#### Abstract

We investigate -approximate contractibility and -approximate amenability of Banach algebras, which are extensions of usual notions of contractibility and amenability, respectively, where is a dense range or an idempotent bounded endomorphism of the corresponding Banach algebra.

#### 1. Introduction

For a Banach algebra , an -bimodule will always refer to a Banach -bimodule , that is, a Banach space which is algebraically an -bimodule, and for which there is a constant such that for , we have A derivation is a linear map, always taken to be continuous, satisfying A Banach algebra is amenable if for any -bimodule , any derivation is inner, that is, there exists , with Let be a Banach algebra and a bounded endomorphism of , that is, a bounded Banach algebra homomorphism from into . A -derivation from into a Banach -bimodule is a bounded linear map satisfying For each , the mapping defined by , for all , is a -derivation called an inner -derivation.

*Remark 1.1. *Throughout this paper, we will assume that is a Banach algebra, and is a bounded endomorphism of unless otherwise specified. Also, we write (-a.i) for -approximately inner, (-a.a) for -approximately amenable, and (-a.c) for -approximately contractible.

The basic definition for the present paper is as follows.

*Definition 1.2. *A -derivation is -a.i, if there exists a net such that for every , , the limit being in norm and we write . Note that we do not suppose to be bounded.

*Definition 1.3. *A Banach algebra is called -a.c if for any -bimodule , every -derivation is -a.i.

*Definition 1.4. *A Banach algebra is called -a.a if for any -bimodule , every -derivation is -a.i.

*Definition 1.5. *Let be a Banach algebra, and let and be Banach -bimodules. The linear map is called a --bimodule homomorphism if

#### 2. Basic Properties

Proposition 2.1. *Let be a -a.c Banach algebra. Then has a left and right approximate identity.*

*Proof. *Consider as a Banach -bimodule with the trivial right action, that is,
Then defined by is a -derivation, and so there is a net such that . Hence for each ,
which shows that is a right approximate identity for . Similarly, one can find a left approximate identity for .

Corollary 2.2. *Let be a -a.c Banach algebra and a continuous epimorphism of . Then has a left and right approximate identity.*

Proposition 2.3. *Let be a bounded endomorphism of Banach algebra . If is -a.c, then is -a.c.*

*Proof. *Let be a Banach -bimodule and let be a -derivation. Then is an -bimodule with the following module actions:
For each , we have
Thus is a continuous -derivation. Since is -a.c, there exists a net such that . In fact,
Therefore, is a -a.i and so is -a.c.

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

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

*Proof. *Let be a Banach -bimodule and let be a continuous derivation. It is easy to see that is a -derivation. Since is -a.c, there exists a net such that . Now for there exists such that , and, therefore,
which shows that is approximately inner and so is a.c.

Corollary 2.6. *Let be a bounded endomorphism of Banach algebra . If is -a.a then it is -a.a too.*

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

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

Proposition 2.9. *Suppose that is a Banach algebra and is a continuous epimorphism. If is a.c, then is -a.c for each bounded endomorphism of .*

*Proof. *Let be a bounded endomorphism of and a Banach -bimodule, then is an -bimodule with the following module actions:
Now let be a continuous -derivation. It is easy to check that is a derivation. Since is approximately contractible, there exists a net such that . We have
Since is an epimorphism, so for each there exists such that , and we have
which shows that is -a.i and so is -a.c.

Proposition 2.10. *Suppose that and are Banach algebras, and let and be bounded endomorphism of and , respectively. Let be a bounded epimorphism such that . If is -a.c, then is -a.c.*

*Proof. *Let be a Banach -bimodule and a continuous -derivation. Then is an -bimodule with the following actions:
It is easy to check that is a -derivation. Since is -a.c, there exists a net such that , so we have
Since is epimorphism, so for all , and hence is -a.c.

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

In this section we state some properties of -approximate contractibility when 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 is dense in .

Proposition 3.1. *Let be a unital Banach algebra with unit e, dense in , a Banach -bimodule, and a -derivation. Then, there is a -derivation and , such that .*

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

*Definition 3.2. *Let be a unital Banach algebra with identity . Banach -bimodule is called -unital if .

Corollary 3.3. *Let be a unital Banach algebra and dense in . Then, is -a.c (resp., -a.a) if and only if for all -unital Banach -bimodule , every -derivation is -a.i.*

*Proof. *Since is a unit for , and is dense in , we see that , so that is a -unital Banach -bimodule. Now by Proposition 3.1, the proof is complete.

Corollary 3.4. *Suppose that is a unital Banach algebra and is dense in . Let be a Banach -bimodule and a -derivation. If is -a.a, then there exists a net , and , such that .*

*Proof. *By Proposition 3.1, such that and is a -derivation. Since and is -a.a, is -a.i. Hence for some net .

In the following proposition we consider -approximate contractibility when is an idempotent endomorphism of . We can see the proof of the following proposition in [1, Proposition 4.1].

Proposition 3.5. *Assume that has an element which is a unit for and is a Banach -bimodule. If is a bounded idempotent endomorphism of , then for each -derivation there exists a -derivation and , such that .*

Corollary 3.6. *Assume that has an element which is a unit for and is a bounded idempotent endomorphism of , then is -a.c (resp., -a.a) if and only if for all -unital Banach -bimodule, , every -derivation (resp., is -a.i.*

Lemma 3.7. *Assume that is a unital Banach algebra with the identity , and is a -unital Banach -bimodule with the following module actions:
**
If is a -derivation, then .*

*Proof. *We have and
Hence and so . Hence .

Proposition 3.8. *Let be a bounded idempotent endomorphism of Banach algebra . If is -a.a, then is -a.a, where is the endomorphism of induced by , that is, .*

*Proof. *Let be a Banach -bimodule and a continuous -derivation. By Proposition 3.5, there exits and such that . Set . It is easy to check that is a -derivation. Since is -a.a, there exists a net such that . Hence . Since is -unital, by Lemma 3.7, and for each we have
This shows that is -a.i, and so is -a.a.

Proposition 3.9. *Let be a bounded endomorphism of Banach algebra . If is -a.a, then is -a.a.*

*Proof. *Let be a Banach -bimodule and a continuous -derivation. is a Banach -bimodule with the following module actions:
for all . Define with . Clearly, is a continuous -derivation. Hence, there is a net such that . Hence, for each we have
which shows that is -a.i and so is -a.a.

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

Lemma 4.1. *Let be a Banach algebra with bounded approximate identity and a Banach -bimodule with trivial left or right action, then every -derivation is -inner.*

*Proof. *Let be a Banach -bimodule with trivial left action. Hence, is a Banach -bimodule with trivial right action, that is,
Let be a continuous -derivation and a bounded approximate identity of . By Banach Alaoglu's Theorem, has a subnet such that , for some . Since and is continuous, . Hence, .

On the other hand, and so . Hence, and is -inner.

The following definitions extends the definition of the neo-unital and essential Banach -bimodule in the classical sense.

*Definition 4.2. *Let be a Banach -bimodule. Then is called -neo-unital (pseudo-unital), if . Similarly, one defines -neo-unital left and right Banach modules.

*Definition 4.3. *Let be a Banach -bimodule. Then is called -essential if . Similarly, one defines -essential left and right Banach modules.

We recall that a bounded approximate identity in Banach algebra for Banach bimodule is a bounded net in such that for each , and .

Proposition 4.4. *Assume that has a left bounded approximate identity, is a bounded idempotent endomorphism of , and is a left Banach -module. Then is -neo-unital if and only if is -essential.*

*Proof. *Let be a -essential Banach -bimodule. Since is idempotent, is Banach subalgebra of . Let be left approximate identity with bound . First suppose that , so there exist , such that . For , and, therefore, .

Now suppose that . There exists such that . Thus,

On the other hand, for each we have and so . Therefore,
Now we have
which shows that is a left bounded approximate identity for . Now by Cohen factorization Theorem, . So is -neo-unital. The other side is trivial.

Corollary 4.5. *Every -neo-unital left Banach -module is essential.*

*Proof. *Let be a -neo-unital left Banach -module. We have so .

Proposition 4.6. *Let be a Banach algebra with a left bounded approximate identity, be a bounded idempotent endomorphism of , and a left Banach -module. Then is closed weakly complemented submodule of .*

*Proof. *Set , since has a left bounded approximate identity, by Cohen factorization Theorem , and we have , which shows that is -essential by Proposition 4.4, is -neo unital that is, . Hence, and so . Thus is closed submodule of .

Now we prove that is weakly complemented in . Let be a left approximate identity in with bound , and define a net in by setting . We have . Thus is a bounded net in since and ball is -compact, so there exists such that we may suppose that and . For each , , and , we have
and so . On other hand, for each ,
Thus is projection, and is projection. So is weakly complemented in and, we have .

Corollary 4.7. *Let have a bounded approximate identity, and let be a Banach -bimodule and a bounded idempotent endomorphism of . Then*(i)* is a closed weakly complemented submodule of ,*(ii)* is -a.a if and only if for every -neo-unital Banach -bimodule , every -derivation is -approximately inner.*

*Proof. *Set . By Proposition 4.6, is a closed and weakly complemented submodule of , and and are projection maps. Let be a -derivation, so and are -derivations and . Since by Lemma 4.1, is -inner. So there exists such that . Thus and so is -a.i if and only if is -a.i.

Now let . By Proposition 4.6, is a closed weakly complemented in , and and are projection maps. Assume that is a -derivation, thus and are -derivations, and we have . Since , by Lemma 4.1, is -inner and so there exists such that . Therefore, . Thus, is .a.i if and only if is .a.i. Set . Thus, . Therefore, is -a.i, if and only if is .a.i. Recall that is -neo-unital. Thus, is -a.a if and only if for every -neo-unital Banach -bimodul, , every -derivation is -a.i.

Corollary 4.8. *Let have a bounded approximate identity, and let be a Banach -bimodule and a bounded idempotent endomorphism of . Then is -a.a if and only if for every -essential Banach -bimodule , every -derivation is -approximately inner.*

Proposition 4.9. *Suppose that is a bounded idempotent endomorphism of and define with . The following statements are equivalent. *(1) is -a.a. (2)There is a net such that for each , and . (3)There is a net such that for each , and for every , .

*Proof. *(13) Suppose that is -a.a, by Proposition 3.8, is -a.a. Let . is a Banach -bimodule with the following module actions:
Set with definition . is -derivation. Recall that . Since is -a.a, thus there exists such that
Set . We have
and for each ,

(3 2) is clear.

(2 1) By Proposition 3.9, it is sufficient to show that is -a.a.

Let be a derivation. By Corollary 4.7, we may take to be -neo-unital. We run the standard argument, so for each , set , where for , , we have . Then, converging to and noting that for , then
Since is -neo-unital, so . So for each and , we have
Thus,
and, therefore, . It follows that is -a.a and so is -a.a.

Proposition 4.10. *Suppose that is -a.a, and let
**
be an admissible short exact sequence of left -module and left --module homomorphism. Then , -approximately split, that is, there is a net of right inverse maps to such that for , and a net of left inverse maps to such that for .*

*Proof. *Following the proof of [2, Theorem 2.3], for a right inverse for , -approximate amenability gives a net such that

Setting gives the required net. Applying the same argument as [2, Proposition 1.1] provides .

We recall that if is a Banach algebra with a weak left (right) approximate identity, then has a left (right) approximate identity [1, Lemma 2.2].

Corollary 4.11. *Suppose that Banach algebra is -a.a, then has left and right approximate identities. *

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

Lemma 4.13. *Let be a bounded idempotent endomorphism of Banach algebra and a -neo-unital Banach -module. If is a bounded approximate identity in , then is a bounded approximate identity for .*

*Proof. *For every we have . Since is idempotent, . For each , there exists and such that . Therefore,
which shows that is a bounded approximate identity for .

It is often convenient to extend a derivation to a large algebra. If a Banach algebra is contained as a closed ideal in another Banach algebra , then the strict topology on with respect to is defined through the family of seminorms , where Note that the strict topology is Hausdorff only if [3].

Proposition 4.14. *Let be a Banach algebra and a closed ideal in . let be a bounded idempotent endomorphism of and has a bounded approximate identity. Let be a -neo-unital Banach -module and a -derivation. Then, is a Banach -bimodule in a canonical fashion, and there is a unique -derivation such that*(i),(ii) is continuous with respect to the strict topology on and the -topology on .

*Proof. *Since is a -neo-unital Banach -module, so for each , there exists and such that . Define .

We claim that is well defined, that is, independent of the choices of and . Let and be such that , and let be a bounded approximate identity for . For each and we have
It is obvious that this operation of on turns into a left Banach -module. Similarly, one defines a right Banach -module structure on . So that, eventually, becomes a Banach -bimodule. To extend , let
We claim that is well-defined, that is, the limit in (4.19) does exist. Let , and let and such that . By Lemma 4.13, is bounded approximate identity for , and we have
So the limit in (4.19) exists. Furthermore, for ,
so is an extension of . Also for and we have
We claim that is continuous with respect to the strict topology on and the -topology an .

Let in .
For each ,
so is continuous.

It remains to show that is a -derivation. From the definition of the strict topology, we have in the strict topology for all because and so . Therefore,
that is, is -derivation.

Corollary 4.15. *Suppose that is -a.a, where is bounded idempotent endomorphism of is a closed ideal in . If has a bounded approximate identity, then is -a.a.*

*Proof. *Suppose that has a bounded approximate identity, is a -neo-unital Banach -bimodule, and is a -derivation. By Proposition 4.14, becomes to a Banach -bimodule and has a unique extension which is a -derivation. Since is -a.a,
So we have , which shows that is -a.i, and is -a.a.

Corollary 4.16. *Let be an a.a Banach algebra and a closed ideal of . Then is -a.a for each bounded endomorphism of .*

Proposition 4.17. *Let be a closed ideal of such that . If is -a.a, then is -a.c, where is an endomorphism of induced by (i.e., for .*

*Proof. *Let be a Banach -bimodule and a -derivation. Then becomes an -bimodule with the following module actions:
where is the canonical homomorphism . It is easy to see that becomes a -derivation. Since is -a.c, there exists a net such that . Therefore, for each ,
Thus, is -a.c.

Proposition 4.18. *Suppose that is a closed ideal in . If is -amenable and is a.a, then is -a.a.*

*Proof. *Let be a Banach -bimodule and a -derivation. is a Banach -bimodule too.

Clearly, is a -derivation, and by -amenability of there exists such that , and, therefore, for each we have . Set . Clearly, is -derivation and . Now let is a Banach -bimodule via the following module actions:
Now we define
Let and for some and . So , and we have . Thus, . Now we have
Thus, , and, therefore,
It is enough to show that is zero on . Suppose that , we have
So for all , on and so for all , on . Since , therefore which shows that is well defined. We claim that is a derivation;
So there exists a net such that . Let be the quotient map. For every , . Set . We have
So , and, therefore, . Which shows that is -a.i and so is -a.a.

*Example 4.19. *Let be a Banach algebra and let . Then with the product becomes a Banach algebra. We denote this algebra with . It is easy to see that has a left identity , while it has not right approximate identity, so is not contractible and is not approximately contractible. Also is biprojective. Now suppose that be defined by . We have
Thus is idempotent. It is easy to see that is identity for , and since is biprojective by [1, Corollary 5.3], is -biprojective. Thus by [1, Theorem 4.3], is -contractible and so