŠΟƒξ€‚aeα·y, Di
Ξ±
βˆ’β†’ ξ€Šy, Diaξ€ƒξ€‹βˆ’ξ€ŠΟƒξ€‚a·y, Dia∈A
.
4.20
So the limit in 4.19exists. Furthermore, for i∈I,

Diξ€ƒξ€„Ο‰βˆ—βˆ’lim
αDieΞ±ξ€ƒβˆ’Οƒξ€‚i·Deα
ξ€„Ο‰βˆ—βˆ’lim
Ξ±DieΞ±ξ€ƒβˆ’DieαDiσeαDi,
4.21
so 
Dis an extension of D.Alsofora∈Aand i∈Iwe have
ξ€ˆξ€
Da·σiξ€ƒξ€„Ο‰βˆ—βˆ’lim
αDaeα·σiξ€ƒβˆ’Οƒξ€‚a·Deα·σi
ξ€„Ο‰βˆ—βˆ’lim
αDaeΞ±iξ€ƒβˆ’Οƒξ€‚aeα·Diξ€ƒβˆ’Οƒξ€‚a·DeΞ±iσaσeα·Di
ξ€„Ο‰βˆ—βˆ’lim
αDaeΞ±iξ€ƒβˆ’Οƒξ€‚a·DeΞ±i Daiξ€ƒβˆ’Οƒξ€‚a·Di.
4.22
We claim that 
Dis continuous with respect to the strict topology on Aand the Ο‰βˆ—-topology
an Xβˆ—.
Let an
strict
β†’ain A.
βˆ€i∈I, aniianξ€„βˆ’β†’ aiia.4.23
For each x∈X,


ξ€ξ€Œx, 
Danξ€ƒξ€βˆ’ξ€Œx, 
Da


lim
Ξ±|ξ€Šx, DaneΞ±ξ€ƒβˆ’Οƒξ€‚an·DeΞ±ξ€ƒξ€‹βˆ’ξ€Šx, DaeΞ±ξ€ƒβˆ’Οƒξ€‚a·Deα|
lim
Ξ±|ξ€Šx, DaneΞ±ξ€ƒβˆ’DaeΞ±ξ€ƒβˆ’Οƒξ€‚anDeασaDeα|
≀lim
αxDaneΞ±ξ€ƒβˆ’Daeα βˆ’ξ€‚Οƒξ€‚a0·DeΞ±ξ€ƒβˆ’Οƒξ€‚anDeα
≀lim
αxDaneΞ±βˆ’aeασanξ€ƒβˆ’Οƒξ€‚aDeα
≀lim
αxDanβˆ’aeασanβˆ’aDeΞ±ξ€ƒξ€„ξ€ƒβˆ’β†’ 0,
4.24
so 
Dis continuous.
14 Abstract and Applied Analysis
It remains to show that 
Dis a Οƒ-derivation. From the definition of the strict topology,
we have aeΞ±β†’ain the strict topology for all a∈Abecause aeΞ±iiaeαα
→aiiai∈
Iand so 
Daeαwβˆ—
→
Da. Therefore,

Dabξ€ƒξ€„Ο‰βˆ—βˆ’lim
Ξ±lim
β
Daeαbeβ
ξ€„Ο‰βˆ—βˆ’lim
Ξ±lim
Ξ²Daeαbeβ
ξ€„Ο‰βˆ—βˆ’lim
Ξ±lim
βσaeαDbeβDaeα·σbeβ
ξ€„Ο‰βˆ—βˆ’lim
Ξ±lim
Ξ²ξ€ˆΟƒξ€‚aeα
Dbeβ
Daeα·σbeβ
σa
Db
Daσb,
4.25
that is, 
Dis Οƒ-derivation.
Corollary 4.15. Suppose that Ais Οƒ-a.a, where Οƒis bounded idempotent endomorphism of A,I is a
closed ideal in A.IfIhas a bounded approximate identity, then Iis Οƒ-a.a.
Proof. Suppose that Ihas a bounded approximate identity, Xis a Οƒ-neo-unital Banach I-
bimodule, and D:I→X
βˆ—is a Οƒ-derivation. By Proposition 4.14,Xbecomes to a Banach
A-bimodule and Dhas a unique extension 
D:A→X
βˆ—which is a Οƒ-derivation. Since Ais
Οƒ-a.a,
βˆƒ{xβˆ—
Ξ±}βŠ†X
βˆ—s.t.
Dalim
ασa·xβˆ—
Ξ±βˆ’xβˆ—
α·σa
a∈A
.4.26
So we have Di 
Dilimασi·xβˆ—
Ξ±βˆ’xβˆ—
α·σi, which shows that Dlimαδσ
xβˆ—
Ξ±is Οƒ-a.i, and
Iis Οƒ-a.a.
Corollary 4.16. Let Abe an a.a Banach algebra and Ia closed ideal of A.ThenA/I is Οƒ-a.a for each
bounded endomorphism Οƒof A/I.
Proposition 4.17. Let Ibe a closed ideal of Asuch that σIξ€ƒβŠ†I.IfAis Οƒ-a.a, then A/I is σ-a.c,
where σis an endomorphism of A/I induced by Οƒ(i.e., σaξ€…IσaIfor a∈A.
Proof. Let Xbe a Banach A/I-bimodule and D:A/I β†’Xaσ-derivation. Then Xbecomes
an A-bimodule with the following module actions:
aΒ·xπa·x, x Β·ax·πa
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 Οƒ-derivation. Since Ais Οƒ-a.c, there exists a net {xΞ±}βŠ†Xsuch that Doπa
limασa·xΞ±βˆ’xα·σaa∈A. Therefore, for each a∈A,
Daξ€…IDoπalim
ασa·xΞ±βˆ’xα·σa
lim
απσa Β·xΞ±βˆ’xα·πσa
lim
ασaI·xΞ±βˆ’xα·σaI
lim
ασaξ€…IxΞ±βˆ’xασaξ€…I.
4.28
Thus, A/I is σ-a.c.
Proposition 4.18. Suppose that Iis a closed ideal in A.IfIis Οƒ-amenable and A/I is a.a, then Ais
Οƒ-a.a.
Proof. Let Xbe a Banach A-bimodule and D:A→X
βˆ—aΟƒ-derivation. Xis a Banach I-
bimodule too.
Clearly, dD|I:Iβ†’X
βˆ—is a Οƒ-derivation, and by Οƒ-amenability of Ithere exists
xβˆ—
0∈X
βˆ—such that Dδσ
xβˆ—
0, and, therefore, for each i∈Iwe have diσi·xβˆ—
0βˆ’xβˆ—
0·σi.Set
D1Dβˆ’Ξ΄Οƒ
xβˆ—
0. Clearly, D1is Οƒ-derivation and D1|I0. Now let X0spanX·σIβˆͺσI·X·
X/X0is a Banach A/I-bimodule via the following module actions:
aξ€…Ixξ€…X0σaxξ€…X0,xξ€…X0aξ€…IxσaX0x∈X,a∈A
.4.29
Now we define

D:A
Iβˆ’β†’ X
X0ξ€‡βˆ—
;ξ€Œxξ€…X0,
Daξ€…Iξ€ƒξ€ξ€„ξ€Šx, D1aa∈A,x ∈X
.4.30
Let aξ€…IaIand xξ€…X0xX0for some a, aξ€‘βˆˆAand x, xξ€‘βˆˆX.Soaβˆ’aξ€‘βˆˆI, and we
have D1aβˆ’a0. Thus, D1aDa. Now we have
ξ€Œxξ€…X0,
Daξ€…Iξ€ƒξ€ξ€„ξ€ŒxX0,
DaI.4.31
Thus, ξ€Šx, D1aξ€ƒξ€‹ξ€„ξ€Šx,D
1aξ€‘ξ€ƒξ€‹ξ€„ξ€Šx,D
1a, and, therefore,
ξ€Šxβˆ’x,D
1a0.4.32
It is enough to show that D1ais zero on X0. Suppose that σix∈X
0, we have
ξ€ŠΟƒξ€‚ix, D1aξ€ƒξ€‹ξ€„ξ€Šx, D1aσiξ€ƒξ€‹ξ€„ξ€Šx, D1aiξ€ƒβˆ’Οƒξ€‚aD1i0,
ξ€Šxσi,D
1aξ€ƒξ€‹ξ€„ξ€Šx, σiD1aξ€ƒξ€‹ξ€„ξ€Šx, D1iaξ€ƒβˆ’D1iσa0.
4.33
16 Abstract and Applied Analysis
So for all a∈A,D1a0onσI·XβˆͺX·σIand so for all a∈A,D1a0onX0. Since
xβˆ’xξ€‘βˆˆX
0, therefore ξ€Šxβˆ’x,D
1a0 which shows that D1is well defined. We claim that

Dis a derivation;
ξ€Œxξ€…X0,
Daξ€…Ibξ€…Iξ€ƒξ€ƒξ€ξ€„ξ€Šx, D1ab
ξ€„ξ€Šx, σaD1bD1aσb
ξ€„ξ€Šxσa,D
1bξ€ƒξ€‹ξ€…ξ€ŠΟƒξ€‚bx, D1a
ξ€„ξ€ŒxσaX0,
Dbξ€…I
ξ€…ξ€ŒΟƒξ€‚bxξ€…X0,
Daξ€…I
ξ€„ξ€Œξ€‚xξ€…X0aξ€…I,
Dbξ€…I
ξ€…ξ€Œξ€‚bξ€…Ixξ€…X0,
Daξ€…I.
4.34
So there exists a net ξ€‚Ο•Ξ±ξ€ƒβŠ†ξ€‚X/X0ξ€ƒβˆ—such that 
Dlimαδϕα.Letq:Xβ†’X/X0be the
quotient map. For every Ξ±,ϕαoqξ€ƒβˆˆX
βˆ—.Setxβˆ—
αϕαoqξ€ƒβŠ†X
βˆ—. We have
ξ€Šx, D1aξ€ƒξ€‹ξ€„ξ€Œxξ€…X0,
Daξ€…I
xξ€…X0,lim
αaξ€…Iξ€ƒΟ•Ξ±βˆ’Ο•Ξ±ξ€‚aξ€…I
lim
Ξ±ξ€ŠxσaX0,Ο•
Ξ±ξ€‹βˆ’ξ€ŠΟƒξ€‚axξ€…X0,Ο•
α
lim
Ξ±ξ€Šqxσa,Ο•
Ξ±ξ€‹βˆ’ξ€Šqσax,Ο•
α
lim
αϕαoqxσaξ€ƒβˆ’Οƒξ€‚axξ€ƒξ€„ξ€Šxσaξ€ƒβˆ’Οƒξ€‚ax, xβˆ—
α
lim
Ξ±ξ€Šx, σaxβˆ—
Ξ±βˆ’xβˆ—
ασa
x, lim
αδσ
xβˆ—
αa.
4.35
So D1Dβˆ’Ξ΄Οƒ
xβˆ—ξ€„limαδσ
xβˆ—
Ξ±, and, therefore, Dlimαδσ
xβˆ—
Ξ±βˆ’xβˆ—
0. Which shows that Dis Οƒ-a.i and
so Ais Οƒ-a.a.
Example 4.19. Let Abe a Banach algebra and let 0 /
ξ€„Ο•βˆˆBallAβˆ—ξ€ƒ. Then Awith the product
aΒ·aϕaabecomes a Banach algebra. We denote this algebra with AΟ•.Itiseasytoseethat
AΟ•has a left identity e, while it has not right approximate identity, so AΟ•is not contractible
and is not approximately contractible. Also AΟ•is biprojective. Now suppose that Οƒ:AΟ•β†’
AΟ•be defined by σaϕae. We have
Οƒ2aσϕaeϕaσeϕaϕeeϕaeσa.4.36
Abstract and Applied Analysis 17
Thus Οƒis idempotent. It is easy to see that eis identity for σAϕ, and since Ais biprojective
by 1, Corollary 5.3,AΟ•is Οƒ-biprojective. Thus by 1, Theorem 4.3,AΟ•is Οƒ-contractible and
so AΟ•is Οƒ-a.c.
It is easy to see that ker Ο•and all subspaces of ker Ο•are all ideals of AΟ•and σker Ο•ξ€ƒβŠ†
ker Ο•so σIξ€ƒβŠ†Ifor each ideal of A. Therefore, by Proposition 4.17,AΟ•/I is σ-a.c for each
ideal Iof AΟ•, where σaξ€…IσaIϕaeξ€…I.
Corollary 4.20. Suppose that Οƒis a bounded idempotent endomorphism of Banach algebra A.Then
Ais Οƒ-a.a if and only if there are nets μ
αin A
βŠ—Aξ€ƒβˆ—βˆ— and Fα,GΞ±ξ€ƒβŠ†A
βˆ—βˆ—, such that for each
a∈A,
1σa·μ
Ξ±βˆ’ΞΌξ€‘ξ€‘
α·σaFΞ±βŠ—Οƒξ€‚aξ€ƒβˆ’Οƒξ€‚aξ€ƒβŠ—GΞ±β†’0,
2σa·Fα→σa,G
α·σa→σa,
3ξ€ƒΟ€βˆ—βˆ—ξ€‚ΞΌξ€‘ξ€‘
α·σaξ€ƒβˆ’Fα·σaξ€ƒβˆ’Gα·σa→0.
Proof. Suppose that Ais Οƒ-a.a, take the net μαgiven in Proposition 4.9 and write
μαμ
Ξ±βˆ’FΞ±βŠ—ξ€„eβˆ’ξ€„eβŠ—GΞ±ξ€…cαeβŠ—ξ€„e, 4.37
where μ
Ξ±ξ€ƒβŠ†ξ€‚A
βŠ—Aξ€ƒβˆ—βˆ—,Fα,GΞ±ξ€ƒβŠ†Aβˆ—βˆ— ,andcΞ±ξ€ƒβŠ†C. Applying Ο€βˆ—βˆ— ,Ο€βˆ—βˆ—ξ€‚ΞΌξ€‘ξ€‘
Ξ±ξ€ƒβˆ’FΞ±βˆ’GΞ±ξ€…cαeβ†’
e, hence cΞ±β†’1, then
Ο€βˆ—βˆ—ξ€‚ΞΌξ€‘ξ€‘
α·σaξ€ƒβˆ’Fα·σaξ€ƒβˆ’Gα·σae·σaξ€ƒβˆ’β†’ e·σa
a∈A
.4.38
So we have iiifurther, by Proposition 4.9,fora∈A
#,
σa·μ
Ξ±βˆ’ξ€„Οƒξ€‚a·FΞ±βŠ—ξ€„eβˆ’ξ€„Οƒξ€‚aξ€ƒβŠ—Gασaξ€ƒβŠ—ξ€„e
μ
α·σaFΞ±βŠ—ξ€„Οƒξ€‚aeβŠ—Gα·σaξ€ƒβˆ’ξ€„eβŠ—ξ€„Οƒξ€‚aξ€ƒβˆ’β†’ 0.
4.39
Thus σa·μ
Ξ±βˆ’ΞΌξ€‘ξ€‘
α·σaFΞ±βŠ—ξ€„Οƒξ€‚aξ€ƒβˆ’ξ€„Οƒξ€‚aξ€ƒβŠ—GΞ±β†’0, and σa·Fα→σa,G
α·σa→σa.
So for a∈A,
σa·μ
Ξ±βˆ’ΞΌξ€‘ξ€‘
α·σaFΞ±βŠ—Οƒξ€‚aξ€ƒβˆ’Οƒξ€‚aξ€ƒβŠ—GΞ±βˆ’β†’ 0,
σa·FΞ±βˆ’β†’ σa,G
α·σaξ€ƒβˆ’β†’ σa.
4.40
18 Abstract and Applied Analysis
Conversely, set cα1andμαμ
Ξ±βˆ’FΞ±βŠ—ξ€„eβˆ’ξ€„eβŠ—GαeβŠ—ξ€„e. We have
σaξ€…Ξ±ξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·ξ€„Οƒξ€‚aασaξ€ƒξ€…Ξ±ξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·ξ€‚Οƒξ€‚aα
σaξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·Οƒξ€‚aaΞΌΞ±βˆ’Ξ±ΞΌΞ±
σaξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·Οƒξ€‚a
σa·μ
Ξ±βˆ’Οƒξ€‚aFΞ±βŠ—eβˆ’Οƒξ€‚aξ€ƒβŠ—GΞ±
σaξ€ƒβŠ—eξ€‚βˆ’ΞΌξ€‘ξ€‘
α·σa
ξ€…FΞ±βŠ—Οƒξ€‚aeβŠ—Gασaξ€ƒβˆ’eβŠ—Οƒξ€‚a
σa·μ
Ξ±βˆ’ΞΌξ€‘ξ€‘
α·σa
ξ€…FΞ±βŠ—Οƒξ€‚aξ€ƒβˆ’Οƒξ€‚aξ€ƒβŠ—GΞ±β†’0a∈A
.
4.41
So σaξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·ξ€„Οƒξ€‚a→0a∈A
#.Also
Ο€βˆ—βˆ—ξ€‚ΞΌΞ±ξ€ƒΒ·Οƒξ€‚aξ€ƒξ€„Ο€βˆ—βˆ— μ
Ξ±βˆ’FΞ±βŠ—ξ€„eβˆ’ξ€„eβŠ—GαeβŠ—ξ€„eσa
ξ€„Ο€βˆ—βˆ—ξ€‚ΞΌξ€‘ξ€‘
ασaξ€ƒβˆ’Fα·σa
βˆ’Gα·σaσaξ€ƒβˆ’β†’ σa
a∈A
,
4.42
and so Ο€βˆ—βˆ—ξ€‚ΞΌΞ±ξ€ƒβ†’ξ€„e.Now,byProposition 4.9,Ais Οƒ-a.a.
For Οƒ-approximate contractibility we have the following parallel result.
Proposition 4.21. Ais Οƒ-a.c if and only if any of the following equivalent conditions hold:
1there is a net ξ€‚ΞΌΞ±ξ€ƒβŠ‚A
#
βŠ—A#such that for each a∈A
#,σaξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·Οƒξ€‚a→0and
πμα→e;
2there is a net μ
Ξ±ξ€ƒβŠ‚A
#
βŠ—A#such that for each a∈A
#,σa·μ
Ξ±βˆ’ΞΌξ€‘
α·σa→0and
πμ
αe;
3there are nets μ
Ξ±ξ€ƒβŠ‚A

βŠ—A,Fα,GΞ±ξ€ƒβŠ‚A, such that for each a∈A,
iσa·μ
Ξ±βˆ’ΞΌξ€‘ξ€‘
α·σaFΞ±βŠ—Οƒξ€‚aξ€ƒβˆ’Οƒξ€‚aξ€ƒβŠ—GΞ±β†’0;
iiσi·Fα→σa,G
α·σa→σa;
iiiπμ
α·σaξ€ƒβˆ’Fα·σaξ€ƒβˆ’Gα·σa→0.
We know Banach algebra Ais amenable if and only if Ahas bounded approximate
diagonal 3.
Proposition 4.22. Banach algebra Ais Οƒ-amenable if and only if Ahas bounded approximate Οƒ-
diagonal, that is, there is a bounded net ξ€‚ΞΌΞ±ξ€ƒβŠ†A

βŠ—A such that for each a∈A,σaξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·Οƒξ€‚a→
0and πμα·σa→σa.
Proposition 4.23. If Banach algebra Ais Οƒ-amenable, then Ais Οƒ-a.c.
Abstract and Applied Analysis 19
Proof. Suppose that Ais Οƒ-amenable. Then there exists a bounded net μαin AβŠ—Asuch that
for each a∈A,
σaξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·Οƒξ€‚aξ€ƒβˆ’β†’ 0,Ο€
μα·σaξ€ƒβˆ’β†’ σa.4.43
Set fαπμα. It is easy to see that fαis a bounded approximate identity. Then μ
α
ΞΌΞ±ξ€…fΞ±βŠ—fΞ±and FαGαfΞ±satisfy i–iiiof Proposition 4.21, because
iσa·μ
Ξ±βˆ’ΞΌξ€‘ξ€‘
α·σafΞ±βŠ—Οƒξ€‚aξ€ƒβˆ’Οƒξ€‚aξ€ƒβŠ—fασaξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·Οƒξ€‚aσafΞ±βŠ—fΞ±βˆ’fΞ±βŠ—
σafΞ±βŠ—Οƒξ€‚aξ€ƒβˆ’Οƒξ€‚aξ€ƒβŠ—fΞ±β†’0a∈A,
iiσa·fασa·πμα→σa,fα·σaπμα·σa→σa,
iiiπμ
α·σaπμαfΞ±βŠ—fα·σafα·σaf2
α·σa.
So
πμ
α·σaξ€ƒβˆ’Fα·σaξ€ƒβˆ’Gα·σafα·σaf2
α·σaξ€ƒβˆ’fα·σaξ€ƒβˆ’fα·σaξ€ƒβˆ’β†’ 0.4.44
Note that f2
Ξ±is a bounded approximate identity too, thus, by Proposition 4.21,Ais Οƒ-a.c.
Corollary 4.24. Suppose that Ais a Οƒ-a.a Banach algebra where Οƒis an idempotent endomorphism of
Aand Iis a closed two-sided ideal of Awhich σIhas a bounded approximate identity and σIξ€ƒβŠ†I.
Then, Iis Οƒ-a.a.
Proof. Let {eΞ±}be a bounded approximate identity in σI,so{eΞ±}is bounded net in σIξ€ƒβˆ—βˆ—,
and so by Banach-Alaoglu theorem there exists a subnet {eΞ²}βŠ†{eΞ±}and EβˆˆΟƒξ€‚Iξ€ƒβˆ—βˆ— such that
eΞ²
wβˆ—
β†’E.Eis a right identity in σIξ€ƒβˆ—βˆ— because for each FβˆˆΟƒξ€‚Iξ€ƒβˆ—βˆ— and fβˆˆΟƒξ€‚Iξ€ƒβˆ—,
ξ€Šf, FEξ€‹ξ€„ξ€ŠfΒ·F, Elim
Ξ²ξ€ŠeΞ²,fFlim
Ξ²ξ€ŠeΞ²f, Fξ€‹ξ€„ξ€Šf, F.4.45
Also EactsasanidentityonσIitself. Let μα,Fα,Gαbe the nets given by
Corollary 4.20 for A. Define μ
αEΒ·ΞΌΞ±Β·Eβˆˆξ€‚I
βŠ—Iξ€ƒβˆ—βˆ—,F
αEΒ·Fα∈Iβˆ—βˆ—,andG
αGΞ±Β·E∈Iβˆ—βˆ—.
Then, for i∈I,
iwe consider
σi·μ
Ξ±βˆ’ΞΌξ€‘
α·σiF
Ξ±βŠ—Οƒξ€‚iξ€ƒβˆ’Οƒξ€‚iξ€ƒβŠ—G
Ξ±
σi·EΒ·ΞΌΞ±Β·Eβˆ’EΒ·ΞΌΞ±Β·E·σiEΒ·FΞ±βŠ—Οƒξ€‚iξ€ƒβˆ’Οƒξ€‚iξ€ƒβŠ—GΞ±Β·E
σi·μα·Eβˆ’E·μασiEΒ·FΞ±βŠ—Οƒξ€‚iξ€ƒβˆ’Οƒξ€‚iξ€ƒβŠ—GΞ±Β·E
E·σi·μα·Eβˆ’E·μα·σi·E
ξ€…EΒ·FΞ±βŠ—Οƒξ€‚i·Eβˆ’E·σiξ€ƒβŠ—GΞ±Β·E
Eσiξ€ƒΒ·ΞΌΞ±βˆ’ΞΌΞ±Β·Οƒξ€‚iFΞ±βŠ—Οƒξ€‚iξ€ƒβˆ’Οƒξ€‚iξ€ƒβŠ—Gα·Eβˆ’β†’ 0,
4.46
20 Abstract and Applied Analysis
iiwe consider
σi·F
ασi·EΒ·Fασi·FΞ±βˆ’β†’ σi,
G
α·σiGΞ±Β·E·σiGα·σiξ€ƒβˆ’β†’ σi
4.47
iiiwe consider
Ο€βˆ—βˆ—ξ€‚ΞΌξ€‘
α·
σaξ€ƒβˆ’F
α·
σaξ€ƒβˆ’G
Ξ±βˆ’ξ€‚
σa
ξ€„Ο€βˆ—βˆ—ξ€‚EΒ·ΞΌΞ±Β·E·σaξ€ƒβˆ’EΒ·Fα·σaξ€ƒβˆ’GΞ±Β·E·σa
EΒ·Ο€βˆ—βˆ—ξ€‚ΞΌΞ±ξ€ƒΒ·E·σaξ€ƒβˆ’EΒ·Fα·σaξ€ƒβˆ’Gα·σa
EΒ·Ο€βˆ—βˆ—ξ€‚ΞΌΞ±ξ€ƒΒ·Οƒξ€‚aξ€ƒβˆ’EΒ·Fα·σaξ€ƒβˆ’Gα·σaξ€ƒβˆ’EΒ·GασaEΒ·Gασa
EΒ·ξ€‚Ο€βˆ—βˆ—ξ€‚ΞΌΞ±ξ€ƒΒ·Οƒξ€‚aξ€ƒβˆ’FΞ±.σaξ€ƒβˆ’GασaEβˆ’ξ€„eGασaξ€ƒβˆ’β†’ 0.
4.48
An alternative proof would be to follow the standard argument stated in Corollary 4.15.
References
1P. C. Curtis Jr. and R. J. Loy, β€œThe structure of amenable Banach algebras,” Journal of the London
Mathematical Society, vol. 40, no. 1, pp. 89–104, 1989.
2M. Eshaghi Gordji, β€œPoint derivations on second duals and unitization of Banach algebras,” Nonlinear
Functional Analysis and Applications, vol. 13, no. 2, pp. 271–275, 2008.
3M. Eshaghi Gordji, β€œHomomorphisms, amenability and weak amenability of Banach algebras,”
Vietnam Journal of Mathematics, vol. 36, no. 3, pp. 253–260, 2008.
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