Abstract

In the present paper, we study the approximate biprojectivity and weak approximate biprojectivity of -Munn Banach algebras when the related sandwich matrix is regular over . In fact, we show that a -Munn Banach algebra with the regular sandwich matrix over is approximately biprojective (weak approximately biprojective) if and only if is approximately biprojective (weak approximately biprojective), respectively. We also study approximate biprojectivity of upper triangular Banach algebra when the associated sandwich matrix with elements in is invertible. Finally, we apply our results to Rees semigroup algebras.

1. Introduction

Biprojectivity is a significant notion in homology theory of Banach algebras which was introduced by Helemskii [1]. Indeed, a Banach algebra is called biprojective if there exists a bounded -bimodule morphism satisfying for every , where is the product morphism given by for every . The first use of the terminology of approximate biprojectivity was initiated by Grønbæk for modules [2], and Zhang defined an approximate version of biprojectivity for Banach algebras [3]. In fact, a Banach algebra is called approximately biprojective if there exists a net of continuous -bimodule morphisms from into , satisfying for every . Aghababa introduced a weaker notion than approximately biprojective called weak approximately biprojective [4]. A Banach algebra is weak approximately biprojective if there exists a net of approximate -bimodule maps from into such that

Note that a net of linear maps from into is called approximate -bimodule, if for every , we have

Module amenable-like properties of Banach algebras which are the generalizations of the classical case have been introduced and studied in Refs. [5, 6].

The -Munn Banach algebras have been introduced by Esslamzadeh [7]. Esslamzadeh and other authors studied some cohomological properties of this class of Banach algebras like amenability, weak amenability, contractibility and Connes-amenability [8]. Recently, Habibian et al. [9] investigated biprojectivity of these algebras when the related sandwich matrix is regular over . They showed that the -Munn Banach algebra with the regular sandwich matrix over is biprojective if and only if the unital Banach algebra is biprojective [9].

Forrest and Marcoux studied the class of -upper triangular Banach algebras [10]. Also, they investigated some notions of amenability and homological properties of triangular Banach algebras [11].

In Section 2, we show that the -Munn Banach algebra with the regular sandwich matrix over is approximately biprojective (weak approximately biprojective) if and only if the unital Banach algebra is approximately biprojective (weak approximately biprojective) and this is equivalent to pseudo contractibility (pseudo amenability) of , respectively. In Section 3, we show that the upper triangular Banach algebra, , with an invertible sandwich matrix over and totally ordered (finite) set with the smallest element is approximately biprojective if and only if is singleton and the unital Banach algebra is approximately biprojective, respectively. In the last section, as the application of these results, we expose some examples.

2. Approximate Biprojectivity of -Munn Algebras

For a unital Banach algebra , arbitrary index sets and , and a nonzero The sandwich matrix over with , and we regard the Banach algebra of all -matrices with elements in equipped with the -norm and the product as

This Banach algebra which is called -Munn Banach algebra is denoted by . It is worthwhile to mention that the component-wise product is given bywhere , , and the notation means the -th entry of is m. Throughout this work, we use the notations of Ref. [9]. Consider those as follows:(i)We denote by the set of all matrices such that -th entry is nonzero and zero elsewhere(ii)We denote by the set of all matrices such that for every , -th entry is nonzero and zero elsewhere(iii)We denote by the set of all matrices such that for every , -th entry is nonzero and zero elsewhere

By (4), the linear map , defined by , is an isomorphism if , and if . Therefore, admits a relatively nice decomposition to its subalgebras as . Suppose that is the set of all invertible elements of . The matrix is called a regular sandwich matrix over if every row and every column of contains at least one nonzero entry. So for every , there is such that is a nonzero invertible element in , and we obtain an idempotent in , namely, . It is an easy verification that by (4), for every , is a left identity for . Similarly, for each , is a right identity of .

Note that for a Banach algebra , the projective tensor product is a Banach -bimodule with

Habibian et al. [9] studied biprojectivity of , and in this paper, we aim to investigate approximate biprojectivity of . For this implication, we need the following definition.

A Banach algebra is pseudo contractible if there is a net satisfying and for every [12].Throughout this section, we assume that is the -Munn Banach algebra, where is a regular sandwich matrix over .

Theorem 1. The following statements are equivalent:(i) is approximately biprojective(ii) is approximately biprojective(iii) is pseudo contractible

Proof. Suppose that is pseudo contractible and there exists , which satisfies and for every . Since , there exist and in such that . Fix an idempotent . Definewhere . We show that is a -bimodule morphism. For and in , we haveNote that for every , and therefore, . Applying this fact, we show that for every α, is a right -bimodule morphism. Indeed, we getAlso, for , we haveSo .
Let be a net of not necessarily bounded -bimodule morphism as required in the definition of approximately biprojective. Consider their restriction for each and regard an idempotent . For each α, one obtainswhere we have taken into account the fact that is a -bimodule morphism, and e is an idempotent. In turn, we getTherefore,with . It follows that for all , we getFor each , choose and such thatConsider the projective tensor product of Banach algebras with the following multiplication:and defineWe haveSo . By (13), for , and by (17), we getThen, is pseudo contractible.
It is proved in ([12] Proposition 3.8).
Now, we recall a definition which we shall use it. A Banach algebra is pseudo amenable if there is a net satisfying and for every [12].

Theorem 2. The following statements are equivalent:(i) is weak approximately biprojective(ii) is weak approximately biprojective(iii) is pseudo amenable

Proof. Suppose that is pseudo amenable. Then there exists which satisfies and . Since , there exist and such that . Define bywhere . For , we haveandThus, for each α, is an approximate -bimodule map. And also, we getSo is weak approximately biprojective.
Suppose that is weak approximately biprojective and fix an idempotent . Let be approximate -bimodule map such that for every . Since , we haveand for each α denote their restrictions by . By definition of , for , we haveTherefore,butsoThen, for and , we getFor every , choose and such thatNow consider the projective tensor product of Banach algebras with the following multiplication:and takeBy easy calculation, we have , and by (28), we havealsothen by (33), we getTherefore, is pseudo amenable.
It is proved in [4] Corollary 3.6.

3. Applications to Upper Triangular Banach Algebras

Let be a unital Banach algebra, be an arbitrary index set, and let be a sandwich matrix, which is a nonzero -upper triangular matrix over with . The vector space of all -upper triangular matrices over with -norm is such that , and with the product, we getfor is a Banach algebra, which is called upper triangular Banach algebra and is denoted by . It should be noted that the components of matrix is given by

For the identity -matrix over , the Banach algebra is denoted by . Taking , the set of all matrices in the sense that -entry is nonzero and zero elsewhere. Since for each , then is decomposition of into -direct sum of subalgebras.

Suppose that is the set of all invertible elements of . When we say is an invertible sandwich matrix over , we mean an -matrix such that every row and every column contains exactly one nonzero entry with infinitely many of them in the main diagonal.

The following lemma holds similar to the case -Munn Banach algebra proved by Esslamzadeh [7].

Lemma 1. Suppose that is finite and is an invertible -upper triangular matrix over . Then and and also are isometrical isomorphisms.

Proof. Let be as in assumption. Define the map by , where is the inverse of . It is easy to see that is a linear bijection. For , we haveand by open mapping theorem, is bounded; thus, it is an isomorphism. Similarly for the other case, consider isomorphism by .
Note that if is finite, then by previous Lemma, we have . Also, since is unital, is unital too.

Lemma 2. Let be a Banach algebra with an (a left/right) identity, be a totally ordered set, and let be an invertible sandwich matrix over . Then has an (a left/right) approximate identity, respectively.

Proof. Let be the set of all finite subsets of . Suppose that and are arbitrary. Then there exists such that . Define by whenever , zero elsewhere and also define by whenever and zero elsewhere, where is the (left/right) identity of and . Thus, we getand similarly, we getso has an (a left/right) approximate identity.
We remind that a Banach algebra is called approximate right -amenable if there exists a not necessarily bounded net in such that and for every and , where is the character space of . The left case is defined similarly. For more information, see Refs. [13, 14].

Theorem 3. Let be a unital Banach algebra, be a totally ordered set with a smallest element and , and let be an invertible sandwich matrix over such that the -th entry of is nonzero. Then is approximately biprojective if and only if is singleton and is approximately biprojective.

Proof. Suppose that is approximately biprojective. Let be the smallest element of . Definewhere is the -th entry of . Note that , then we have , and, therefore, is nonzero. Also, for , consider -th entry of that is equal tosoThus, . Since is unital by Lemma 2, has a right approximate identity and by [15] Theorem 3.9, we can conclude that is approximately right -amenable. Define a closed ideal in byWe have . Now by [16] Proposition 5.1, is approximately right -amenable. So there exists a net such thatWe go toward a contradiction and assume that has at least two elements. Setwhere is the identity of . For every α, has the form , where for every . As every row and every column in contains exactly one nonzero entry, by setting and in (44), we conclude that , where is nonzero element of -th row of . Since is continuous, we have , and therefore gives a contradiction. So is singleton.

Proposition 1. Let be a weak approximately biprojective Banach algebra and . If there exists an element in the sense that for every and , then is approximately right -amenable.

Proof. Suppose that is a net of approximate -bimodule maps such that for every . DefineIt is readily seen that is a bounded linear map andfor every and . Let be as in assumption. Set , then we haveandSo is approximately right -amenable.

Proposition 2. Suppose that is a unital Banach algebra, and is an invertible sandwich matrix over . If is a finite set which has at least two elements, then is not weak approximately biprojective.

Proof. Assume a contradiction that is weak approximately biprojective and is a finite set. So has an identity. Without loss of generality, we may assume that . Defineand then, . By Proposition 1, is approximately right -amenable. Define a closed ideal in byWe have . Now by [16] Proposition 5.1, is approximately right -amenable. So there exists a net such thatA similar argument as in the proof of Theorem 3, we obtain a contradiction and must be singleton.

Theorem 4. Let be a unital Banach algebra, and let be finite. If is an invertible sandwich matrix over , then is weak approximately biprojective if and only if is singleton and is weak approximately biprojective.

Proof. By Proposition 2 the proof is trivial.

Corollary 1. Let be a unital Banach algebra, and let be finite. Then is weak approximately biprojective if and only if is singleton and is weak approximately biprojective.

4. Some Applied Examples

In this section, we apply our results to some special semigroups, which are called Rees matrix semigroups. Let be a group, and be index sets, and be the group with zero arising from by adjunction of a zero element. Let be a regular sandwich matrix over , so each row and each column of contains at least one nonzero entry. The associated Rees semigroup is defined bywhere denotes the element of with in the -th place and 0 elsewhere, and is the zero matrix. The binary operation on is defined byand

This semigroup is denoted by and called Rees matrix semigroup over . Further, by [7]Proposition 5.6, we getwhere the zero of is identified with the zero of the -Munn algebra , and is considered as a matrix over . Furthermore, the multiplication in satisfies the following equations for all and :