𝜎-Approximately Contractible Banach Algebras

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.

1. Introduction

For a Banach algebra 𝒜, 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 ‖𝑎⋅𝑥‖≤𝑐‖𝑎‖‖𝑥‖,‖𝑥⋅𝑎‖≤𝑐‖𝑎‖‖𝑥‖. (1.1) A derivation 𝐷∶𝒜→𝒳 is a linear map, always taken to be continuous, satisfying 𝐷(𝑎𝑏)=𝐷(𝑎)⋅𝑏+𝑎⋅𝐷(𝑏)(𝑎,𝑏∈𝒜). (1.2) 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 𝐷(𝑎)=𝑎⋅𝑥^∗−𝑥^∗⋅𝑎=𝛿_𝑥∗(𝑎)(𝑎∈𝒜). (1.3) 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 𝐷(𝑎𝑏)=𝜎(𝑎)⋅𝐷(𝑏)+𝐷(𝑎)⋅𝜎(𝑏)(𝑎,𝑏∈𝒜). (1.4) For each 𝑥∈𝒳, the mapping 𝛿^𝜎_𝑥∶𝒜⟶𝒳 (1.5) defined by 𝛿^𝜎_𝑥(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 that 𝒜 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. A 𝜎-derivation 𝐷∶𝒜→𝒳 is 𝜎-a.i, if there exists a net (𝑥_𝛼)⊆𝒳 such that for every 𝑎∈𝒜, 𝐷(𝑎)=lim_𝛼𝜎(𝑎)⋅𝑥_𝛼−𝑥_𝛼⋅𝜎(𝑎), the limit being in norm and we write 𝐷=lim𝛿^𝜎_𝑥𝛼. Note that we do not suppose (𝑥_𝛼) to be bounded.

Definition 1.3. A Banach algebra 𝒜 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 algebra 𝒜 is called 𝜎-a.a if for any 𝒜-bimodule 𝒳, every 𝜎-derivation 𝐷∶𝒜→𝒳^∗ is 𝜎-a.i.

Definition 1.5. Let 𝒜 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 𝑇(𝑎⋅𝑥)=𝜎(𝑎)⋅𝑇(𝑥),𝑇(𝑥⋅𝑎)=𝑇(𝑥)⋅𝜎(𝑎)(𝑎∈𝒜,𝑥∈𝒳). (1.6)

2. Basic Properties

Proposition 2.1. Let 𝒜 be a 𝜎-a.c Banach algebra. Then 𝜎(𝒜) has a left and right approximate identity.

Proof. Consider 𝒳=𝒜 as a Banach 𝒜-bimodule with the trivial right action, that is, 𝑎⋅𝑥=𝑎𝑥,𝑥⋅𝑎=0(𝑎∈𝒜,𝑥∈𝒳). (2.1) Then 𝐷∶𝒜→𝒳 defined by 𝐷(𝑎)=𝜎(𝑎) is a 𝜎-derivation, and so there is a net {𝑢_𝛼}⊆𝒳(=𝒜) such that 𝐷=lim_𝛼𝛿^𝜎_𝑢𝛼. Hence for each 𝑎∈𝐴, 𝜎(𝑎)=𝐷(𝑎)=lim_𝛼𝛿^𝜎_𝑢𝛼(𝑎)=lim_𝛼𝜎(𝑎)⋅𝑢_𝛼−𝑢_𝛼⋅𝜎(𝑎)=lim_𝛼𝜎(𝑎)𝑢_𝛼, (2.2) which shows that {𝑢_𝛼} is a right approximate identity for 𝜎(𝒜). Similarly, one can find a left approximate identity for 𝜎(𝒜).

Corollary 2.2. Let 𝒜 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. Let 𝜑 be a bounded endomorphism of Banach algebra 𝒜. If 𝒜 is 𝜎-a.c, then 𝒜 is (𝜑𝑜𝜎)-a.c.

Proof. Let 𝒳 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: 𝑎∗𝑥=𝜑(𝑎)⋅𝑥,𝑥∗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(𝑎)∗𝐷(𝑏)+𝐷(𝑎)∗𝜎(𝑏). (2.4) Thus 𝐷∶𝒜→(𝒳,∗) is a continuous 𝜎-derivation. Since 𝒜 is 𝜎-a.c, there exists a net {𝑥_𝛼}⊆𝒳 such that 𝐷=lim𝛿^𝜎_𝑥𝛼. In fact, 𝐷(𝑎)=lim_𝛼𝜎(𝑎)∗𝑥_𝛼−𝑥_𝛼∗𝜎(𝑎)=lim_𝛼𝜑𝑜𝜎(𝑎)⋅𝑥_𝛼−𝑥_𝛼⋅𝜑𝑜𝜎(𝑎)=lim_𝛼𝛿_𝑥^𝜑𝑜𝜎_𝛼(𝑎)(𝑎∈𝒜). (2.5) Therefore, 𝐷 is a (𝜑𝑜𝜎)-a.i and so 𝒜 is (𝜑𝑜𝜎)-a.c.

Corollary 2.4. Let 𝒜 be an a.c Banach algebra. Then 𝒜 is 𝜎-a.c for each bounded endomorphism 𝜎 of 𝒜.

Proposition 2.5. Let 𝒜 be a 𝜎-a.c Banach algebra, where 𝜎 is a bounded epimorphism of 𝒜. Then 𝒜 is a.c.

Proof. Let 𝒳 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 {𝑥_𝛼}⊆𝑋 such that 𝑑𝑜𝜎(𝑎)=lim_𝛼𝜎(𝑎)𝑥_𝛼−𝑥_𝛼𝜎(𝑎). Now for 𝑏∈𝒜 there exists 𝑎∈𝒜 such that 𝑏=𝜎(𝑎), and, therefore, 𝑑(𝑏)=𝑑(𝜎(𝑎))=lim_𝛼𝑥_𝛼𝜎(𝑎)−𝜎(𝑎)𝑥_𝛼=lim_𝛼𝑥_𝛼𝑏−𝑏𝑥_𝛼, (2.6) which shows that 𝑑 is approximately inner and so 𝒜 is a.c.

Corollary 2.6. Let 𝜑 be a bounded endomorphism of Banach algebra 𝒜. If 𝒜 is 𝜎-a.a then it is (𝜑𝑜𝜎)-a.a too.

Corollary 2.7. Let 𝒜 be an a.a Banach algebra. For each bounded endomorphism 𝜎, 𝒜 is 𝜎-a.a.

Corollary 2.8. Let 𝒜 be a 𝜎-a.a Banach algebra, where 𝜎 is a bounded epimorphism of 𝒜. Then 𝒜 is a.a.

Proposition 2.9. Suppose that ℬ 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. Let 𝜎∶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: 𝑎∗𝑥=𝜎𝜑(𝑎)⋅𝑥,𝑥∗𝑎=𝑥⋅𝜎𝜑(𝑎)(𝑎∈𝒜,𝑥∈𝒳). (2.7) 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 {𝑥_𝛼}⊆𝒳 such that 𝐷𝑜𝜑(𝑎)=lim_𝛼𝛿_𝑥𝛼(𝑎). We have 𝐷𝜑(𝑎)=𝐷𝑜𝜑(𝑎)=lim_𝛼𝛿_𝑥𝛼(𝑎)=lim_𝛼𝑎∗𝑥_𝛼−𝑥_𝛼∗𝑎=lim_𝛼𝜎𝜑(𝑎)𝑥_𝛼−𝑥_𝛼𝜎𝜑(𝑎)(𝑎∈𝒜). (2.8) Since 𝜑 is an epimorphism, so for each 𝑏∈ℬ there exists 𝑎∈𝐴 such that 𝑏=𝜑(𝑎), and we have 𝐷(𝑏)=lim_𝛼𝜎(𝑏)𝑥_𝛼−𝑥_𝛼𝜎(𝑏), (2.9) which shows that 𝐷 is 𝜎-a.i and so 𝐵 is 𝜎-a.c.

Proposition 2.10. Suppose that 𝒜 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. Let 𝒳 be a Banach ℬ-bimodule and 𝐷∶ℬ→𝒳 a continuous 𝜏-derivation. Then (𝒳,∗) is an 𝒜-bimodule with the following actions: 𝑎∗𝑥=𝜑(𝑎)⋅𝑥,𝑥∗𝑎=𝑥⋅𝜑(𝑎)(𝑎∈𝒜,𝑥∈𝒳). (2.10) It is easy to check that 𝐷𝑜𝜑∶𝒜→(𝒳,∗) is a 𝜎-derivation. Since 𝒜 is 𝜎-a.c, there exists a net {𝑥_𝛼}⊆𝒳 such that 𝐷𝑜𝜑(𝑎)=lim_𝛼𝛿^𝜎_𝑥𝛼(𝑎), so we have 𝐷𝜑(𝑎)=lim_𝛼𝜎(𝑎)∗𝑥_𝛼−𝑥_𝛼∗𝜎(𝑎)=lim_𝛼𝜑(𝜎(𝑎))⋅𝑥_𝛼−𝑥_𝛼⋅𝜑(𝜎(𝑎))=lim_𝛼𝜏𝜑(𝑎)⋅𝑥_𝛼−𝑥_𝛼⋅𝜏𝜑(𝑎)(𝑎∈𝒜). (2.11) Since 𝜑 is epimorphism, so 𝐷(𝑏)=lim_𝛼𝜏(𝑏)𝑥_𝛼−𝑥_𝛼𝜏(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 of 𝜎-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. Let 𝒜 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+𝛿_𝜂.

The following definition extends the definition of the unital Banach 𝒜-module in the classical sense.

Definition 3.2. Let 𝒜 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. Let 𝒜 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. Since 𝜎(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 that 𝒜 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 (𝜂_𝛼)⊆𝑒⋅𝒳^∗⋅𝑒, and 𝜂∈𝒳^∗, such that 𝐷=lim_𝛼𝛿^𝜎_𝜂𝛼+𝛿_𝜂.

Proof. By Proposition 3.1, 𝐷=𝐷₁+𝛿_𝜂 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=lim_𝛼𝛿^𝜎_𝜂𝛼 for some net (𝜂_𝛼)⊆𝑒⋅𝒳^∗⋅𝑒.

In the following proposition we consider 𝜎-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 that 𝒜 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+𝛿_𝜂.

Corollary 3.6. Assume that 𝒜 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 that 𝒜 is a unital Banach algebra with the identity 𝑒, and (𝒳,∗) is a 𝜎-unital Banach 𝒜-bimodule with the following module actions: 𝑎∗𝑥=𝜎(𝑎)𝑥,𝑥∗𝑎=𝑥𝜎(𝑎)(𝑎∈𝒜,𝑥∈𝒳). (3.1) If 𝐷∶𝒜→𝒳^∗ is a 𝜎-derivation, then 𝐷(𝑒)=0.

Proof. We have 𝐷(𝑒)=𝐷(𝑒𝑒)=𝜎(𝑒)𝐷(𝑒)+𝐷(𝑒)𝜎(𝑒) and ⟨(𝑒)𝜎(𝑒=⟨(𝑒)𝜎(𝑒)⟩=⟨(𝑒)𝜎(𝑒)𝜎(𝑒=⟨(𝑒)𝜎(𝑒=⟨(𝑒).𝑒∗𝑥,𝐷)⟩𝑥,𝐷∗𝑒𝑥,𝐷)⟩𝑥,𝐷)⟩𝑒∗𝑥,𝐷)⟩(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 𝜎 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. Let 𝒳 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+𝛿_𝜂. 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 (𝑥^∗_𝛾)⊆𝒳^∗ such that 𝑑=lim_𝛾𝛿^𝜎_𝑥∗𝛾. Hence 𝐷₁(𝑎)=lim_𝛾𝜎(𝑎)𝑥^∗_𝛾−𝑥^∗_𝛾𝜎(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 𝐷₁(𝑎+𝛼)=𝐷₁(𝑎)+𝛼𝐷₁(𝑒)=𝐷₁(𝑎)=lim_𝛾𝜎(𝑎)𝑥^∗_𝛾−𝑥^∗_𝛾𝜎(𝑎)=lim_𝛾(𝜎(𝑎+𝛼)−𝛼)𝑥^∗_𝛾−𝑥^∗_𝛾(𝜎(𝑎+𝛼)−𝛼)=lim_𝛾𝜑(𝑎+𝛼)𝑥^∗_𝛾−𝑥^∗_𝛾𝜑(𝑎+𝛼). (3.3) This shows that 𝐷₁ is 𝜎-a.i, and so 𝒜^# is 𝜎-a.a.

Proposition 3.9. Let 𝜎 be a bounded endomorphism of Banach algebra 𝒜. If 𝐴^# is 𝜎-a.a, then 𝒜 is 𝜎-a.a.

Proof. Let 𝒳 be a Banach 𝒜-bimodule and 𝐷∶𝒜→𝒳^∗ a continuous 𝜎-derivation. 𝒳 is a Banach 𝐴^#-bimodule with the following module actions: (𝑎+𝛼)⋅𝑥=𝑎⋅𝑥+𝛼𝑥,𝑥⋅(𝑎+𝛼)=𝑥⋅𝑎+𝛼𝑥, (3.4) 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 (𝑥^∗_𝛾)⊆𝒳^∗ such that 𝐷^#=lim𝜎_𝛾𝛿_𝑥∗𝛾. Hence, for each 𝑎∈𝒜 we have 𝐷(𝑎)=𝐷^#(𝑎+𝛼)=lim_𝛾𝜎(𝑎+𝛼)𝑥^∗_𝛾−𝑥^∗_𝛾𝜎(𝑎+𝛼)=lim_𝛾𝜎(𝑎)𝑥^∗_𝛾−𝑥^∗_𝛾𝜎(𝑎) (3.5) 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. Let 𝒜 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. Let 𝒳 be a Banach 𝒜-bimodule with trivial left action. Hence, 𝒳^∗ is a Banach 𝒜-bimodule with trivial right action, that is, 𝑥^∗⋅𝑎=0,𝑎⋅𝑥^∗=𝑎𝑥^∗𝑥^∗∈𝒳^∗,𝑎∈𝒜. (4.1) Let 𝐷∶𝒜→𝒳^∗ be a continuous 𝜎-derivation and (𝑒_𝛼) a bounded approximate identity of 𝒜. By Banach Alaoglu's Theorem, (𝐷(𝑒_𝛼)) has a subnet (𝐷(𝑒_𝛽)) such that 𝐷(𝑒_𝛽)−→^𝑤∗𝑥^∗₀, for some 𝑥^∗₀∈𝒳^∗. Since 𝑎⋅𝑒−→_𝛽^‖⋅‖𝑎 and 𝐷 is continuous, 𝐷(𝑎⋅𝑒_𝛽)−→^‖⋅‖𝐷(𝑎). Hence, 𝐷(𝑎⋅𝑒_𝛽)−→^𝑤∗𝐷(𝑎).
On the other hand, 𝐷(𝑎⋅𝑒_𝛽)=𝜎(𝑎)𝐷(𝑒_𝛽)−→^𝑤∗𝜎(𝑎)𝑥^∗₀ and so 𝐷(𝑎)=𝜎(𝑎)𝑥^∗₀. Hence, 𝐷(𝑎)=𝜎(𝑎)𝑥^∗₀−𝑥^∗₀𝜎(𝑎) and 𝐷 is 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 𝒳 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. Let 𝒳 be a Banach 𝒜-bimodule. Then 𝒳 is called 𝜎-essential if 𝒳=𝜎(𝒜)𝒳𝜎(𝒜)=span𝜎(𝒜)⋅𝒳⋅𝜎(𝒜). Similarly, one defines 𝜎-essential left and right Banach modules.
We recall that a bounded approximate identity in Banach algebra 𝒜 for Banach 𝒜-bimodule 𝒳 is a bounded net (𝑒_𝛼) in 𝒜 such that for each 𝑥∈𝒳, 𝑒_𝛼𝑥→𝑥 and 𝑥𝑒_𝛼→𝑥.

Proposition 4.4. Assume that 𝒜 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. Let 𝒳 be a 𝜎-essential Banach 𝒜-bimodule. Since 𝜎 is idempotent, 𝜎(𝒜) is Banach subalgebra of 𝒜. Let (𝑒_𝛼)⊆𝒜 be left approximate identity with bound 𝑚. First suppose that 𝑧∈span𝜎(𝒜)⋅𝒳, so there exist 𝑎₁,…,𝑎_𝑛∈𝒜, 𝑥₁,…,𝑥_𝑛∈𝒳 such that 𝑧=∑^𝑛_𝑖=1𝜎(𝑎_𝑖)𝑥_𝑖. For 1≤𝑖≤𝑛, 𝑒_𝛼𝑎_𝑖→𝑎_𝑖 and, therefore, 𝜎(𝑒_𝛼)𝑧→𝑧.
Now suppose that 𝑧∈𝜎(𝒜)𝒳. There exists {𝑧_𝑛}⊆span𝜎(𝒜)⋅𝒳 such that 𝑧_𝑛→𝑧. Thus, ∃𝑛₀∈ℕs.t.∀𝑛𝑛≥𝑛₀;𝑧_𝑛<𝜀−𝑧2(‖𝜎‖𝑚+1) (4.2)
On the other hand, for each 𝑛∈ℕ we have 𝜎(𝑒_𝛼)𝑧−→_𝑛^𝛼𝑧_𝑛 and so 𝜎(𝑒_𝛼)𝑧−→_𝑛0^𝛼𝑧_𝑛0. Therefore, ∃𝛼₀;∀𝛼𝛼≥𝛼₀;𝜎𝑒_𝛼𝑧_𝑛0−𝑧_𝑛0<𝜀2. (4.3) Now we have 𝜎𝑒_𝛼𝑧−𝑧≤𝜎𝑒_𝛼𝑧−𝜎𝑒_𝛼𝑧_𝑛0+𝜎𝑒_𝛼𝑧_𝑛0−𝑧_𝑛0+𝑧_𝑛0≤‖𝜎𝑒−𝑧‖_𝛼𝑧−𝑧_𝑛0+𝜎𝑒_𝛼𝑧_𝑛0−𝑧_𝑛0+𝑧_𝑛0<‖𝜎𝑒−𝑧‖_𝛼+1𝑧−𝑧_𝑛0+𝜀2<𝜎‖)𝜀(‖𝑚+1𝜎‖)2+𝜀(‖𝑚+12=𝜀, (4.4) which shows that (𝜎(e_𝛼))⊆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. Every 𝜎-neo-unital left Banach 𝒜-module is essential.

Proof. Let 𝒳 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. Let 𝒜 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. Set 𝒴=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 that 𝜎(𝒜)⋅𝒳 is weakly complemented in 𝒳. Let (𝑒_𝛼) be a left approximate identity in 𝒜 with bound m, and define a net (𝑇_𝛼) in ℬ(𝒳^∗) by setting 𝑇_𝛼(𝑥^∗)=𝑥^∗⋅𝜎(𝑒_𝛼)(𝑥^∗∈𝒳^∗). We have ‖𝑇_𝛼‖≤‖𝜎‖𝑚. Thus (𝑇_𝛼) 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^∗−lim_𝛼𝑇_𝛼=𝑇 and ‖𝑇‖≤‖𝜎‖𝑚. For each 𝑎∈𝒜, 𝑥∈𝒳, and 𝑥^∗∈𝒳^∗, we have 𝜎(𝑎)⋅𝑥,𝑇𝑥^∗=lim_𝛼𝜎(𝑎)⋅𝑥,𝑥^∗⋅𝜎𝑒_𝛼=lim_𝛼𝜎𝑒_𝛼𝜎(𝑎)⋅𝑥,𝑥^∗=𝜎(𝑎)⋅𝑥,𝑥^∗, (4.5) and so 𝑥^∗−𝑇𝑥^∗∈(𝜎(𝒜)⋅𝒳)^⟂. On other hand, for each 𝑥^∗∈𝒳^∗, 𝑇²𝑥^∗=𝑇𝑇𝑥^∗=lim_𝛼𝑇𝑥^∗𝜎𝑒_𝛼=lim_𝛼𝑥^∗𝜎𝑒_𝛼=𝑇𝑥^∗. (4.6) Thus 𝑇 is projection, and 𝐼_𝒳∗−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. Let 𝒜 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. Set 𝒴=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=𝛿^𝜎_𝐽0. Thus 𝐷=𝑇𝑜𝐷+𝛿^𝜎_𝐽0 and so 𝐷 is 𝜎-a.i if and only if 𝑇𝑜𝐷∶𝒜→𝒴^∗ is 𝜎-a.i.
Now let 𝒵=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=𝛿^𝜎_𝑍0. Therefore, 𝐷₁=𝑇′𝑜𝐷₁+𝛿^𝜎_𝑍0. Thus, 𝐷₁ is 𝜎.a.i if and only if 𝑇′𝑜𝐷₁ is 𝜎.a.i. Set 𝐷𝑜𝑇=𝐷₁. Thus, 𝐷=𝑇′𝑜𝐷₁+𝛿^𝜎_𝑍0+𝛿^𝜎_𝐽0. 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. Let 𝒜 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 that 𝜎 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 (𝜇_𝛼)⊆(𝐴^#⊗𝒜^#)^∗∗ such that for each 𝑎∈𝒜^#, 𝜎(𝑎)⋅𝜇_𝛼−𝜇_𝛼⋅𝜎(𝑎)→0 and 𝜋^∗∗(𝜇_𝛼)→𝑒̂. (3)There is a net (𝜇^′_𝛼)⊆(𝐴^#⊗𝒜^#)^∗∗ such that for each 𝑎∈𝒜^#, 𝜎(𝑎)⋅𝜇_𝛼−𝜇_𝛼⋅𝜎(𝑎)→0 and for every 𝛼, 𝜋^∗∗(𝜇^′_𝛼)=𝑒̂.

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: 𝑎⋅(𝑏⊗𝑐)=𝜎(𝑎)(𝑏⊗𝑐),(𝑏⊗𝑐)⋅𝑎=(𝑏⊗𝑐)𝜎(𝑎)𝑎,𝑏,𝑐∈𝒜^#. (4.7) 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 (𝑒_𝛼)⊆ker𝜋^∗∗ such that 𝛿𝑢(𝑎)=lim_𝛼𝜎(𝑎)𝑒_𝛼−𝑒_𝛼𝜎(𝑎)𝑎∈𝒜^#. (4.8) Set 𝜇^′_𝛼=𝑢−𝑒_𝛼∈(𝒜^#⊗𝒜^#)^∗∗. We have 𝜎(𝑎)𝜇^′_𝛼−𝜇^′_𝛼𝜎(𝑎)=𝜎(𝑎)𝑢−𝑢𝜎(𝑎)−𝜎(𝑎)𝑒_𝛼−𝑒_𝛼𝜎(𝑎)⟶0, (4.9) and for each 𝛼, 𝜋^∗∗𝜇^′_𝛼=𝜋^∗∗𝑢−𝑒_𝛼=𝜋^∗∗(𝑢)−𝜋^∗∗𝑒_𝛼=𝜋(𝑢)=𝑒. (4.10)
(3 ⇒2) is clear.
(2 ⇒1) By Proposition 3.9, it is sufficient to show that 𝐴^# is 𝜎-a.a.
Let 𝐷∶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 𝑓_𝛼(𝑥)=𝜇_𝛼(𝜓_𝑥), where for 𝑎,𝑏∈𝒜^#, 𝑥∈𝒳, we have 𝜓_𝑥(𝑎⊗𝑏)=⟨𝑥,𝜎(𝑎)𝐷(𝑏)⟩. Then, (𝑚^𝛾_𝛼)⊂𝐴^#⊗𝒜^# converging 𝜔^∗ to 𝜇_𝛼(𝛼∈𝐼) and noting that for 𝑚∈𝒜^#⊗𝒜^#,𝑎∈𝒜^#,𝑥∈𝒳, then 𝜓𝜎_{(𝑎)𝑥−𝑥}𝜎_(𝑎)(𝑚)=𝜎(𝑎)𝜓_𝑥−𝜓_𝑥𝜎(𝑎)(𝑚)−⟨𝑥,𝜎(𝜋(𝑚))𝐷(𝑎)⟩. (4.11) Since 𝒳 is 𝜎-neo-unital, so 𝒳=𝒳𝜎(𝒜^#). So for each 𝑎∈𝐴 and 𝑥∈𝒳, we have 𝜎(𝑎)𝑥−𝑥𝜎(𝑎),𝑓_𝛼=𝜓𝜎_{(𝑎)𝑥−𝑥}𝜎_(𝑎),𝜇_𝛼=lim_𝛾𝑚^𝛾_𝛼,𝜓𝜎_{(𝑎)𝑥−𝑥}𝜎_(𝑎)=𝜎(𝑎)𝜓_𝑥−𝜓_𝑥𝜎(𝑎),𝜇_𝛼−lim_𝛾𝑥,𝜎𝜋𝑚^𝛾_𝛼𝐷(𝑎)=𝜓_𝑥,𝜇_𝛼𝜎(𝑎)−𝜎(𝑎)𝜇_𝛼−𝑥,𝜋^∗∗𝜇_𝛼𝐷(𝑎). (4.12) Thus, 𝑥,𝜎(𝑎)𝑓_𝛼−𝑓_𝛼𝜎(𝑎−⟨(𝑎≤𝜓)𝑥,𝐷)⟩_𝑥,𝜎(𝑎)𝜇_𝛼−𝜇_𝛼𝜎(𝑎)+‖𝑥‖𝜋^∗∗𝜇_𝛼𝐷(𝑎≤‖𝐷‖⋅‖𝑥𝜎(𝑎)𝜇−𝑒̂‖)‖‖_𝛼−𝜇_𝛼𝜎(𝑎)+‖𝑥‖𝜋^∗𝜇_𝛼−𝑒̂‖𝐷(𝑎)‖, (4.13) and, therefore, 𝐷=lim_𝛼𝛿𝜎_𝑓𝛼. It follows that 𝒜^# is 𝜎-a.a and so 𝒜 is 𝜎-a.a.

Proposition 4.10. Suppose that 𝒜 is 𝜎-a.a, and let Σ∶0⟶𝒳⟶^∗𝑓𝒴⟶^𝑔𝒵⟶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 𝐺_𝛼∶𝒵→𝒴 of right inverse maps to 𝑔 such that lim_𝛼(𝜎(𝑎)𝐺_𝛼−𝐺_𝛼𝜎(𝑎))=0 for 𝑎∈𝒜, and a net 𝐹_𝛼∶𝒴→𝒳^∗ of left inverse maps to 𝑓 such that lim_𝛼(𝜎(𝑎)𝑓_𝛼−𝑓_𝛼𝜎(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 (𝜑_𝛼)⊆ℬ(𝒵,𝒳^∗) such that 𝜎(𝑎)⋅𝐺−𝐺⋅𝜎(𝑎)=lim_𝛼𝜎(𝑎)⋅𝑓𝐺_𝛼−𝑓𝐺_𝛼⋅𝜎(𝑎)(𝑎∈𝒜). (4.15)
Setting 𝐺_𝛼=𝐺−𝑓𝜑_𝛼 gives the required net. Applying the same argument as [2, Proposition 1.1] provides (𝐹_𝛼).

We recall that if 𝒜 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 algebra 𝒜 is 𝜎-a.a, then 𝜎(𝒜) has left and right approximate identities.

Corollary 4.12. Suppose that Banach algebra 𝒜 is 𝜎-a.a and 𝜎 is a bounded epimorphism of 𝒜, then 𝒜 has left and right approximate identities.

Lemma 4.13. Let 𝜎 be a bounded idempotent endomorphism of Banach algebra 𝒜 and 𝒳 a 𝜎-neo-unital Banach 𝒜-module. If (𝑒_𝛼)_𝛼 is a bounded approximate identity in 𝒜, then (𝜎(𝑒_𝛼))_𝛼 is a bounded approximate identity for 𝒳.

Proof. For every 𝑎∈𝒜 we have 𝑒_𝛼𝜎(𝑎)→𝜎(𝑎). Since 𝜎 is idempotent, 𝜎(𝑒_𝛼)𝜎(𝑎)→𝜎(𝑎). For each 𝑥∈𝒳, there exists 𝑎∈𝒜 and 𝑦∈𝒳 such that 𝑥=𝜎(𝑎)⋅𝑦. Therefore, 𝜎𝑒_𝛼⋅𝑥=𝜎𝑒_𝛼𝜎(𝑎)⋅𝑦⟶𝜎(𝑎)⋅𝑦=𝑥, (4.16) which shows that (𝜎(𝑒_𝛼)) is a bounded approximate identity for 𝒳.

It is often convenient to extend a derivation to a large algebra. If a Banach algebra 𝐼 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 (𝑃_𝑖)_𝑖∈𝐼, where 𝑃_𝑖(𝑎)∶=‖𝑎𝑖‖+‖𝑖𝑎‖(𝑎∈𝒜). (4.17) Note that the strict topology is Hausdorff only if {𝑎∈𝒜∶𝑎⋅𝐼=𝐼⋅𝑎={0}}={0} [3].

Proposition 4.14. Let 𝒜 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. Since 𝒳 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 that 𝑎⋅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 (𝑒_𝛼)_𝛼 be a bounded approximate identity for 𝐼. For each 𝑎∈𝒜 and 𝑥∈𝒳 we have 𝑎⋅𝑥=𝜎(𝑎𝑖)⋅𝑦=lim_𝛼𝜎𝑎𝑒_𝛼𝑖⋅𝑦=lim_𝛼𝜎𝑎𝑒_𝛼𝜎(𝑖)⋅𝑦=lim_𝛼𝜎𝑎𝑒_𝛼𝑥=lim_𝛼𝜎𝑎𝑒_𝛼𝜎𝑖^′⋅𝑦′=lim_𝛼𝜎𝑎𝑒_𝛼𝑖^′⋅𝑦′=𝜎𝑎𝑖^′⋅𝑦^′. (4.18) 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 𝐷∶𝒜⟶𝒳^∗,𝑎⟶𝜔^∗−lim_𝛼𝐷𝑎𝑒_𝛼−𝜎(𝑎)⋅𝐷𝑒_𝛼. (4.19) 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(𝑒_𝛼) is bounded approximate identity for 𝒳, and we have 𝑥,𝐷𝑎𝑒_𝛼−𝜎(𝑎)⋅𝐷𝑒_𝛼=𝑦⋅𝜎(𝑖),𝐷𝑎𝑒_𝛼−𝜎(𝑎)⋅𝐷𝑒_𝛼=𝑦,𝜎(𝑖)𝐷𝑎𝑒_𝛼−𝜎(𝑖𝑎)⋅𝐷𝑒_𝛼=𝑦,𝐷𝑖𝑎𝑒_𝛼−𝐷(𝑖)𝜎𝑎𝑒_𝛼−𝐷𝑖𝑎𝑒_𝛼+𝐷(𝑖𝑎)𝜎𝑒_𝛼=𝜎𝑒_𝛼⋅𝑦,𝐷(𝑖𝑎)−𝜎𝑎𝑒_𝛼⋅𝑦,𝐷(𝑖)_𝛼⟶𝑦,𝐷(𝑖𝑎)−𝜎(𝑎)⋅𝑦,𝐷(𝑖)(𝑎∈𝒜). (4.20) So the limit in (4.19) exists. Furthermore, for 𝑖∈𝐼, 𝐷(𝑖)=𝜔^∗−lim_𝛼𝐷𝑖𝑒_𝛼−𝜎(𝑖)⋅𝐷𝑒_𝛼=𝜔^∗−lim_𝛼𝐷𝑖𝑒_𝛼−𝐷𝑖𝑒_𝛼+𝐷(𝑖)𝜎𝑒_𝛼=𝐷(𝑖), (4.21) so 𝐷 is an extension of 𝐷. Also for 𝑎∈𝒜 and 𝑖∈𝐼 we have 𝐷𝑎⋅𝜎(𝑖)=𝜔^∗−lim_𝛼𝐷𝑎𝑒_𝛼⋅𝜎(𝑖)−𝜎(𝑎)⋅𝐷𝑒_𝛼⋅𝜎(𝑖)=𝜔^∗−lim_𝛼𝐷𝑎𝑒_𝛼𝑖−𝜎𝑎𝑒_𝛼⋅𝐷(𝑖)−𝜎(𝑎)⋅𝐷𝑒_𝛼𝑖+𝜎(𝑎)𝜎𝑒_𝛼⋅𝐷(𝑖)=𝜔^∗−lim_𝛼𝐷𝑎𝑒_𝛼𝑖−𝜎(𝑎)⋅𝐷𝑒_𝛼𝑖=𝐷(𝑎𝑖)−𝜎(𝑎)⋅𝐷(𝑖). (4.22) We claim that 𝐷 is continuous with respect to the strict topology on 𝒜 and the 𝜔^∗-topology an 𝒳^∗.
Let 𝑎−−−→_𝑛^strict𝑎 in 𝒜. ∀𝑖∈𝐼,𝑎_𝑛𝑖+𝑖𝑎_𝑛⟶‖𝑎𝑖‖+‖𝑖𝑎‖. (4.23) For each 𝑥∈𝒳, 𝑥,𝐷𝑎_𝑛−𝑥,𝐷(𝑎)=lim_𝛼𝑥,𝐷𝑎_𝑛𝑒_𝛼−𝜎𝑎_𝑛⋅𝐷𝑒_𝛼−𝑥,𝐷𝑎𝑒_𝛼−𝜎(𝑎)⋅𝐷𝑒_𝛼=lim_𝛼𝑥,𝐷𝑎_𝑛𝑒_𝛼−𝐷𝑎𝑒_𝛼−𝜎𝑎_𝑛𝐷𝑒_𝛼+𝜎(𝑎)𝐷𝑒_𝛼≤lim_𝛼‖𝑥‖𝐷𝑎_𝑛𝑒_𝛼−𝐷𝑎𝑒_𝛼−𝜎𝑎₀⋅𝐷𝑒_𝛼−𝜎𝑎_𝑛𝐷𝑒_𝛼≤lim_𝛼‖𝑥‖‖𝐷‖𝑎_𝑛𝑒_𝛼−𝑎𝑒_𝛼+𝜎𝑎_𝑛−𝜎(𝑎)𝐷𝑒_𝛼≤lim_𝛼‖𝑥‖‖𝐷‖𝑎_𝑛−𝑎𝑒_𝛼+‖𝜎‖𝑎_𝑛−𝑎𝐷𝑒_𝛼⟶0, (4.24) so 𝐷 is continuous.
It remains to show that 𝐷 is a 𝜎-derivation. From the definition of the strict topology, we have 𝑎𝑒_𝛼→𝑎 in the strict topology for all 𝑎∈𝒜 because ‖𝑎𝑒_𝛼𝑖‖+‖𝑖𝑎𝑒_𝛼‖−→^𝛼‖𝑎𝑖‖+‖𝑖𝑎‖(𝑖∈𝐼) and so 𝐷(𝑎𝑒_𝛼)−→^𝑤∗𝐷(𝑎). Therefore, 𝐷(𝑎𝑏)=𝜔^∗−lim_𝛼lim_𝛽𝐷𝑎𝑒_𝛼𝑏𝑒_𝛽=𝜔^∗−lim_𝛼lim_𝛽𝐷𝑎𝑒_𝛼𝑏𝑒_𝛽=𝜔^∗−lim_𝛼lim_𝛽𝜎𝑎𝑒_𝛼𝐷𝑏𝑒_𝛽+𝐷𝑎𝑒_𝛼⋅𝜎𝑏𝑒_𝛽=𝜔^∗−lim_𝛼lim_𝛽𝜎𝑎𝑒_𝛼𝐷𝑏𝑒_𝛽+𝐷𝑎𝑒_𝛼⋅𝜎𝑏𝑒_𝛽=𝜎(𝑎)𝐷(𝑏)+D(𝑎)𝜎(𝑏), (4.25) that is, 𝐷 is 𝜎-derivation.

Corollary 4.15. Suppose that 𝒜 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 that 𝐼 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, ∃𝑥^∗_𝛼⊆𝒳^∗s.t.𝐷(𝑎)=lim_𝛼𝜎(𝑎)⋅𝑥^∗_𝛼−𝑥^∗_𝛼⋅𝜎(𝑎)(𝑎∈𝒜). (4.26) So we have 𝐷(𝑖)=𝐷(𝑖)=lim_𝛼𝜎(𝑖)⋅𝑥^∗_𝛼−𝑥^∗_𝛼⋅𝜎(𝑖), which shows that 𝐷=lim_𝛼𝛿^𝜎_𝑥∗𝛼 is 𝜎-a.i, and 𝐼 is 𝜎-a.a.

Corollary 4.16. Let 𝒜 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. Let 𝐼 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. Let 𝒳 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: 𝑎⋅𝑥=𝜋(𝑎)⋅𝑥,𝑥⋅𝑎=𝑥⋅𝜋(𝑎)(𝑎∈𝒜,𝑥∈𝒳), (4.27) 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 {𝑥_𝛼}⊆𝒳 such that 𝐷𝑜𝜋(𝑎)=lim_𝛼𝜎(𝑎)⋅𝑥_𝛼−𝑥_𝛼⋅𝜎(𝑎)(𝑎∈𝒜). Therefore, for each (𝑎∈𝐴), 𝐷(𝑎+𝐼)=𝐷𝑜𝜋(𝑎)=lim_𝛼𝜎(𝑎)⋅𝑥_𝛼−𝑥_𝛼⋅𝜎(𝑎)=lim_𝛼𝜋(𝜎(𝑎))⋅𝑥_𝛼−𝑥_𝛼⋅𝜋(𝜎(𝑎))=lim_𝛼(𝜎(𝑎)+𝐼)⋅𝑥_𝛼−𝑥_𝛼⋅(𝜎(𝑎)+𝐼)=lim_𝛼𝜎(𝑎+𝐼)𝑥_𝛼−𝑥_𝛼𝜎(𝑎+𝐼). (4.28) Thus, 𝒜/𝐼 is 𝜎-a.c.

Proposition 4.18. Suppose that 𝐼 is a closed ideal in 𝒜. If 𝐼 is 𝜎-amenable and 𝐴/𝐼 is a.a, then 𝒜 is 𝜎-a.a.

Proof. Let 𝒳 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 𝑥^∗₀∈𝒳^∗ such that 𝐷=𝛿^𝜎_𝑥^∗₀, and, therefore, for each 𝑖∈𝐼 we have 𝑑(𝑖)=𝜎(𝑖)⋅𝑥^∗₀−𝑥^∗₀⋅𝜎(𝑖). Set 𝐷₁=𝐷−𝛿^𝜎_𝑥^∗₀. 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: (𝑎+𝐼)𝑥+𝒳₀=𝜎(𝑎)𝑥+𝒳₀,𝑥+𝒳₀(𝑎+𝐼)=𝑥𝜎(𝑎)+𝒳₀(𝑥∈𝒳,𝑎∈𝒜). (4.29) Now we define 𝒜𝐷∶𝐼⟶𝒳𝒳₀^∗;𝑥+𝒳₀,𝐷(𝑎+𝐼)=𝑥,𝐷₁(𝑎)(𝑎∈𝒜,𝑥∈𝒳). (4.30) 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 𝑥+𝑋₀,𝐷(𝑎+𝐼)=𝑥′+𝑋₀,𝐷𝑎^′+𝐼. (4.31) Thus, ⟨𝑥,𝐷₁(𝑎)⟩=⟨𝑥′,𝐷₁(𝑎′)⟩=⟨𝑥′,𝐷₁(𝑎)⟩, and, therefore, 𝑥−𝑥^′,𝐷₁(𝑎)=0. (4.32) It is enough to show that 𝐷₁(𝑎) is zero on 𝒳₀. Suppose that 𝜎(𝑖)𝑥∈𝒳₀, we have 𝜎(𝑖)𝑥,𝐷₁(𝑎)=𝑥,𝐷₁(𝑎)𝜎(𝑖)=𝑥,𝐷₁(𝑎𝑖)−𝜎(𝑎)𝐷₁(𝑖(𝑖))=0,𝑥𝜎,𝐷₁(𝑎)=𝑥,𝜎(𝑖)𝐷₁(𝑎)=𝑥,𝐷₁(𝑖𝑎)−𝐷₁(𝑖)𝜎(𝑎)=0. (4.33) 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; 𝑥+𝒳₀,𝐷((𝑎+𝐼)(𝑏+𝐼))=𝑥,𝐷₁(=(𝑎)𝐷𝑎𝑏)𝑥,𝜎₁(𝑏)+𝐷₁(𝑎)𝜎(𝑏=(𝑎))𝑥𝜎,𝐷₁(𝑏)+𝜎(𝑏)𝑥,𝐷₁(𝑎=(𝑎))𝑥𝜎+𝒳₀,𝐷(+𝜎(𝑏)𝑏+𝐼)𝑥+𝒳₀,𝐷(𝑎+I=)𝑥+𝒳₀(),𝐷(+)𝑎+𝐼𝑏+𝐼)(𝑏+𝐼𝑥+𝒳₀,𝐷(𝑎+𝐼). (4.34) So there exists a net (𝜑_𝛼)⊆(𝒳/𝒳₀)^∗ such that 𝐷=lim_𝛼𝛿_𝜑𝛼. Let 𝑞∶𝒳→𝒳/𝒳₀ be the quotient map. For every 𝛼, (𝜑_𝛼𝑜𝑞)∈𝒳^∗. Set (𝑥^∗_𝛼)=(𝜑_𝛼𝑜𝑞)⊆𝒳^∗. We have 𝑥,𝐷₁(𝑎)=𝑥+𝑋₀,𝐷(=𝑎+𝐼)𝑥+𝑋₀,lim_𝛼(𝑎+𝐼)𝜑_𝛼−𝜑_𝛼(𝑎+𝐼)=lim_𝛼𝑥𝜎(𝑎)+𝑋₀,𝜑_𝛼−𝜎(𝑎)𝑥+𝑋₀,𝜑_𝛼=lim_𝛼𝑞(𝑥𝜎(𝑎)),𝜑_𝛼−𝑞(𝜎(𝑎)𝑥),𝜑_𝛼=lim_𝛼𝜑_𝛼𝑜𝑞(𝑥𝜎(𝑎)−𝜎(𝑎)𝑥)=𝑥𝜎(𝑎)−𝜎(𝑎)𝑥,𝑥^∗_𝛼=lim_𝛼𝑥,𝜎(𝑎)𝑥^∗_𝛼−𝑥^∗_𝛼𝜎(𝑎=)𝑥,lim_𝛼𝛿^𝜎_𝑥∗𝛼(𝑎). (4.35) So 𝐷₁=𝐷−𝛿^𝜎_𝑥∗=lim_𝛼𝛿^𝜎_𝑥∗𝛼, and, therefore, 𝐷=lim_𝛼𝛿^𝜎_{(𝑥∗𝛼−𝑥}^∗₀₎. Which shows that 𝐷 is 𝜎-a.i and so 𝒜 is 𝜎-a.a.

Example 4.19. Let 𝒜 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 𝒜_𝜑. It is easy to see that 𝒜_𝜑 has a left identity 𝑒, while it has not right approximate identity, so 𝒜_𝜑 is not contractible and is not approximately contractible. Also 𝒜_𝜑 is biprojective. Now suppose that 𝜎∶𝒜_𝜑→𝒜_𝜑 be defined by 𝜎(𝑎)=𝜑(𝑎)𝑒. We have 𝜎²(𝑎)=𝜎𝜑(𝑎)𝑒=𝜑(𝑎)𝜎(𝑒)=𝜑(𝑎)𝜑(𝑒)𝑒=𝜑(𝑎)𝑒=𝜎(𝑎). (4.36) Thus 𝜎 is idempotent. It is easy to see that 𝑒 is identity for 𝜎(𝒜_𝜑), and since 𝒜 is biprojective by [1, Corollary 5.3], 𝒜_𝜑 is 𝜎-biprojective. Thus by [1, Theorem 4.3], 𝒜_𝜑 is 𝜎-contractible and so 𝒜_𝜑 is 𝜎-a.c.
It is easy to see that ker𝜑 and all subspaces of ker𝜑 are all ideals of 𝒜_𝜑 and 𝜎(ker𝜑)⊆ker𝜑 so 𝜎(𝐼)⊆𝐼 for each ideal of 𝒜. Therefore, by Proposition 4.17, 𝒜_𝜑/𝐼 is 𝜎-a.c for each ideal 𝐼 of 𝒜_𝜑, where 𝜎(𝑎+𝐼)=𝜎(𝑎)+𝐼=𝜑(𝑎)e+𝐼.

Corollary 4.20. Suppose that 𝜎 is a bounded idempotent endomorphism of Banach algebra 𝒜. Then 𝒜 is 𝜎-a.a if and only if there are nets (𝜇_𝛼^′′) in (𝒜⊗𝒜)^∗∗ and (𝐹_𝛼),(𝐺_𝛼)⊆𝒜^∗∗, such that for each 𝑎∈𝒜, (1)𝜎(𝑎)⋅𝜇_𝛼^′′−𝜇_𝛼^′′⋅𝜎(𝑎)+𝐹_𝛼⊗𝜎(𝑎)−𝜎(𝑎)⊗𝐺_𝛼→0, (2)𝜎(𝑎)⋅𝐹_𝛼→𝜎(𝑎),𝐺_𝛼⋅𝜎(𝑎)→𝜎(𝑎), (3)𝜋^∗∗(𝜇_𝛼^′′)⋅𝜎(𝑎)−𝐹_𝛼⋅𝜎(𝑎)−𝐺_𝛼⋅𝜎(𝑎)→0.

Proof. Suppose that 𝒜 is 𝜎-a.a, take the net (𝜇_𝛼) given in Proposition 4.9 and write 𝜇_𝛼=𝜇_𝛼^′′−𝐹_𝛼⊗𝑒̂−𝑒̂⊗𝐺_𝛼+𝑐_𝛼𝑒̂⊗𝑒,̂ (4.37) where (𝜇_𝛼^′′)⊆(𝐴⊗𝐴)^∗∗, (𝐹_𝛼),(𝐺_𝛼)⊆𝐴^∗∗, and (𝑐_𝛼)⊆ℂ. Applying 𝜋^∗∗, 𝜋^∗∗(𝜇_𝛼^′′)−𝐹_𝛼−𝐺_𝛼+𝑐_𝛼𝑒̂→𝑒̂, hence 𝑐_𝛼→1, then 𝜋^∗∗𝜇_𝛼^′′⋅𝜎(𝑎)−𝐹_𝛼⋅𝜎(𝑎)−𝐺_𝛼⋅𝜎(𝑎)+𝑒̂⋅𝜎(𝑎)⟶𝑒̂⋅𝜎(𝑎)(𝑎∈𝒜). (4.38) So we have (iii) further, by Proposition 4.9, for 𝑎∈𝒜^#, 𝜎(𝑎)⋅𝜇_𝛼^′′−𝜎(𝑎)⋅𝐹_𝛼⊗𝑒̂−𝜎(𝑎)⊗𝐺_𝛼+𝜎(𝑎)⊗𝑒̂+𝜇_𝛼^′′⋅𝜎(𝑎)+𝐹_𝛼⊗𝜎(𝑎)+𝑒̂⊗𝐺_𝛼⋅𝜎(𝑎)−𝑒̂⊗𝜎(𝑎)⟶0. (4.39) Thus 𝜎(𝑎)⋅𝜇_𝛼^′′−𝜇_𝛼^′′⋅𝜎(𝑎)+𝐹_𝛼⊗𝜎(𝑎)−𝜎(𝑎)⊗𝐺_𝛼→0, and 𝜎(𝑎)⋅𝐹_𝛼→𝜎(𝑎),𝐺_𝛼⋅𝜎(𝑎)→𝜎(𝑎). So for 𝑎∈𝒜, 𝜎(𝑎)⋅𝜇_𝛼^′′−𝜇_𝛼^′′⋅𝜎(𝑎)+𝐹_𝛼⊗𝜎(𝑎)−𝜎(𝑎)⊗𝐺_𝛼𝜎(𝑎)⟶0,⋅𝐹_𝛼⟶𝜎(𝑎),𝐺_𝛼⋅𝜎(𝑎)⟶𝜎(𝑎). (4.40)
Conversely, set 𝑐_𝛼=1 and 𝜇_𝛼=𝜇_𝛼^′′−𝐹_𝛼⊗𝑒̂−𝑒̂⊗𝐺_𝛼+𝑒̂⊗𝑒̂. We have 𝜎(𝑎+𝛼)⋅𝜇_𝛼−𝜇_𝛼⋅𝜎(𝑎+𝛼)=(𝜎(𝑎)+𝛼)⋅𝜇_𝛼−𝜇_𝛼⋅(𝜎(𝑎))(𝑎)+𝛼=𝜎⋅𝜇_𝛼−𝜇_𝛼⋅𝜎(𝑎)+𝑎𝜇_𝛼−𝛼𝜇_𝛼=𝜎(𝑎)⋅𝜇_𝛼−𝜇_𝛼(𝑎)(𝑎)⋅𝜎=𝜎⋅𝜇_𝛼^′′−𝜎(𝑎)𝐹_𝛼⊗𝑒−𝜎(𝑎)⊗𝐺_𝛼+𝜎(𝑎)⊗𝑒−𝜇_𝛼^′′⋅𝜎(𝑎)+𝐹_𝛼⊗𝜎(𝑎)+𝑒⊗𝐺_𝛼𝜎(𝑎)(𝑎)(𝑎)−𝑒⊗𝜎=𝜎⋅𝜇_𝛼^′′−𝜇_𝛼^′′⋅𝜎(𝑎)+𝐹_𝛼⊗𝜎(𝑎)−𝜎(𝑎)⊗𝐺_𝛼→0(𝑎∈𝒜). (4.41) So 𝜎(𝑎)⋅𝜇_𝛼−𝜇_𝛼⋅𝜎(𝑎)→0(𝑎∈𝒜^#). Also 𝜋^∗∗𝜇_𝛼⋅𝜎(𝑎)=𝜋^∗∗𝜇_𝛼^′′−𝐹_𝛼⊗𝑒̂−𝑒̂⊗𝐺_𝛼+𝑒̂⊗𝑒̂𝜎(𝑎)=𝜋^∗∗𝜇_𝛼^′′𝜎(𝑎)−𝐹_𝛼⋅𝜎(𝑎)−𝐺_𝛼⋅𝜎(𝑎)+𝜎(𝑎)⟶𝜎(𝑎)(𝑎∈𝒜), (4.42) and so 𝜋^∗∗(𝜇_𝛼)→𝑒̂. Now, by Proposition 4.9, 𝒜 is 𝜎-a.a.

For 𝜎-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 (𝜇_𝛼)⊂𝒜^#⊗𝒜^# such that for each 𝑎∈𝒜^#, 𝜎(𝑎)⋅𝜇_𝛼−𝜇_𝛼⋅𝜎(𝑎)→0 and 𝜋(𝜇_𝛼)→𝑒; (2)there is a net (𝜇^′_𝛼)⊂𝒜^#⊗𝒜^# such that for each 𝑎∈𝒜^#, 𝜎(𝑎)⋅𝜇^′_𝛼−𝜇^′_𝛼⋅𝜎(𝑎)→0 and 𝜋(𝜇^′_𝛼)=𝑒; (3)there are nets (𝜇_𝛼^′′)⊂𝒜⊗𝒜, (𝐹_𝛼),(𝐺_𝛼)⊂𝒜, such that for each 𝑎∈𝒜,(i)𝜎(𝑎)⋅𝜇_𝛼^′′−𝜇_𝛼^′′⋅𝜎(𝑎)+𝐹_𝛼⊗𝜎(𝑎)−𝜎(𝑎)⊗𝐺_𝛼→0;(ii)𝜎(𝑖)⋅𝐹_𝛼→𝜎(𝑎),𝐺_𝛼⋅𝜎(𝑎)→𝜎(𝑎); (iii)𝜋(𝜇_𝛼^′′)⋅𝜎(𝑎)−𝐹_𝛼⋅𝜎(𝑎)−𝐺_𝛼⋅𝜎(𝑎)→0.

We know Banach algebra 𝒜 is amenable if and only if 𝒜 has bounded approximate diagonal [3].

Proposition 4.22. Banach algebra 𝒜 is 𝜎-amenable if and only if 𝒜 has bounded approximate 𝜎-diagonal, that is, there is a bounded net (𝜇_𝛼)⊆𝒜⊗𝒜 such that for each 𝑎∈𝒜, 𝜎(𝑎)⋅𝜇_𝛼−𝜇_𝛼⋅𝜎(𝑎)→0 and 𝜋(𝜇_𝛼)⋅𝜎(𝑎)→𝜎(𝑎).

Proposition 4.23. If Banach algebra 𝒜 is 𝜎-amenable, then 𝒜 is 𝜎-a.c.

Proof. Suppose that 𝒜 is 𝜎-amenable. Then there exists a bounded net (𝜇_𝛼) in 𝒜⊗𝒜 such that for each 𝑎∈𝒜, 𝜎(𝑎)⋅𝜇_𝛼−𝜇_𝛼⋅𝜎(𝑎)⟶0,𝜋𝜇_𝛼⋅𝜎(𝑎)⟶𝜎(𝑎). (4.43) Set 𝑓_𝛼=𝜋(𝜇_𝛼). It is easy to see that (𝑓_𝛼) is a bounded approximate identity. Then 𝜇_𝛼^′′=𝜇_𝛼+𝑓_𝛼⊗𝑓_𝛼 and 𝐹_𝛼=𝐺_𝛼=𝑓_𝛼 satisfy (i)–(iii) of Proposition 4.21, because(i)𝜎(𝑎)⋅𝜇_𝛼^′′−𝜇_𝛼^′′⋅𝜎(𝑎)+𝑓_𝛼⊗𝜎(𝑎)−𝜎(𝑎)⊗𝑓_𝛼=𝜎(𝑎)⋅𝜇_𝛼−𝜇_𝛼⋅𝜎(𝑎)+𝜎(𝑎)𝑓_𝛼⊗𝑓_𝛼−𝑓_𝛼⊗𝜎(𝑎) + 𝑓_𝛼⊗𝜎(𝑎)−𝜎(𝑎)⊗𝑓_𝛼→0(𝑎∈𝒜),(ii)𝜎(𝑎)⋅𝑓_𝛼=𝜎(𝑎)⋅𝜋(𝜇_𝛼)→𝜎(𝑎), 𝑓_𝛼⋅𝜎(𝑎)=𝜋(𝜇_𝛼)⋅𝜎(𝑎)→𝜎(𝑎),(iii)𝜋(𝜇_𝛼^′′)⋅𝜎(𝑎)=𝜋(𝜇_𝛼+𝑓_𝛼⊗𝑓_𝛼)⋅𝜎(𝑎)=𝑓_𝛼⋅𝜎(𝑎)+𝑓²_𝛼⋅𝜎(𝑎).
So 𝜋𝜇_𝛼^′′⋅𝜎(𝑎)−𝐹_𝛼⋅𝜎(𝑎)−𝐺_𝛼⋅𝜎(𝑎)=𝑓_𝛼⋅𝜎(𝑎)+𝑓²_𝛼⋅𝜎(𝑎)−𝑓_𝛼⋅𝜎(𝑎)−𝑓_𝛼⋅𝜎(𝑎)⟶0. (4.44) Note that 𝑓²_𝛼 is a bounded approximate identity too, thus, by Proposition 4.21, 𝒜 is 𝜎-a.c.

Corollary 4.24. Suppose that 𝒜 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 {𝑒_𝛼} be a bounded approximate identity in 𝜎(𝐼), so {𝑒_𝛼̂} is bounded net in 𝜎(𝐼)^∗∗, and so by Banach-Alaoglu theorem there exists a subnet {𝑒_𝛽̂}⊆{𝑒_𝛼̂} and 𝐸∈𝜎(𝐼)^∗∗ such that 𝑒̂−→_𝛽^𝑤∗𝐸. 𝐸 is a right identity in 𝜎(𝐼)^∗∗ because for each 𝐹∈𝜎(𝐼)^∗∗ and 𝑓∈𝜎(𝐼)^∗, 𝑓,𝐹□𝐸=𝑓⋅𝐹,𝐸=lim_𝛽𝑒_𝛽,𝑓𝐹=lim_𝛽𝑒_𝛽𝑓,𝐹=𝑓,𝐹. (4.45)
Also 𝐸 acts as an identity on 𝜎(𝐼) itself. Let (𝜇_𝛼),(𝐹_𝛼),(𝐺_𝛼) be the nets given by Corollary 4.20 for 𝒜. Define 𝜇^′_𝛼=𝐸⋅𝜇_𝛼⋅𝐸∈(𝐼⊗𝐼)^∗∗,𝐹^′_𝛼=𝐸⋅𝐹_𝛼∈𝐼^∗∗, and 𝐺^′_𝛼=𝐺_𝛼⋅𝐸∈𝐼^∗∗. Then, for 𝑖∈𝐼,
(i) we consider 𝜎(𝑖)⋅𝜇^′_𝛼−𝜇^′_𝛼⋅𝜎(𝑖)+𝐹^′_𝛼⊗𝜎(𝑖)−𝜎(𝑖)⊗𝐺^′_𝛼=𝜎(𝑖)⋅𝐸⋅𝜇_𝛼⋅𝐸−𝐸⋅𝜇_𝛼⋅𝐸⋅𝜎(𝑖)+𝐸⋅𝐹_𝛼⊗𝜎(𝑖)−𝜎(𝑖)⊗𝐺_𝛼(𝑖)⋅𝐸=𝜎⋅𝜇_𝛼⋅𝐸−𝐸⋅𝜇_𝛼𝜎(𝑖)+𝐸⋅𝐹_𝛼⊗𝜎(𝑖)−𝜎(𝑖)⊗𝐺_𝛼(𝑖)⋅𝐸=𝐸⋅𝜎⋅𝜇_𝛼⋅𝐸−𝐸⋅𝜇_𝛼⋅𝜎(𝑖)⋅𝐸+𝐸⋅𝐹_𝛼⊗𝜎(𝑖)⋅𝐸−𝐸⋅𝜎(𝑖)⊗𝐺_𝛼𝜎(𝑖)⋅𝐸=𝐸⋅𝜇_𝛼−𝜇_𝛼⋅𝜎(𝑖)+𝐹_𝛼⊗𝜎(𝑖)−𝜎(𝑖)⊗𝐺_𝛼⋅𝐸⟶0, (4.46)
(ii) we consider 𝜎(𝑖)⋅𝐹^′_𝛼=𝜎(𝑖)⋅𝐸⋅𝐹_𝛼=𝜎(𝑖)⋅𝐹_𝛼(𝑖),𝐺⟶𝜎_′𝛼⋅𝜎(𝑖)=𝐺_𝛼⋅𝐸⋅𝜎(𝑖)=𝐺_𝛼⋅𝜎(𝑖)⟶𝜎(𝑖) (4.47)
(iii) we consider 𝜋^∗∗𝜇^′_𝛼⋅𝜎(𝑎)−𝐹^′_𝛼⋅𝜎(𝑎)−𝐺^′_𝛼−𝜎(𝑎)=𝜋^∗∗𝐸⋅𝜇_𝛼⋅𝐸⋅𝜎(𝑎)−𝐸⋅𝐹_𝛼⋅𝜎(𝑎)−𝐺_𝛼⋅𝐸⋅𝜎(𝑎)=𝐸⋅𝜋^∗∗𝜇_𝛼⋅𝐸⋅𝜎(𝑎)−𝐸⋅𝐹_𝛼⋅𝜎(𝑎)−𝐺_𝛼⋅𝜎(𝑎)=𝐸⋅𝜋^∗∗𝜇_𝛼⋅𝜎(𝑎)−𝐸⋅𝐹_𝛼⋅𝜎(𝑎)−𝐺_𝛼⋅𝜎(𝑎)−𝐸⋅𝐺_𝛼𝜎(𝑎)+𝐸⋅𝐺_𝛼𝜎(𝑎)=𝐸⋅𝜋^∗∗𝜇_𝛼⋅𝜎(𝑎)−𝐹_𝛼.𝜎(𝑎)−𝐺_𝛼𝜎(𝑎)+(𝐸−𝑒)̂𝐺_𝛼𝜎(𝑎)⟶0. (4.48) An alternative proof would be to follow the standard argument stated in Corollary 4.15.