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## Abstract

We investigate σ -approximate contractibility and σ -approximate amenability of Banach algebras, which are extensions of usual notions of contractibility and amenability, respectively, where σ 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
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
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 c0
such that for a∈A,x ∈X, we have
a·xcax,x·acax.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·xx·aδxa
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
xaσa·xx·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,Dalimασa·xαxα·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 Daσais a σ-derivation, and so there is a net {uα}⊆X
Asuch that Dlimαδσ
uα. Hence for each aA,
αδσ
uαalim
ασa·uαuα·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:
axϕa·x, x ax·ϕa
a∈A,x ∈X
.2.3
For each a, b ∈A, we have
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
ασaxαxασa
lim
αϕoσa·xαxα·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
doσalimασaxαxασa.Nowforb∈Athere exists a∈Asuch that bσa,and,
therefore,
dbdσa lim
αxασaσaxα
lim
αxαbbxα,
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:
axσϕa·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
Doϕalimαδxαa. We have
αδxαalim
αaxαxαa
lim
ασϕaxαxασϕaa∈A
.
2.8
Since ϕis an epimorphism, so for each b∈Bthere exists aAsuch that bϕa, and we
have
Dblim
ασbxαxασb,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:
axϕa·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 Doϕalimαδσ
xαa, so we have
Dϕalim
ασaxαxασa
lim
αϕσa ·xαxα·ϕσa
lim
ατϕa·xαxα·τϕaa∈A
.
2.11
Since ϕis epimorphism, so Dblimατbxαxατbfor 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,weseethatσee,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 ηαe·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·eand Ais σ-a.a, D1:A→e·X·eis σ-a.i. Hence
D1limαδσ
ηαfor some net ηαe·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:
axσax, x aa
a∈A,x ∈X
.3.1
If D:A→X
is a σ-derivation, then De0.
Proof. We have DeDeeσeDeDeσeand
ex, Deσex, Deσeex, Deσeσe
x, Deσeex, Dex∈X
.
3.2
Hence DeσeDeand so σeDe0. Hence De0.
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, σaασaα.
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 D1alimγσax
γx
γσa,a∈A.
Since σe·X
·σeis σ-unital, by Lemma 3.7,D1e0 and for each aα∈A
#we have
D1aαD1aαD1eD1alim
γσax
γx
γσa
lim
γσaααx
γx
γσaαα
lim
γϕaαx
γx
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 ∈XC. Deﬁne D#:A#→X
with D#aαDa. 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
γσaαx
γx
γσaαlim
γσax
γx
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, Xis a Banach A-
bimodule with trivial right action, that is,
x·a0,a·xaxx∈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.
σax
0and so Daσax
0. Hence,
Daσax
0x
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
σAXσAspan 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,eαxxand xeαx.
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
zspan σA·X, so there exist a1,...,a
n∈A,x1,...,x
n∈Xsuch that zn
i1σaixi. For
1in,eαaiaiand, therefore, σeαzz.
Now suppose that zσAX. There exists {zn}⊆span σA·Xsuch that znz.
Thus,
n0Ns.t.nnn0;znz<ε
2σm14.2
On the other hand, for each nNwe have σeαzn
α
znand so σeαzn0
α
zn0.
Therefore,
α0;ααα0;σeαzn0zn0<ε
2.4.3
Now we have
σeαzzσeαzσeαzn0σeαzn0zn0zn0z
σeαzzn0σeαzn0zn0zn0z
<σeα1zzn0ε
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·YσA·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 BXby setting Tαx
x·σeαx∈X
. We have Tα≤σm.ThusTαis a bounded net in BXsince
BXX⊗Xand ball BXis w-compact, so there exists T∈BXsuch that we
Abstract and Applied Analysis 9
may suppose that wlimαTαTand T≤σm. For each a∈A,x∈X,andx∈X
,we
have
σa·x, T xlim
ασa·x, x·σeα
lim
ασeασa·x, x
σa·x, x,
4.5
and so xTxσA·X. On other hand, for each x∈X
,
T2xTTxlim
αTxσeαlim
αxσeαTx.4.6
Thus Tis projection, and IXT:XσA·Xis projection. So σA·Xis weakly
complemented in Xand, we have XσA·XσA·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 IT:X→Y
are projection maps. Let D:A→X
be a σ-derivation, so ToD and IToD are σ-derivations and DToDIToD.
Since A·X/Y{0}by Lemma 4.1,IToD is σ-inner. So there exists J0∈Y
such that
IToD δσ
J0.ThusDToD δσ
J0and so Dis σ-a.i if and only if ToD :A→Y
is σ-a.i.
Now let ZσA.ByProposition 4.6,Zis a closed weakly complemented in Y,
and T:Y→Z
and IT:Y→Z
are projection maps. Assume that D1:A→Y
is a σ-
derivation, thus ToD and IToD are σ-derivations, and we have D1ToD1IT·D1.
Since Y/Z·A {0},byLemma 4.1,IT·D1is σ-inner and so there exists z0Z
such that IToD1δσ
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:AXis
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 σaασaα. The following statements are equivalent.
1Ais σ-a.a.
2ThereisanetμαA#
⊗A#∗∗ such that for each a∈A
#,σa·μαμα·σa0and
π∗∗μαe.
10 Abstract and Applied Analysis
3Thereisanetμ
αA#
⊗A#∗∗ such that for each a∈A
#,σa·μαμα·σa0and
for every α,π∗∗μ
αe.
Proof. 13Suppose that Ais σ-a.a, by Proposition 3.8,A#is σ-a.a. Let uee∈A
#
⊗A#.
A#
⊗A#is a Banach A#-bimodule with the following module actions:
a·bcσabc,bc·abcσaa, b, c ∈A
#.4.7
Set δu:A#ker π∗∗ with deﬁnition δuaσa·uu·σaa∈A
#.δuis σ-derivation.
Recall that ker π∗∗ ker π∗∗. Since A#is σ-a.a, thus there exists eαker π∗∗ such that
δualim
ασaeαeασaa∈A
#.4.8
Set μ
αueαA#
⊗A#∗∗. We have
σaμ
αμ
ασaσauuσaσaeαeασa −→ 0,4.9
and for each α,
π∗∗μ
απ∗∗ueαπ∗∗ uπ∗∗ eαπue. 4.10
32is clear.
21By Proposition 3.9,itissucient 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 αI,setfαxμαψx, where for a, b ∈A
#,
x∈X, we have ψxabx, σaDb. Then, mγ
αA#
⊗A#converging ωto μααI
and noting that for m∈A
#
⊗A#,a ∈A
#,x∈X, then
ψσaxxσamσaψxψxσamx, σπmDa.4.11
Since Xis σ-neo-unital, so XXσA#. So for each aAand x∈X, we have
σaxxσa,f
αψσaxxσa
α
lim
γmγ
ασaxxσa
σaψxψxσa
αlim
γx, σπmγ
αDa
ψx
ασaσaμαx, π∗∗ μαDa.
4.12
Abstract and Applied Analysis 11
Thus,
x, σafαfασax, Da
ψx,σaμαμασa
x
π∗∗μαe
Da
D·x
σaμαμασa
x
πμαe
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
limασaGαGασa  0for a∈A, and a net Fα:Y→X
of left inverse maps to fsuch that
limασafαfασa  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,Xsuch that
σa·GG·σalim
ασa·fGαfGα·σaa∈A
.4.15
Setting GαGαgives 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,thenσeααis a bounded
approximate identity for X.
Proof. For every a∈Awe have eασaσa. Since σis idempotent, σeασaσa.
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 PiiI, where
Pia:aiiaa∈A
.4.17
Note that the strict topology is Hausdoronly 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 iIand
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
iIand 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−→ ωlim
αDaeασa·Deα.4.19
We claim that
Dis well-deﬁned, that is, the limit in 4.19does exist. Let x∈X,andletiI
and y∈Xsuch that xy·σi.ByLemma 4.13,σeαis bounded approximate identity for
X, and we have
x, Daeασa·Deαy·σi,D
aeασa·Deα
y, σiDaeασia·Deα
Abstract and Applied Analysis 13
y, DiaeαDiσaeαDiaeαDiaσeα
σeα·y, Diaσaeα·y, Di
α
−→ y, Diaσa·y, Dia∈A
.
4.20
So the limit in 4.19exists. Furthermore, for iI,
Diωlim
αDieασi·Deα
ωlim
αDieαDieαDiσeαDi,
4.21
so
Dis an extension of D.Alsofora∈Aand iIwe have
Da·σiωlim
αDaeα·σiσa·Deα·σi
ωlim
αDaeαiσaeα·Diσa·Deαiσaσeα·Di
ωlim
αDaeαiσa·Deαi Daiσa·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.
iI, aniian−→ aiia.4.23
For each x∈X,
x,
Danx,
Da
lim
α|x, Daneασan·Deαx, Daeασa·Deα|
lim
lim
αxDaneαDaeα σa0·DeασanDeα
lim
αxDaneαaeασanσaDeα
lim
αxDanaeασanaDeα−→ 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 aeαain the strict topology for all a∈Abecause aeαiiaeαα
→aiiai
Iand so
Daeαw
Da. Therefore,
Dabωlim
αlim
β
Daeαbeβ
ωlim
αlim
βDaeαbeβ
ωlim
αlim
βσaeαDbeβDaeα·σbeβ
ωlim
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
αx
α·σa
a∈A
.4.26
So we have Di
Dilimασi·x
αx
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 σII.IfAis σ-a.a, then A/I is σ-a.c,
where σis an endomorphism of A/I induced by σ(i.e., σaIσaIfor 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 :AX
becomes a σ-derivation. Since Ais σ-a.c, there exists a net {xα}⊆Xsuch that Doπa
limασa·xαxα·σaa∈A. Therefore, for each aA,
DaIDoπalim
ασa·xαxα·σa
lim
απσa ·xαxα·πσa
lim
ασaI·xαxα·σaI
lim
ασaIxαxασaI.
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 iIwe have diσi·x
0x
0·σi.Set
D1Dδσ
x
0. Clearly, D1is σ-derivation and D1|I0. Now let X0spanσIσI·X·
X/X0is a Banach A/I-bimodule via the following module actions:
aIxX0σaxX0,xX0aIaX0x∈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.SoaaI, and we
have D1aa0. Thus, D1aDa. Now we have
xX0,
DaIxX0,
DaI.4.31
Thus, x, D1ax,D
1ax,D
1a, and, therefore,
xx,D
1a0.4.32
It is enough to show that D1ais zero on X0. Suppose that σix∈X
0, we have
i,D
4.33
16 Abstract and Applied Analysis
So for all a∈A,D1a0onσI·X∪X·σIand so for all a∈A,D1a0onX0. Since
xx∈X
0, therefore xx,D
1a0 which shows that D1is well deﬁned. We claim that
Dis a derivation;
xX0,
DaIbIx, D1ab
a,D
1bσbx, D1a
aX0,
DbI
σbxX0,
DaI
xX0aI,
DbI
bIxX0,
DaI.
4.34
So there exists a net ϕαX/X0such that
Dlimαδϕα.Letq:X→X/X0be the
quotient map. For every α,ϕαoq∈X
.Setx
αϕαoq⊆X
. We have
x, D1axX0,
DaI
xX0,lim
αaIϕαϕαaI
lim
αaX0
ασaxX0
α
lim
αqa
αqσax
α
lim
αϕαoqaσaxaσax, x
α
lim
αx, σax
αx
ασa
x, lim
αδσ
x
αa.
4.35
So D1Dδσ
xlimαδσ
x
α, and, therefore, Dlimαδσ
x
αx
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 /
ϕBallA. 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 σaϕae. 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 σIIfor each ideal of A. Therefore, by Proposition 4.17,Aϕ/I is σ-a.c for each
ideal Iof Aϕ, where σaIσaIϕaeI.
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
αμ
α·σaFασaσaGα0,
2σa·Fασa,G
α·σaσa,
3π∗∗μ
α·σaFα·σaGα·σa0.
Proof. Suppose that Ais σ-a.a, take the net μαgiven in Proposition 4.9 and write
μαμ
αFαeeGαcαee, 4.37
where μ
αA
A∗∗,Fα,GαA∗∗ ,andcαC. Applying π∗∗ ,π∗∗μ
αFαGαcαe
e, hence cα1, then
π∗∗μ
α·σaFα·σaGα·σae·σa−→ e·σa
a∈A
.4.38
So we have iiifurther, by Proposition 4.9,fora∈A
#,
σa·μ
ασa·FαeσaGασae
μ
α·σaFασaeGα·σaeσa−→ 0.
4.39
Thus σa·μ
αμ
α·σaFασaσaGα0, and σa·Fασa,G
α·σaσa.
So for a∈A,
σa·μ
αμ
α·σaFασaσaGα−→ 0,
σa·Fα−→ σa,G
α·σa−→ σa.
4.40
18 Abstract and Applied Analysis
Conversely, set cα1andμαμ
αFαeeGαee. We have
σaα·μαμα·σaασaα·μαμα·σaα
σa·μαμα·σaααμα
σa·μαμα·σa
σa·μ
ασaFαeσaGα
σaeμ
α·σa
FασaeGασaeσa
σa·μ
αμ
α·σa
FασaσaGα0a∈A
.
4.41
So σa·μαμα·σa0a∈A
#.Also
π∗∗μα·σaπ∗∗ μ
αFαeeGαeeσa
π∗∗μ
ασaFα·σa
Gα·σaσa−→ σa
a∈A
,
4.42
and so π∗∗μαe.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·μαμα·σa0and
πμαe;
2there is a net μ
α⊂A
#
⊗A#such that for each a∈A
#,σa·μ
αμ
α·σa0and
πμ
αe;
3there are nets μ
α⊂A
⊗A,Fα,Gα⊂A, such that for each a∈A,
iσa·μ
αμ
α·σaFασaσaGα0;
iiσi·Fασa,G
α·σaσa;
iiiπμ
α·σaFα·σaGα·σa0.
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·μαμα·σa
0and πμα·σaσa.
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·μαμα·σa−→ 0
μα·σa−→ σa.4.43
Set fαπμα. It is easy to see that fαis a bounded approximate identity. Then μ
α
μαfαfαand FαGαfαsatisfy iiiiof Proposition 4.21, because
iσa·μ
αμ
α·σafασaσafασa·μαμα·σaσafαfαfα
σafασaσafα0a∈A,
iiσa·fασa·πμασa,fα·σaπμα·σaσa,
iiiπμ
α·σaπμαfαfα·σafα·σaf2
α·σa.
So
πμ
α·σaFα·σaGα·σafα·σaf2
α·σafα·σafα·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 σII.
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 EσI∗∗ such that
eβ
w
E.Eis a right identity in σI∗∗ because for each FσI∗∗ and fσI,
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·EI
I∗∗,F
αE·FαI∗∗,andG
αGα·EI∗∗.
Then, for iI,
iwe consider
σi·μ
αμ
α·σiF
ασiσiG
α
σi·E·μα·EE·μα·E·σiE·FασiσiGα·E
σi·μα·EE·μασiE·FασiσiGα·E
E·σi·μα·EE·μα·σi·E
E·Fασi·EE·σiGα·E
Eσi·μαμα·σiFασiσiGα·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·
σaF
α·
σaG
α
σa
π∗∗E·μα·E·σaE·Fα·σaGα·E·σa
E·π∗∗μα·E·σaE·Fα·σaGα·σa
E·π∗∗μα·σaE·Fα·σaGα·σaE·GασaE·Gασa
E·π∗∗μα·σaFαaGασaEeGασa−→ 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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