# σ-Approximately Contractible Banach Algebras

Authors: M. Momeni; T. Yazdanpanah; M. R. Mardanbeigi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We investigateσ-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.


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## Body

## 1. Introduction

For a Banach algebra𝒜, an 𝒜-bimodule will always refer to a Banach 𝒜-bimodule 𝒳, that is, 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
(1.1)‖a⋅x‖≤c‖a‖‖x‖,‖x⋅a‖≤c‖a‖‖x‖.
A derivation D:𝒜→𝒳 is a linear map, always taken to be continuous, satisfying
(1.2)D(ab)=D(a)⋅b+a⋅D(b)(a,b∈A).
A Banach algebra 𝒜 is amenable if for any 𝒜-bimodule 𝒳, any derivation D:𝒜→𝒳* is inner, that is, there exists x*∈𝒳*, with
(1.3)D(a)=a⋅x*-x*⋅a=δx*(a)(a∈A).
Let 𝒜 be a Banach algebra and σ a bounded endomorphism of 𝒜, that is, a bounded Banach algebra homomorphism from 𝒜 into 𝒜. A σ-derivation from 𝒜 into a Banach 𝒜-bimodule 𝒳 is a bounded linear map D:𝒜→𝒳 satisfying
(1.4)D(ab)=σ(a)⋅D(b)+D(a)⋅σ(b)(a,b∈A).
For each x∈𝒳, the mapping
(1.5)δxσ:A⟶X
defined by δxσ(a)=σ(a)·x-x·σ(a), for all a∈𝒜, is a σ-derivation called an inner σ-derivation.Remark 1.1.
Throughout this paper, we will assume that𝒜 is a Banach algebra, and σ is a bounded endomorphism of 𝒜 unless otherwise specified. Also, we write (σ-a.i) for σ-approximately inner, (σ-a.a) for σ-approximately amenable, and (σ-a.c) for σ-approximately contractible.The basic definition for the present paper is as follows.Definition 1.2.
Aσ-derivation D:𝒜→𝒳 is σ-a.i, if there exists a net (xα)⊆𝒳 such that for every a∈𝒜, D(a)=limασ(a)·xα-xα·σ(a), the limit being in norm and we write D=limδxασ. Note that we do not suppose (xα) to be bounded.Definition 1.3.
A Banach algebra𝒜 is called σ-a.c if for any 𝒜-bimodule 𝒳, every σ-derivation D:𝒜→𝒳 is σ-a.i.Definition 1.4.
A Banach algebra𝒜 is called σ-a.a if for any 𝒜-bimodule 𝒳, every σ-derivation D:𝒜→𝒳* is σ-a.i.Definition 1.5.
Let𝒜 be a Banach algebra, and let 𝒳 and 𝒴 be Banach 𝒜-bimodules. The linear map T:𝒳→𝒴 is called a σ-𝒜-bimodule homomorphism if
(1.6)T(a⋅x)=σ(a)⋅T(x),T(x⋅a)=T(x)⋅σ(a)(a∈A,x∈X).


## 2. Basic Properties

Proposition 2.1.
Let𝒜 be a σ-a.c Banach algebra. Then σ(𝒜) has a left and right approximate identity.Proof.
Consider𝒳=𝒜 as a Banach 𝒜-bimodule with the trivial right action, that is,
(2.1)a⋅x=ax,x⋅a=0(a∈A,x∈X).
Then D:𝒜→𝒳 defined by D(a)=σ(a) is a σ-derivation, and so there is a net {uα}⊆𝒳(=𝒜) such that D=limαδuασ. Hence for each a∈A,
(2.2)σ(a)=D(a)=lim⁡αδuασ(a)=lim⁡ασ(a)⋅uα-uα⋅σ(a)=lim⁡ασ(a)uα,
which shows that {uα} is a right approximate identity for σ(𝒜). Similarly, one can find a left approximate identity for σ(𝒜).Corollary 2.2.
Let𝒜 be a σ-a.c Banach algebra and σ a continuous epimorphism of 𝒜. Then 𝒜 has a left and right approximate identity.Proposition 2.3.
Letφ be a bounded endomorphism of Banach algebra 𝒜. If 𝒜 is σ-a.c, then 𝒜 is (φoσ)-a.c.Proof.
Let𝒳 be a Banach 𝒜-bimodule and let D:𝒜→𝒳 be a (φoσ)-derivation. Then (𝒳,*) is an 𝒜-bimodule with the following module actions:
(2.3)a*x=φ(a)⋅x,x*a=x⋅φ(a)(a∈A,x∈X).
For each a,b∈𝒜, we have
(2.4)D(ab)=(φoσ(a))⋅D(b)+D(a)⋅(φoσ(b))=σ(a)*D(b)+D(a)*σ(b).
Thus D:𝒜→(𝒳,*) is a continuous σ-derivation. Since 𝒜 is σ-a.c, there exists a net {xα}⊆𝒳 such that D=limδxασ. In fact,
(2.5)D(a)=lim⁡α(σ(a)*xα-xα*σ(a))=lim⁡α(φoσ(a)⋅xα-xα⋅φoσ(a))=lim⁡αδxαφoσ(a)(a∈A).
Therefore, D is a (φoσ)-a.i and so 𝒜 is (φoσ)-a.c.Corollary 2.4.
Let𝒜 be an a.c Banach algebra. Then 𝒜 is σ-a.c for each bounded endomorphism σ of 𝒜.Proposition 2.5.
Let𝒜 be a σ-a.c Banach algebra, where σ is a bounded epimorphism of 𝒜. Then 𝒜 is a.c.Proof.
Let𝒳 be a Banach 𝒜-bimodule and let d:𝒜→𝒳 be a continuous derivation. It is easy to see that doσ is a σ-derivation. Since 𝒜 is σ-a.c, there exists a net {xα}⊆X such that doσ(a)=limασ(a)xα-xασ(a). Now for b∈𝒜 there exists a∈𝒜 such that b=σ(a), and, therefore,
(2.6)d(b)=d(σ(a))=lim⁡αxασ(a)-σ(a)xα=lim⁡αxαb-bxα,
which shows that d is approximately inner and so 𝒜 is a.c.Corollary 2.6.
Letφ be a bounded endomorphism of Banach algebra 𝒜. If 𝒜 is σ-a.a then it is (φoσ)-a.a too.Corollary 2.7.
Let𝒜 be an a.a Banach algebra. For each bounded endomorphism σ, 𝒜 is σ-a.a.Corollary 2.8.
Let𝒜 be a σ-a.a Banach algebra, where σ is a bounded epimorphism of 𝒜. Then 𝒜 is a.a.Proposition 2.9.
Suppose thatℬ is a Banach algebra and φ:𝒜→ℬ is a continuous epimorphism. If 𝒜 is a.c, then ℬ is σ-a.c for each bounded endomorphism σ of ℬ.Proof.
Letσ:ℬ→ℬ be a bounded endomorphism of ℬ and 𝒳 a Banach ℬ-bimodule, then (𝒳,*) is an 𝒜-bimodule with the following module actions:
(2.7)a*x=σ(φ(a))⋅x,x*a=x⋅σ(φ(a))(a∈A,x∈X).
Now let D:ℬ→𝒳 be a continuous σ-derivation. It is easy to check that Doφ:𝒜→(𝒳,*) is a derivation. Since 𝒜 is approximately contractible, there exists a net {xα}⊆𝒳 such that Doφ(a)=limαδxα(a). We have
(2.8)D(φ(a))=Doφ(a)=lim⁡αδxα(a)=lim⁡α(a*xα-xα*a)=lim⁡ασ(φ(a))xα-xασ(φ(a))(a∈A).
Since φ is an epimorphism, so for each b∈ℬ there exists a∈A such that b=φ(a), and we have
(2.9)D(b)=lim⁡ασ(b)xα-xασ(b),
which shows that D is σ-a.i and so B is σ-a.c.Proposition 2.10.
Suppose that𝒜 and ℬ are Banach algebras, and let σ and τ be bounded endomorphism of 𝒜 and ℬ, respectively. Let φ:𝒜→ℬ be a bounded epimorphism such that φoσ=τoφ. If 𝒜 is σ-a.c, then ℬ is τ-a.c.Proof.
Let𝒳 be a Banach ℬ-bimodule and D:ℬ→𝒳 a continuous τ-derivation. Then (𝒳,*) is an 𝒜-bimodule with the following actions:
(2.10)a*x=φ(a)⋅x,x*a=x⋅φ(a)(a∈A,x∈X).
It is easy to check that Doφ:𝒜→(𝒳,*) is a σ-derivation. Since 𝒜 is σ-a.c, there exists a net {xα}⊆𝒳 such that Doφ(a)=limαδxασ(a), so we have
(2.11)D(φ(a))=lim⁡ασ(a)*xα-xα*σ(a)=lim⁡αφ(σ(a))⋅xα-xα⋅φ(σ(a))=lim⁡ατ(φ(a))⋅xα-xα⋅τ(φ(a))(a∈A).
Since φ is epimorphism, so D(b)=limατ(b)xα-xατ(b) for all b∈ℬ, and hence ℬ is τ-a.c.


## 3.σ-Approximate Contractibility for Unital Banach Algebras

In this section we state some properties ofσ-approximate contractibility when 𝒜 has 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.
Let𝒜 be a unital Banach algebra with unit e, σ(𝒜) dense in 𝒜, 𝒳 a Banach 𝒜-bimodule, and D:𝒜→𝒳 a σ-derivation. Then, there is a σ-derivation D1:𝒜→e·𝒳·e and η∈𝒳, such that D=D1+δη.The following definition extends the definition of the unital Banach𝒜-module in the classical sense.Definition 3.2.
Let𝒜 be a unital Banach algebra with identity e. Banach 𝒜-bimodule 𝒳 is called σ-unital if 𝒳=σ(e)·𝒳·σ(e).Corollary 3.3.
Let𝒜 be a unital Banach algebra and σ(𝒜) dense in 𝒜. Then, 𝒜 is 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.Proof.
Sinceσ(e) is a unit for σ(𝒜), and σ(A) is dense in 𝒜, we see that σ(e)=e, so that e·𝒳·e is a σ-unital Banach 𝒜-bimodule. Now by Proposition 3.1, the proof is complete.Corollary 3.4.
Suppose that𝒜 is a unital Banach algebra and σ(A) is dense in 𝒜. Let 𝒳 be a Banach 𝒜-bimodule and D:𝒜→𝒳* a σ-derivation. If 𝒜 is σ-a.a, then there exists a net (ηα)⊆e·𝒳*·e, and η∈𝒳*, such that D=limαδηασ+δη.Proof.
By Proposition3.1, D=D1+δη such that η∈𝒳* and D1:𝒜→e·𝒳*·e is a σ-derivation. Since e·𝒳*·e≅(e·𝒳·e)* and 𝒜 is σ-a.a, D1:𝒜→(e·𝒳·e)* is σ-a.i. Hence D1=limαδηασ for some net (ηα)⊆e·𝒳*·e.In the following proposition we considerσ-approximate contractibility when σ is an idempotent endomorphism of 𝒜. We can see the proof of the following proposition in [1, Proposition 4.1].Proposition 3.5.
Assume that𝒜 has an element e which is a unit for σ(𝒜) and 𝒳 is a Banach 𝒜-bimodule. If σ is a bounded idempotent endomorphism of 𝒜, then for each σ-derivation D:𝒜→𝒳 there exists a σ-derivation D1:𝒜→σ(e)·𝒳·σ(e) and η∈𝒳, such that D=D1+δη.Corollary 3.6.
Assume that𝒜 has an element e which is a unit for σ(𝒜) and σ is a bounded idempotent endomorphism of 𝒜, then 𝒜 is 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 that𝒜 is a unital Banach algebra with the identity e, and (𝒳,*) is a σ-unital Banach 𝒜-bimodule with the following module actions:
(3.1)a*x=σ(a)x,x*a=xσ(a)(a∈A,x∈X).
If D:𝒜→𝒳* is a σ-derivation, then D(e)=0.Proof.
We haveD(e)=D(ee)=σ(e)D(e)+D(e)σ(e) and
(3.2)〈e*x,D(e)σ(e)〉=〈x,D(e)σ(e)*e〉=〈x,D(e)σ(e)σ(e)〉=〈x,D(e)σ(e)〉=〈e*x,D(e)〉(x∈X).
Hence D(e)σ(e)=D(e) and so σ(e)D(e)=0. Hence D(e)=0.Proposition 3.8.
Letσ be a bounded idempotent endomorphism of Banach algebra 𝒜. If 𝒜 is σ-a.a, then 𝒜# is σ^-a.a, where σ^ is the endomorphism of 𝒜# induced by σ, that is, σ^(a+α)=σ(a)+α.Proof.
Let𝒳 be a Banach 𝒜#-bimodule and D:𝒜#→𝒳* a continuous σ^-derivation. By Proposition 3.5, there exits η∈𝒳* and D1:𝒜#→σ^(e)·𝒳*·σ^(e) such that D=D1+δη. Set d:D1|𝒜:𝒜→σ^(e)·𝒳*·σ^(e). It is easy to check that d is a σ-derivation. Since 𝒜 is σ-a.a, there exists a net (xγ*)⊆𝒳* such that d=limγδxγ*σ. Hence D1(a)=limγσ(a)xγ*-xγ*σ(a),(a∈𝒜). Since σ^(e)·𝒳*·σ^(e) is σ^-unital, by Lemma 3.7, D1(e)=0 and for each a+α∈𝒜# we have
(3.3)D1(a+α)=D1(a)+αD1(e)=D1(a)=lim⁡γσ(a)xγ*-xγ*σ(a)=lim⁡γ(σ^(a+α)-α)xγ*-xγ*(σ^(a+α)-α)=lim⁡γφ(a+α)xγ*-xγ*φ(a+α).
This shows that D1 is σ^-a.i, and so 𝒜# is σ^-a.a.Proposition 3.9.
Letσ be a bounded endomorphism of Banach algebra 𝒜. If A# is σ^-a.a, then 𝒜 is σ-a.a.Proof.
Let𝒳 be a Banach 𝒜-bimodule and D:𝒜→𝒳* a continuous σ-derivation. 𝒳 is a Banach A#-bimodule with the following module actions:
(3.4)(a+α)⋅x=a⋅x+αx,x⋅(a+α)=x⋅a+αx,
for all a∈𝒜,x∈𝒳,α∈ℂ. Define D#:𝒜#→𝒳* with D#(a+α)=D(a). Clearly, D# is a continuous σ^-derivation. Hence, there is a net (xγ*)⊆𝒳* such that D#=limγσ^δxγ*. Hence, for each a∈𝒜 we have
(3.5)D(a)=D#(a+α)=lim⁡γσ^(a+α)xγ*-xγ*σ^(a+α)=lim⁡γσ(a)xγ*-xγ*σ(a)
which shows that D is σ-a.i and so 𝒜 is σ-a.a.


## 4.σ-Approximate Amenability When 𝒜 Has a Bounded Approximate Identity

Lemma 4.1.
Let𝒜 be a Banach algebra with bounded approximate identity and 𝒳 a Banach 𝒜-bimodule with trivial left or right action, then every σ-derivation D:𝒜→𝒳* is σ-inner.Proof.
Let𝒳 be a Banach 𝒜-bimodule with trivial left action. Hence, 𝒳* is a Banach 𝒜-bimodule with trivial right action, that is,
(4.1)x*⋅a=0,a⋅x*=ax*(x*∈X*,a∈A).
Let D:𝒜→𝒳* be a continuous σ-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)→w*x0*, for some x0*∈𝒳*. Since a·eβ→∥·∥a and D is continuous, D(a·eβ)→∥·∥D(a). Hence, D(a·eβ)→w*D(a).
On the other hand,D(a·eβ)=σ(a)D(eβ)→w*σ(a)x0* and so D(a)=σ(a)x0*. Hence, D(a)=σ(a)x0*-x0*σ(a) and D is σ-inner.The following definitions extends the definition of the neo-unital and essential Banach𝒜-bimodule in the classical sense.Definition 4.2.
Let𝒳 be a Banach 𝒜-bimodule. Then 𝒳 is called σ-neo-unital (σ-pseudo-unital), if 𝒳=σ(𝒜)·𝒳·σ(𝒜). Similarly, one defines σ-neo-unital left and right Banach modules.Definition 4.3.
Let𝒳 be a Banach 𝒜-bimodule. Then 𝒳 is called σ-essential if 𝒳=σ(𝒜)𝒳σ(𝒜)=span¯σ(𝒜)·𝒳·σ(𝒜). Similarly, one defines σ-essential left and right Banach modules.
We recall that a bounded approximate identity in Banach algebra𝒜 for Banach 𝒜-bimodule 𝒳 is a bounded net (eα) in 𝒜 such that for each x∈𝒳, eαx→x and xeα→x.Proposition 4.4.
Assume that𝒜 has a left bounded approximate identity, σ is a bounded idempotent endomorphism of 𝒜, and 𝒳 is a left Banach 𝒜-module. Then 𝒳 is σ-neo-unital if and only if 𝒳 is σ-essential.Proof.
Let𝒳 be a σ-essential Banach 𝒜-bimodule. Since σ is idempotent, σ(𝒜) is Banach subalgebra of 𝒜. Let (eα)⊆𝒜 be left approximate identity with bound m. First suppose that z∈spanσ(𝒜)·𝒳, so there exist a1,…,an∈𝒜, x1,…,xn∈𝒳 such that z=∑i=1n‍σ(ai)xi. For 1≤i≤n, eαai→ai and, therefore, σ(eα)z→z.
Now suppose thatz∈σ(𝒜)𝒳. There exists {zn}⊆spanσ(𝒜)·𝒳 such that zn→z. Thus,
(4.2)∃n0∈Ns.t.∀n(n≥n0;‖zn-z‖<ɛ2(‖σ‖m+1))
On the other hand, for eachn∈ℕ we have σ(eα)zn→αzn and so σ(eα)zn0→αzn0. Therefore,
(4.3)∃α0;∀α(α≥α0;‖σ(eα)zn0-zn0‖<ɛ2).
Now we have
(4.4)‖σ(eα)z-z‖≤‖σ(eα)z-σ(eα)zn0+σ(eα)zn0-zn0+zn0-z‖≤‖σ‖‖eα‖‖z-zn0‖+‖σ(eα)zn0-zn0‖+‖zn0-z‖<(‖σ‖‖eα‖+1)‖z-zn0‖+ɛ2<(‖σ‖m+1)ɛ(‖σ‖m+1)2+ɛ2=ɛ,
which shows that (σ(eα))⊆σ(𝒜) is a left bounded approximate identity for 𝒳. Now by Cohen factorization Theorem, 𝒳=σ(𝒜)·𝒳. So 𝒳 is σ-neo-unital. The other side is trivial.Corollary 4.5.
Everyσ-neo-unital left Banach 𝒜-module is essential.Proof.
Let𝒳 be a σ-neo-unital left Banach 𝒜-module. We have 𝒳=σ(𝒜)·𝒳⊆𝒜·𝒳⊆𝒜𝒳⊆𝒳 so 𝒳=𝒜𝒳.Proposition 4.6.
Let𝒜 be a Banach algebra with a left bounded approximate identity, σ be a bounded idempotent endomorphism of 𝒜, and 𝒳 a left Banach 𝒜-module. Then σ(𝒜)·𝒳 is closed weakly complemented submodule of 𝒳.Proof.
Set𝒴=σ(𝒜)𝒳, since 𝒜 has 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 𝒴 is σ-essential by Proposition 4.4, 𝒴 is σ-neo unital that is, 𝒴=σ(𝒜)·𝒴. Hence, σ(𝒜)𝒳=𝒴=σ(𝒜)·𝒴⊆σ(𝒜)·𝒳 and so σ(𝒜)𝒳=σ(𝒜)·𝒳. Thus σ(𝒜)·𝒳 is closed submodule of 𝒳.
Now we prove thatσ(𝒜)·𝒳 is weakly complemented in 𝒳. Let (eα) be a left approximate identity in 𝒜 with bound m, and define a net (Tα) in ℬ(𝒳*) by setting Tα(x*)=x*·σ(eα)(x*∈𝒳*). We have ∥Tα∥≤∥σ∥m. Thus (Tα) is a bounded net in ℬ(𝒳*) since ℬ(𝒳*)=(𝒳*⊗𝒳)* and ball ℬ(𝒳*) is w*-compact, so there exists T∈ℬ(𝒳*) such that we may suppose that w*-limαTα=T and ∥T∥≤∥σ∥m. For each a∈𝒜, x∈𝒳, and x*∈𝒳*, we have
(4.5)〈σ(a)⋅x,T(x*)〉=lim⁡α〈σ(a)⋅x,x*⋅σ(eα)〉=lim⁡α〈σ(eα)σ(a)⋅x,x*〉=〈σ(a)⋅x,x*〉,
and so x*-Tx*∈(σ(𝒜)·𝒳)⊥. On other hand, for each x*∈𝒳*,
(4.6)T2x*=T(Tx*)=lim⁡αT(x*)σ(eα)=lim⁡αx*σ(eα)=T(x*).
Thus T is projection, and I𝒳*-T:𝒳*→(σ(𝒜)·𝒳)⊥ is projection. So σ(𝒜)·𝒳 is weakly complemented in 𝒳 and, we have 𝒳*=(σ(𝒜)·𝒳)⊥⊕(σ(𝒜)·𝒳)*.Corollary 4.7.
Let𝒜 have a bounded approximate identity, and let 𝒳 be a Banach 𝒜-bimodule and σ a bounded idempotent endomorphism of 𝒜. Then(i)
σ(𝒜)·𝒳·σ(𝒜) is a closed weakly complemented submodule of 𝒳,(ii)
𝒜 is 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.
Set𝒴=σ(𝒜)·𝒳. By Proposition 4.6, 𝒴 is a closed and weakly complemented submodule of 𝒳, and T:𝒳*→𝒴* and I-T:𝒳*→𝒴⊥ are projection maps. Let D:𝒜→𝒳* be a σ-derivation, so ToD and (I-T)oD are σ-derivations and D=(ToD)+(I-T)oD. Since A·(X/Y)={0} by Lemma 4.1, (I-T)oD is σ-inner. So there exists J0∈𝒴⊥ such that (I-T)oD=δJ0σ. Thus D=ToD+δJ0σ and so D is σ-a.i if and only if ToD:𝒜→𝒴* is σ-a.i.
Now let𝒵=𝒴·σ(𝒜). By Proposition 4.6, 𝒵 is 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 T′oD and (I-T′)oD are σ-derivations, and we have D1=T′oD1+(I-T′)·D1. Since (𝒴/𝒵)·𝒜={0}, by Lemma 4.1, (I-T′)·D1 is σ-inner and so there exists z0∈Z⊥ such that (I-T′)oD1=δZ0σ. Therefore, D1=T′oD1+δZ0σ. Thus, D1 is σ.a.i if and only if T′oD1 is σ.a.i. Set DoT=D1. Thus, D=T′oD1+δZ0σ+δJ0σ. Therefore, D is σ-a.i, if and only if T′oD1:𝒜→𝒵*=(σ(𝒜)·𝒳·σ(𝒜))* is σ.a.i. Recall that 𝒵 is σ-neo-unital. Thus, 𝒜 is 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.
Let𝒜 have a bounded approximate identity, and let 𝒳 be a Banach 𝒜-bimodule and σ a bounded idempotent endomorphism of 𝒜. Then 𝒜 is 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 thatσ is a bounded idempotent endomorphism of 𝒜 and define σ^:𝒜#→𝒜# with σ^(a+α)=σ(a)+α. The following statements are equivalent. (1)
𝒜 is σ-a.a.(2)
There is a net(μα)⊆(A#⊗^𝒜#)** such that for each a∈𝒜#, σ^(a)·μα-μα·σ^(a)→0 and π**(μα)→e^.(3)
There is a net(μα′)⊆(A#⊗^𝒜#)** such that for each a∈𝒜#, σ^(a)·μα-μα·σ^(a)→0 and for every α, π**(μα′)=e^.Proof.
(1⇒3) Suppose that 𝒜 is σ-a.a, by Proposition 3.8, 𝒜# is σ^-a.a. Let u=e⊗e∈𝒜#⊗^𝒜#. 𝒜#⊗^𝒜# is a Banach 𝒜#-bimodule with the following module actions:
(4.7)a⋅(b⊗c)=σ^(a)(b⊗c),(b⊗c)⋅a=(b⊗c)σ^(a)(a,b,c∈A#).
Set δu^:𝒜#→kerπ** with definition δu^(a)=σ^(a)·u^-u^·σ^(a)(a∈𝒜#). δu^ is σ^-derivation. Recall that kerπ**=(kerπ)**. Since 𝒜# is σ^-a.a, thus there exists (eα)⊆kerπ** such that
(4.8)δu^(a)=lim⁡ασ^(a)eα-eασ^(a)(a∈A#).
Set μα′=u^-eα∈(𝒜#⊗^𝒜#)**. We have
(4.9)σ^(a)μα′-μα′σ^(a)=σ^(a)u^-u^σ^(a)-(σ^(a)eα-eασ^(a))⟶0,
and for each α,
(4.10)π**(μα′)=π**(u^-eα)=π**(u^)-π**(eα)=π(u)=e.
(3⇒2) is clear.
(2⇒1) By Proposition 3.9, it is sufficient to show that A# is σ^-a.a.
LetD:A#→𝒳* be a derivation. By Corollary 4.7, we may take 𝒳 to be σ-neo-unital. We run the standard argument, so for each α∈I, set fα(x)=μα(ψx), where for a,b∈𝒜#, x∈𝒳, we have ψx(a⊗b)=〈x,σ^(a)D(b)〉. Then, (mαγ)⊂A#⊗^𝒜# converging ω* to μα(α∈I) and noting that for m∈𝒜#⊗^𝒜#,a∈𝒜#,x∈𝒳, then
(4.11)ψσ^(a)x-xσ^(a)(m)=(σ^(a)ψx-ψxσ^(a))(m)-〈x,σ^(π(m))D(a)〉.
Since 𝒳 is σ^-neo-unital, so 𝒳=𝒳σ^(𝒜#). So for each a∈A and x∈𝒳, we have
(4.12)〈σ^(a)x-xσ^(a),fα〉=〈ψσ^(a)x-xσ^(a),μα〉=lim⁡γ〈mαγ,ψσ^(a)x-xσ^(a)〉=〈σ^(a)ψx-ψxσ^(a),μα〉-lim⁡γ〈x,σ^(π(mαγ)D(a))〉=〈ψx,μασ^(a)-σ^(a)μα〉-〈x,π**(μα)D(a)〉.
Thus,
(4.13)‖〈x,σ^(a)fα-fασ^(a)〉-〈x,D(a)〉‖σ^(a)≤‖〈ψx,σ^(a)μα-μασ^(a)〉‖+‖x‖‖π**(μα)-e^‖‖D(a)‖σ^(a)≤‖D‖⋅‖x‖‖σ^(a)μα-μασ^(a)‖+‖x‖‖π*(μα)-e^‖‖D(a)‖,
and, therefore, D=limαδfασ^. It follows that 𝒜# is σ^-a.a and so 𝒜 is σ-a.a.Proposition 4.10.
Suppose that𝒜 is σ-a.a, and let
(4.14)Σ:0⟶X*⟶fY⟶gZ⟶0,
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→𝒴 of right inverse maps to g such that limα(σ(a)Gα-Gασ(a))=0 for a∈𝒜, and a net Fα:𝒴→𝒳* of left inverse maps to f such that limα(σ(a)fα-fασ(a))=0 for a∈𝒜.Proof.
Following the proof of [2, Theorem 2.3], for a right inverse G for g, σ-approximate amenability gives a net (φα)⊆ℬ(𝒵,𝒳*) such that
(4.15)σ(a)⋅G-G⋅σ(a)=lim⁡α(σ(a)⋅fGα-fGα⋅σ(a))(a∈A).
SettingGα=G-fφα gives the required net. Applying the same argument as [2, Proposition 1.1] provides (Fα).We recall that if𝒜 is a Banach algebra with a weak left (right) approximate identity, then 𝒜 has a left (right) approximate identity [1, Lemma 2.2].Corollary 4.11.
Suppose that Banach algebra𝒜 is σ-a.a, then σ(𝒜) has left and right approximate identities.Corollary 4.12.
Suppose that Banach algebra𝒜 is σ-a.a and σ is a bounded epimorphism of 𝒜, then 𝒜 has left and right approximate identities.Lemma 4.13.
Letσ be a bounded idempotent endomorphism of Banach algebra 𝒜 and 𝒳 a σ-neo-unital Banach 𝒜-module. If (eα)α is a bounded approximate identity in 𝒜, then (σ(eα))α is a bounded approximate identity for 𝒳.Proof.
For everya∈𝒜 we have eασ(a)→σ(a). Since σ is idempotent, σ(eα)σ(a)→σ(a). For each x∈𝒳, there exists a∈𝒜 and y∈𝒳 such that x=σ(a)·y. Therefore,
(4.16)σ(eα)⋅x=σ(eα)σ(a)⋅y⟶σ(a)⋅y=x,
which shows that (σ(eα)) is a bounded approximate identity for 𝒳.It is often convenient to extend a derivation to a large algebra. If a Banach algebraI is contained as a closed ideal in another Banach algebra 𝒜, then the strict topology on 𝒜 with respect to I is defined through the family of seminorms (Pi)i∈I, where
(4.17)Pi(a):=‖ai‖+‖ia‖(a∈A).
Note that the strict topology is Hausdorff only if {a∈𝒜:a·I=I·a={0}}={0} [3].Proposition 4.14.
Let𝒜 be a Banach algebra and I a closed ideal in 𝒜. let σ be a bounded idempotent endomorphism of 𝒜 and I has a bounded approximate identity. Let 𝒳 be a σ-neo-unital Banach I-module and D:I→𝒳* a σ-derivation. Then, 𝒳 is a Banach 𝒜-bimodule in a canonical fashion, and there is a unique σ-derivation D~:𝒜→𝒳* such that(i)
D~|I=D,(ii)
D~ is continuous with respect to the strict topology on 𝒜 and the ω*-topology on 𝒳*.Proof.
Since𝒳 is a σ-neo-unital Banach I-module, so for each x∈𝒳, there exists i∈I and y∈𝒳 such that x=σ(i)·y. Define a·x=σ(ai)·y(a∈𝒜).
We claim thata·x is well defined, that is, independent of the choices of i and y. Let i′∈I and y′∈𝒳 be such that x=σ(i′)·y′, and let (eα)α be a bounded approximate identity for I. For each a∈𝒜 and x∈𝒳 we have
(4.18)a⋅x=σ(ai)⋅y=lim⁡ασ(aeαi)⋅y=lim⁡ασ(aeα)σ(i)⋅y=lim⁡ασ(aeα)x=lim⁡ασ(aeα)σ(i′)⋅y′=lim⁡ασ(aeαi′)⋅y′=σ(ai′)⋅y′.
It is obvious that this operation of 𝒜 on 𝒳 turns 𝒳 into a left Banach 𝒜-module. Similarly, one defines a right Banach 𝒜-module structure on 𝒳. So that, eventually, 𝒳 becomes a Banach 𝒜-bimodule. To extend D, let
(4.19)D~:A⟶X*,a⟶ω*-lim⁡α(D(aeα)-σ(a)⋅D(eα)).
We claim that D~ is well-defined, that is, the limit in (4.19) does exist. Let x∈𝒳, and let i∈I and y∈𝒳 such that x=y·σ(i). By Lemma 4.13, σ(eα) is bounded approximate identity for 𝒳, and we have
(4.20)〈x,D(aeα)-σ(a)⋅D(eα)〉=〈y⋅σ(i),D(aeα)-σ(a)⋅D(eα)〉=〈y,σ(i)D(aeα)-σ(ia)⋅D(eα)〉=〈y,D(iaeα)-D(i)σ(aeα)-D(iaeα)+D(ia)σ(eα)〉=〈σ(eα)⋅y,D(ia)〉-〈σ(aeα)⋅y,D(i)〉⟶α〈y,D(ia)〉-〈σ(a)⋅y,D(i)〉(a∈A).
So the limit in (4.19) exists. Furthermore, for i∈I,
(4.21)D~(i)=ω*-lim⁡α(D(ieα)-σ(i)⋅D(eα))=ω*-lim⁡αD(ieα)-D(ieα)+D(i)σ(eα)=D(i),
so D~ is an extension of D. Also for a∈𝒜 and i∈I we have
(4.22)(D~a)⋅σ(i)=ω*-lim⁡α(D(aeα)⋅σ(i)-σ(a)⋅D(eα)⋅σ(i))=ω*-lim⁡α(D(aeαi)-σ(aeα)⋅D(i)-σ(a)⋅D(eαi)+σ(a)σ(eα)⋅D(i))=ω*-lim⁡α(D(aeαi)-σ(a)⋅D(eαi))=D(ai)-σ(a)⋅D(i).
We claim that D~ is continuous with respect to the strict topology on 𝒜 and the ω*-topology an 𝒳*.
Letan→stricta in 𝒜.
(4.23)∀i∈I,‖ani‖+‖ian‖⟶‖ai‖+‖ia‖.
For each x∈𝒳,
(4.24)|〈x,D~(an)〉-〈x,D~(a)〉|x,D~(an)=lim⁡α|〈x,D(aneα)-σ(an)⋅D(eα)〉-〈x,D(aeα)-σ(a)⋅D(eα)〉|x,D~(an)=lim⁡α|〈x,D(aneα)-D(aeα)-σ(an)D(eα)+σ(a)D(eα)〉|x,D~(an)≤lim⁡α‖x‖‖(D(aneα)-D(aeα))-(σ(a0)⋅D(eα)-σ(an)D(eα))‖x,D~(an)≤lim⁡α‖x‖(‖D‖‖aneα-aeα‖+‖σ(an)-σ(a)‖‖D(eα)‖)x,D~(an)≤lim⁡α‖x‖(‖D‖‖an-a‖‖eα‖+‖σ‖‖an-a‖‖D(eα)‖)⟶0,
so D~ is continuous.
It remains to show thatD~ is a σ-derivation. From the definition of the strict topology, we have aeα→a in the strict topology for all a∈𝒜 because ∥aeαi∥+∥iaeα∥→α∥ai∥+∥ia∥(i∈I) and so D~(aeα)→w*D~(a). Therefore,
(4.25)D~(ab)=ω*-lim⁡αlim⁡βD~((aeα)(beβ))=ω*-lim⁡αlim⁡βD((aeα)(beβ))=ω*-lim⁡αlim⁡β(σ(aeα)D(beβ)+D(aeα)⋅σ(beβ))=ω*-lim⁡αlim⁡β(σ(aeα)D~(beβ)+D~(aeα)⋅σ(beβ))=σ(a)D~(b)+D~(a)σ(b),
that is, D~ is σ-derivation.Corollary 4.15.
Suppose that𝒜 is σ-a.a, where σ is bounded idempotent endomorphism of 𝒜,I is a closed ideal in 𝒜. If I has a bounded approximate identity, then I is σ-a.a.Proof.
Suppose thatI has a bounded approximate identity, 𝒳 is a σ-neo-unital Banach I-bimodule, and D:I→𝒳* is a σ-derivation. By Proposition 4.14, 𝒳 becomes to a Banach 𝒜-bimodule and D has a unique extension D~:𝒜→𝒳* which is a σ-derivation. Since 𝒜 is σ-a.a,
(4.26)∃{xα*}⊆X*s.t.D~(a)=lim⁡ασ(a)⋅xα*-xα*⋅σ(a)(a∈A).
So we have D(i)=D~(i)=limασ(i)·xα*-xα*·σ(i), which shows that D=limαδxα*σ is σ-a.i, and I is σ-a.a.Corollary 4.16.
Let𝒜 be an a.a Banach algebra and I a closed ideal of 𝒜. Then 𝒜/I is σ-a.a for each bounded endomorphism σ of 𝒜/I.Proposition 4.17.
LetI be a closed ideal of 𝒜 such that σ(I)⊆I. If 𝒜 is σ-a.a, then 𝒜/I is σ^-a.c, where σ^ is an endomorphism of 𝒜/I induced by σ (i.e., σ^(a+I)=σ(a)+I for a∈𝒜).Proof.
Let𝒳 be a Banach 𝒜/I-bimodule and D:𝒜/I→𝒳 a σ^-derivation. Then 𝒳 becomes an 𝒜-bimodule with the following module actions:
(4.27)a⋅x=π(a)⋅x,x⋅a=x⋅π(a)(a∈A,x∈X),
where π is the canonical homomorphism π:𝒜→𝒜/I. It is easy to see that Doπ:A→X becomes a σ-derivation. Since 𝒜 is σ-a.c, there exists a net {xα}⊆𝒳 such that Doπ(a)=limασ(a)·xα-xα·σ(a)(a∈𝒜). Therefore, for each (a∈A),
(4.28)D(a+I)=Doπ(a)=lim⁡ασ(a)⋅xα-xα⋅σ(a)=lim⁡απ(σ(a))⋅xα-xα⋅π(σ(a))=lim⁡α(σ(a)+I)⋅xα-xα⋅(σ(a)+I)=lim⁡ασ^(a+I)xα-xασ^(a+I).
Thus, 𝒜/I is σ^-a.c.Proposition 4.18.
Suppose thatI is a closed ideal in 𝒜. If I is σ-amenable and A/I is a.a, then 𝒜 is σ-a.a.Proof.
Let𝒳 be a Banach 𝒜-bimodule and D:𝒜→𝒳* a σ-derivation. 𝒳 is a Banach I-bimodule too.
Clearly,d=D|I:I→𝒳* is a σ-derivation, and by σ-amenability of I there exists x0*∈𝒳* such that D=δx0*σ, and, therefore, for each i∈I we have d(i)=σ(i)·x0*-x0*·σ(i). Set D1=D-δx0*σ. Clearly, D1 is σ-derivation and D1|I=0. Now let 𝒳0=span¯(𝒳·σ(I)∪σ(I)·𝒳)·(𝒳/𝒳0) is a Banach 𝒜/I-bimodule via the following module actions:
(4.29)(a+I)(x+X0)=σ(a)x+X0,(x+X0)(a+I)=xσ(a)+X0(x∈X,a∈A).
Now we define
(4.30)D~:AI⟶(XX0)*;〈x+X0,D~(a+I)〉=〈x,D1(a)〉(a∈A,x∈X).
Let a+I=a′+I and x+𝒳0=x′+𝒳0 for some a,a′∈𝒜 and x,x′∈𝒳. So a-a′∈I, and we have D1(a-a′)=0. Thus, D1(a)=D(a′). Now we have
(4.31)〈x+X0,D~(a+I)〉=〈x′+X0,D~(a′+I)〉.
Thus, 〈x,D1(a)〉=〈x′,D1(a′)〉=〈x′,D1(a)〉, and, therefore,
(4.32)〈x-x′,D1(a)〉=0.
It is enough to show that D1(a) is zero on 𝒳0. Suppose that σ(i)x∈𝒳0, we have
(4.33)〈σ(i)x,D1(a)〉=〈x,D1(a)σ(i)〉=〈x,D1(ai)-σ(a)D1(i)〉=0,〈xσ(i),D1(a)〉=〈x,σ(i)D1(a)〉=〈x,D1(ia)-D1(i)σ(a)〉=0.
So for all a∈𝒜, D1(a)=0 on σ(I)·𝒳∪𝒳·σ(I) and so for all a∈𝒜, D1(a)=0 on 𝒳0. Since x-x′∈𝒳0, therefore 〈x-x′,D1(a)〉=0 which shows that D1 is well defined. We claim that D~ is a derivation;
(4.34)〈x+X0,D~((a+I)(b+I))〉=〈x,D1(ab)〉=〈x,σ(a)D1(b)+D1(a)σ(b)〉=〈xσ(a),D1(b)〉+〈σ(b)x,D1(a)〉=〈xσ(a)+X0,D~(b+I)〉+〈σ(b)x+X0,D~(a+I)〉=〈(x+X0)(a+I),D~(b+I)〉+〈(b+I)(x+X0),D~(a+I)〉.
So there exists a net (φα)⊆(𝒳/𝒳0)* such that D~=limαδφα. Let q:𝒳→𝒳/𝒳0 be the quotient map. For every α, (φαoq)∈𝒳*. Set (xα*)=(φαoq)⊆𝒳*. We have
(4.35)〈x,D1(a)〉=〈x+X0,D~(a+I)〉=〈x+X0,lim⁡α(a+I)φα-φα(a+I)〉=lim⁡α〈xσ(a)+X0,φα〉-〈σ(a)x+X0,φα〉=lim⁡α〈q(xσ(a)),φα〉-〈q(σ(a)x),φα〉=lim⁡αφαoq(xσ(a)-σ(a)x)=〈xσ(a)-σ(a)x,xα*〉=lim⁡α〈x,σ(a)xα*-xα*σ(a)〉=〈x,lim⁡αδxα*σ(a)〉.
So D1=D-δx*σ=limαδxα*σ, and, therefore, D=limαδ(xα*-x0*)σ. Which shows that D is σ-a.i and so 𝒜 is σ-a.a.Example 4.19.
Let𝒜 be a Banach algebra and let 0≠φ∈Ball(𝒜*). Then 𝒜 with the product a·a′=φ(a)a′ becomes a Banach algebra. We denote this algebra with 𝒜φ. It is easy to see that 𝒜φ has a left identity e, while it has not right approximate identity, so 𝒜φ is not contractible and is not approximately contractible. Also 𝒜φ is biprojective. Now suppose that σ:𝒜φ→𝒜φ be defined by σ(a)=φ(a)e. We have
(4.36)σ2(a)=σ(φ(a)e)=φ(a)σ(e)=φ(a)φ(e)e=φ(a)e=σ(a).
Thus σ is idempotent. It is easy to see that e is identity for σ(𝒜φ), and since 𝒜 is biprojective by [1, Corollary 5.3], 𝒜φ is σ-biprojective. Thus by [1, Theorem 4.3], 𝒜φ is σ-contractible and so 𝒜φ is σ-a.c.
It is easy to see thatkerφ and all subspaces of kerφ are all ideals of 𝒜φ and σ(kerφ)⊆kerφ so σ(I)⊆I for each ideal of 𝒜. Therefore, by Proposition 4.17, 𝒜φ/I is σ^-a.c for each ideal I of 𝒜φ, where σ^(a+I)=σ(a)+I=φ(a)e+I.Corollary 4.20.
Suppose thatσ is a bounded idempotent endomorphism of Banach algebra 𝒜. Then 𝒜 is σ-a.a if and only if there are nets (μα′′) in (𝒜⊗^𝒜)** and (Fα),(Gα)⊆𝒜**, such that for each a∈𝒜, (1)
σ(a)·μα′′-μα′′·σ(a)+Fα⊗σ(a)-σ(a)⊗Gα→0,(2)
σ(a)·Fα→σ(a),Gα·σ(a)→σ(a),(3)
π**(μα′′)·σ(a)-Fα·σ(a)-Gα·σ(a)→0.Proof.
Suppose that𝒜 is σ-a.a, take the net (μα) given in Proposition 4.9 and write
(4.37)μα=μα′′-Fα⊗e^-e^⊗Gα+cαe^⊗e^,
where (μα′′)⊆(A⊗^A)**, (Fα),(Gα)⊆A**, and (cα)⊆ℂ. Applying π**, π**(μα′′)-Fα-Gα+cαe^→e^, hence cα→1, then
(4.38)π**(μα′′)⋅σ(a)-Fα⋅σ(a)-Gα⋅σ(a)+e^⋅σ(a)⟶e^⋅σ(a)(a∈A).
So we have (iii) further, by Proposition 4.9, for a∈𝒜#,
(4.39)σ^(a)⋅μα′′-σ^(a)⋅Fα⊗e^-σ^(a)⊗Gα+σ^(a)⊗e^=+μα′′⋅σ^(a)+Fα⊗σ^(a)+e^⊗Gα⋅σ^(a)-e^⊗σ^(a)⟶0.
Thus σ^(a)·μα′′-μα′′·σ^(a)+Fα⊗σ^(a)-σ^(a)⊗Gα→0, and σ^(a)·Fα→σ^(a),Gα·σ^(a)→σ^(a). So for a∈𝒜,
(4.40)σ(a)⋅μα′′-μα′′⋅σ(a)+Fα⊗σ(a)-σ(a)⊗Gα⟶0,σ(a)⋅Fα⟶σ(a),Gα⋅σ(a)⟶σ(a).
Conversely, setcα=1 and μα=μα′′-Fα⊗e^-e^⊗Gα+e^⊗e^. We have
(4.41)σ^(a+α)⋅μα-μα⋅σ^(a+α)=(σ(a)+α)⋅μα-μα⋅(σ(a)+α)=σ(a)⋅μα-μα⋅σ(a)+aμα-αμα=σ(a)⋅μα-μα⋅σ(a)=σ(a)⋅μα′′-σ(a)Fα⊗e-σ(a)⊗Gα+σ(a)⊗e(-μα′′⋅σ(a)+σ(a)⊗e=+Fα⊗σ(a)+e⊗Gασ(a)-e⊗σ(a))=σ(a)⋅μα′′-μα′′⋅σ(a)+Fα⊗σ(a)-σ(a)⊗Gα→0(a∈A).
So σ^(a)·μα-μα·σ^(a)→0(a∈𝒜#). Also
(4.42)π**(μα)⋅σ(a)=π**(μα′′-Fα⊗e^-e^⊗Gα+e^⊗e^)σ(a)=π**(μα′′)σ(a)-Fα⋅σ(a)-Gα⋅σ(a)+σ(a)⟶σ(a)(a∈A),
and so π**(μα)→e^. Now, by Proposition 4.9, 𝒜 is σ-a.a.Forσ-approximate contractibility we have the following parallel result.Proposition 4.21.
𝒜 is σ-a.c if and only if any of the following equivalent conditions hold: (1)
there is a net(μα)⊂𝒜#⊗^𝒜# such that for each a∈𝒜#, σ(a)·μα-μα·σ(a)→0 and π(μα)→e;(2)
there is a net(μα′)⊂𝒜#⊗^𝒜# such that for each a∈𝒜#, σ(a)·μα′-μα′·σ(a)→0 and π(μα′)=e;(3)
there are nets(μα′′)⊂𝒜⊗^𝒜, (Fα),(Gα)⊂𝒜, such that for each a∈𝒜,
(i)
σ(a)·μα′′-μα′′·σ(a)+Fα⊗σ(a)-σ(a)⊗Gα→0;(ii)
σ(i)·Fα→σ(a),Gα·σ(a)→σ(a);(iii)
π(μα′′)·σ(a)-Fα·σ(a)-Gα·σ(a)→0.We know Banach algebra𝒜 is amenable if and only if 𝒜 has bounded approximate diagonal [3].Proposition 4.22.
Banach algebra𝒜 is σ-amenable if and only if 𝒜 has bounded approximate σ-diagonal, that is, there is a bounded net (μα)⊆𝒜⊗^𝒜 such that for each a∈𝒜, σ(a)·μα-μα·σ(a)→0 and π(μα)·σ(a)→σ(a).Proposition 4.23.
If Banach algebra𝒜 is σ-amenable, then 𝒜 is σ-a.c.Proof.
Suppose that𝒜 is σ-amenable. Then there exists a bounded net (μα) in 𝒜⊗𝒜 such that for each a∈𝒜,
(4.43)σ(a)⋅μα-μα⋅σ(a)⟶0,π(μα)⋅σ(a)⟶σ(a).
Set fα=π(μα). It is easy to see that (fα) is a bounded approximate identity. Then μα′′=μα+fα⊗fα and Fα=Gα=fα satisfy (i)–(iii) of Proposition 4.21, because(i)
σ(a)·μα′′-μα′′·σ(a)+fα⊗σ(a)-σ(a)⊗fα=σ(a)·μα-μα·σ(a)+σ(a)fα⊗fα-fα⊗σ(a) + fα⊗σ(a)-σ(a)⊗fα→0(a∈𝒜),(ii)
σ(a)·fα=σ(a)·π(μα)→σ(a), fα·σ(a)=π(μα)·σ(a)→σ(a),(iii)
π(μα′′)·σ(a)=π(μα+fα⊗fα)·σ(a)=fα·σ(a)+fα2·σ(a).
So(4.44)π(μα′′)⋅σ(a)-Fα⋅σ(a)-Gα⋅σ(a)=fα⋅σ(a)+fα2⋅σ(a)-fα⋅σ(a)-fα⋅σ(a)⟶0.
Note that fα2 is a bounded approximate identity too, thus, by Proposition 4.21, 𝒜 is σ-a.c.Corollary 4.24.
Suppose that𝒜 is a σ-a.a Banach algebra where σ is an idempotent endomorphism of 𝒜 and I is a closed two-sided ideal of 𝒜 which σ(I) has a bounded approximate identity and σ(I)⊆I. Then, I is σ-a.a.Proof.
Let{eα} be a bounded approximate identity in σ(I), so {e^α} is bounded net in σ(I)**, and so by Banach-Alaoglu theorem there exists a subnet {e^β}⊆{e^α} and E∈σ(I)** such that e^β→w*E. E is a right identity in σ(I)** because for each F∈σ(I)** and f∈σ(I)*,
(4.45)〈f,F□E〉=〈f⋅F,E〉=lim⁡β〈eβ,fF〉=lim⁡β〈eβf,F〉=〈f,F〉.
AlsoE acts as an identity on σ(I) itself. Let (μα),(Fα),(Gα) be the nets given by Corollary 4.20 for 𝒜. Define μα′=E·μα·E∈(I⊗^I)**,Fα′=E·Fα∈I**, and Gα′=Gα·E∈I**. Then, for i∈I,
(i) we consider(4.46)σ(i)⋅μα′-μα′⋅σ(i)+Fα′⊗σ(i)-σ(i)⊗Gα′μα′-μα′=σ(i)⋅E⋅μα⋅E-E⋅μα⋅E⋅σ(i)+E⋅Fα⊗σ(i)-σ(i)⊗Gα⋅Eμα′-μα′=σ(i)⋅μα⋅E-E⋅μασ(i)+E⋅Fα⊗σ(i)-σ(i)⊗Gα⋅Eμα′-μα′=E⋅σ(i)⋅μα⋅E-E⋅μα⋅σ(i)⋅Eμα′-μα′=+E⋅Fα⊗σ(i)⋅E-E⋅σ(i)⊗Gα⋅Eμα′-μα′=E(σ(i)⋅μα-μα⋅σ(i)+Fα⊗σ(i)-σ(i)⊗Gα)⋅E⟶0,
(ii) we consider(4.47)σ(i)⋅Fα′=σ(i)⋅E⋅Fα=σ(i)⋅Fα⟶σ(i),Gα′⋅σ(i)=Gα⋅E⋅σ(i)=Gα⋅σ(i)⟶σ(i)
(iii) we consider(4.48)π**(μα′)⋅σ(a)^-Fα′⋅σ(a)^-Gα′-σ(a)^E⋅σ(a)=π**(E⋅μα⋅E)⋅σ(a)-E⋅Fα⋅σ(a)-Gα⋅E⋅σ(a)E⋅σ(a)=E⋅π**(μα)⋅E⋅σ(a)-E⋅Fα⋅σ(a)-Gα⋅σ(a)E⋅σ(a)=E⋅π**(μα)⋅σ(a)-E⋅Fα⋅σ(a)-Gα⋅σ(a)-E⋅Gασ(a)+E⋅Gασ(a)E⋅σ(a)=E⋅(π**(μα)⋅σ(a)-Fα.σ(a)-Gασ(a))+(E-e^)Gασ(a)⟶0.
An alternative proof would be to follow the standard argument stated in Corollary 4.15.



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*Source: 653140-2012-08-09.xml*