Abstract

Suppose that is a Banach algebra and is its enveloping dual Banach algebra, we show that is approximately contractible (approximately amenable) if has the same property. Also, we study the relation between the pseudoamenability of and the pseudoamenability of the second dual and we also characterize approximate biflatness and approximate biprojectivity of associated with approximate biflatness and approximate biprojectivity of the second dual .

1. Introduction and Preliminaries

Approximate contractibility and approximate amenability for Banach algebras were defined and studied by Ghahramani and Loy by [1]. They proved that a Banach algebra is approximately amenable, if the second dual is approximately amenable by [1], Theorem 2.3. They denoted that the measure algebra is approximately amenable if and only if is discrete and amenable by [1], Theorem 3.1, and also they showed that the group algebra is approximately amenable if and only if is amenable by [1], Theorem 3.2. As shown in [1], Theorem 3.3, is approximately amenable if and only if is finite.

The basic properties of biprojectivity and biflatness are investigated in [2]. In 1999, Zhang introduced the notion of approximate biprojectivity for Banach algebras [3]. It is well known that a Banach algebra is pseudocontractible if and only if is approximately biprojective and has a central approximate identity ([4], Proposition 3.8).

The concept of approximate biflatness for Banach algebra was introduced by Samei et al. [5]. They showed that a Banach algebra A is pseudoamenable whenever it is approximately biflat and has an approximate identity. Also, they studied approximate biflatness for various classes of Segal algebras in both group algebra, , and the Fourier algebra, , of a locally compact group G [5].

The module cohomological properties for Banach algebras, namely, module (approximate) biprojectivity and module (approximate) biflatness for Banach algebras which are generalizations of the classical cases, were introduced in [6, 7]. In these articles, the authors found necessary and sufficient conditions for to be approximately module biprojective and module biflat, where is an inverse semigroup.

For a Banach algebra and a Banach -bimodule , the collection of all elements is such that the module maps ; and are weakly compact denoted by and the -bimodule is denoted by [5]. Hence, we can write . Now, if , then we write and it is well known that is a universal canonical dual Banach algebra [5]. is called the enveloping dual Banach algebra associated with by [8].

Choi et al. defined the concept of -virtual diagonal for a Banach algebra [8]. They showed that has a -virtual diagonal if and only if the dual Banach algebra is Connes-amenable. They also proved that for the group algebra , the existence of a virtual diagonal is equivalent to the existence of a -virtual diagonal [8].

Motivated by the results mentioned above, in this paper, we study the approximate amenability, approximate contractibility, and pseudoamenability of the enveloping dual Banach algebra. We also investigate the cohomological properties of the enveloping dual Banach algebra and , where is a locally compact group. We study the relation between the pseudoamenability of and the pseudoamenability of the second dual and we also characterize approximate biflatness and approximate biprojectivity of associated with the approximate biflatness and approximate biprojectivity of the second dual . By giving examples of some Banach algebras, we investigate the cohomological properties of these Banach algebras with respect to their enveloping dual Banach algebras.

For a Banach algebra and a Banach -bimodule , a derivation is a linear map such thatfor every . A bounded derivation is called approximately inner, if there exists a net such that for all .

Let be a Banach algebra. We remind that the projective tensor product is a Banach -bimodule via the following actions:for every . Also, the map, denotes the product morphism which is defined by .

A Banach algebra is approximately contractible (approximately amenable), if for all -bimodule , every bounded derivation is approximately inner by [1]. We remind that is pseudoamenable, if there is a net , such that and for all [4] and also is called approximately biflat, if there exists a net of bounded -bimodule morphisms such that , where is the weak operator topology on . We recall that the weak operator topology on is a locally convex topology determined by the seminorms , where [5]. A Banach algebra is called approximately biprojective if there is a net of bounded -bimodule morphism such that for all (for more details see [3]).

2. Cohomological Properties of the Dual Banach Algebras

Throughout this section, is the enveloping dual Banach algebra associated with and we will study the cohomological properties of such as approximate amenability and pseudoamenability.

Theorem 1. Let be a Banach algebra. If is approximately contractible, then is approximately contractible.

Proof. Suppose that is a bounded derivation, where is a Banach -bimodule, then we can extend to a bounded derivation by [9], Lemma 2.2. Suppose that is the canonical inclusion of and is the canonical inclusion of and is the adjoint of the inclusion map , then, we define a map by . However, are the homomorphism algebra, so is a derivation. Since is approximately contractible, there exists a net such thatfor all . Since the module action on is weak∗-continuous by [9], Lemma 1.1 and by Goldstein’s theorem, we can imply thatfor . In particular, we havefor . By Goldstein’s theorem, there exists a bounded net such that and sofor .
Suppose that is a directed set with product ordering which is defined bywhere and are the set of all maps . We recall that means that for all .
We set and . By iterated limit theorem ([10], page 69) and by the equation (6), we see thatfor . On the other hand, and . So,for . This follows that is weakly approximately inner and equivalently, approximately inner. Thus, is contractible.

Corollary 2. Let be an approximately amenable Banach algebra. Then, is approximately amenable.

Proof. We know that a Banach algebra is approximately amenable if and only if is approximately contractible ([4], Theorem 3.1). Since is approximately amenable, is approximately contractible. So, by the previous theorem, is approximately contractible, and hence is approximately amenable.
The following example shows us that the converse of Corollary 2 is not true.

Example 1. Let be an amenable locally compact group which is not discrete ([11], Chapter 2). If the converse of Corollary 2 is true, then, we have that is approximately amenable if and only if is approximately amenable. Since has a bounded approximate identity, is unital ([12], Lemma 2.9) and so the authors in [4], Proposition 3.2 show that is pseudoamenable if and only if is pseudoamenable. However, the measure algebra is the quotient of [13]. Hence, is pseudoamenable ([4], Proposition 2.2). On the other hand, the authors in [4], Proposition 4.2 show that is pseudoamenable if and only if is amenable and discrete. So, we can imply that must be amenable and discrete which is a contradiction. Therefore, the converse of Corollary 2 is not true.

Theorem 3. Let be a Banach algebra with a bounded approximate identity and let be approximately amenable. Then, is pseudoamenable.

Proof. Since has a bounded approximate identity, is unital by ([12], Lemma 2.9). However, a unital Banach algebra is pseudoamenable if and only if it is approximately amenable ([4], Proposition 3.2). Since is approximately amenable, Corollary 2 follows that is approximately amenable. However, is a unital Banach algebra. So, is pseudoamenable.

Theorem 4. Let be a unital Banach algebra and let be pseudoamenable. Then, is pseudoamenable.

Proof. Since has an identity, is unital ([12], Lemma 2.9). However, a unital Banach algebra is pseudoamenable if and only if it is approximately amenable ([4], Proposition 3.2). Since is unital and pseudoamenable, is approximately amenable. So, by Corollary 2, we imply that is approximately amenable. Hence, is pseudoamenable.

Theorem 5. Let be a Banach algebra and let the second dual be pseudoamenable. Then, is pseudoamenable.

Proof. Let be the adjoint of the inclusion map , which is a continuous epimorphism. Then, by hypothesis, is pseudoamenable ([4], proposition 2.2).
In the following example, we show that there is a Banach algebra such that is pseudoamenable but is not pseudoamenable.

Example 2. Note that, the second dual is pseudoamenable if and only if is finite ([4], proposition 4.2). Let be an infinite amenable locally compact group. Then, Theorem 3 follows that is pseud amenable, but is not pseudoamenable.

In this section, we study pseudoamenability and approximate amenability of the enveloping dual Banach algebra and , where is a locally compact group.

Proposition 6. Let be a locally compact group. Then, is pseudoamenable if is amenable.

Proof. Let be amenable. Then, is approximately amenable. By Corollary 2, we imply that is approximately amenable. Since has a bounded approximate identity, is unial ([12], Lemma 2.9). So, is pseudoamenable if and only if is approximately amenable ([4], Proposition 3.2). Hence, is pseudoamenable.

Proposition 7. Let be a locally compact group. Then, is pseudoamenable if is discrete and amenable.

Proof. Let be amenable and discrete. Then, is approximately amenable by ([1], Theorem 3.1). By Corollary 2, is approximately amenable. Since has an identity, is unital ([12], Lemma 2.9). Hence, is pseudoamenable if and only if is approximately amenable ([4], Proposition 3.2). Hence, is pseudoamenable.

Proposition 8. Let be a locally compact group. Then, is approximately amenable if is amenable.

Proof. Let be an amenable group. Then, is approximately amenable by ([1], Theorem 3.2). So, by Corollary 2, one can show that is approximately amenable.

Proposition 9. Let be a locally compact group. Then, is approximately amenable if is amenable and discrete.

Proof. We know that is approximately amenable if and only if is amenable and discrete ([1], Theorem 3.1). Hence, Corollary 2 follows that is approximately amenable.

Corollary 10. Let be a locally compact group. Then, is approximately amenable if is finite.

Proof. We know that is approximately amenable if and only if is finite ([1], Theorem 3.3). So, we can imply that is approximately amenable by Corollary 2.
In the following example, we show that there is a Banach algebra such that is pseudoamenable but is not pseudoamenable.

Example 3. Let be an infinite amenable locally compact group. Then, is pseudoamenable by [4], Proposition 4.1. However, since is infinite, is not pseudoamenable ([4], Proposition 4.2).

4. Approximate Biflat and Approximate Biprojective

Now, we study approximate biflatness and approximate biprojectivity of the enveloping dual Banach algebra , associated with the second dual .

Remark 11. By [14], we suppose that is a Banach algebra and is a subspace of . Then, is called a -left invariant if for every . We suppose that is a left invariant subspace. If , then the subspace is called -left introverted. The notation -will often be omitted to simplify.

Remark 12. We consider the left introverted subspaces and of such that and define the map by , for every , which is a continuous homomorphism from onto and its kernel is -closed ideal of by [14], Lemma 1.1, where . Indeed, we have the hollowing direct sum decomposition .
Since is an introverted subspace of [14], there is a direct sum decomposition , where is the enveloping dual Banach algebra associated to by [15], Remark 3.4.

The following result is given in [5], Proposition 2.8.

Proposition 13. Let be a family of (quantized) Banach algebras.(i)If each is (operator) approximately biflat, then for , is (operator) approximately biflat.(ii)If and are (operators) approximately biflat quantized Banach algebras and has an operator space structure such that the projection maps are completely bounded, then is also an operator approximately biflat.(iii)If each is (operator) biflat and (respectively, ), then is (operator) biflat.

Proposition 14. Let be a Banach algebra. If and are approximately biflat, then is approximately biflat.

Proof. It is clear from the previous proposition.

Theorem 15. Let be approximately biflat. Then, is approximately biflat.

Proof. Since is approximately biflat, there is a net of bounded -bimodule morphisms such that .
Let be the projection map and let be defined by for every . Now, we define by , which is a net of bounded -bimodule morphisms such that . To see this, suppose that and , then, we have . Since , we haveHence, is approximately biflat.

Theorem 16. If and are approximately biprojective, then is approximately biprojective.

Proof. Since and are approximately biprojective, there exists a net of bounded -bimodule morphisms and a net of bounded -bimodule morphisms such that and , respectively.
Suppose that and are the projection map and and are defined by and , respectively, for every and , then, we haveSo,Let be directed by the product ordering. For every , we define . By iterated limit theorem ([10], P. 69), equation (12), implies the following:and certainly is a net of bounded -bimodule morphisms. Hence, is approximately biprojective.

Theorem 17. Let be approximately biprojective, then, is approximately biprojective.

Proof. Since is approximately biprojective, there is a net of bounded -bimodule morphism such that . We define by , where and are defined by Theorem 15. Certainly, is a net of bounded -bimodule morphisms. Now, we prove that .
We have . Hence,So, is approximately biprojective.

Example 4. Let be a commutative compact group. Then, with convolution multiplication is an approximately biprojective Banach algebra by [3] and so it is approximately biflat. Since is a Hilbert space, we have . Hence, Theorems 15 and 17 imply that is approximately biflat and approximately biprojective, respectively.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the Iran National Science Foundation (INSF), under the Grant 97000599.