On Amenability-Like Properties of a Class of Matrix Algebras

Rostami, M.; Shariati, S. F.; Sahami, A.

doi:https://doi.org/10.1155/2022/3194715

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Research Article | Open Access

Volume 2022 | Article ID 3194715 | https://doi.org/10.1155/2022/3194715

On Amenability-Like Properties of a Class of Matrix Algebras

M. Rostami,¹S. F. Shariati,¹and A. Sahami²

Academic Editor: Humberto Rafeiro

Received19 Feb 2022

Accepted17 Mar 2022

Published06 Apr 2022

Abstract

In this study, we show that a matrix algebra is a dual Banach algebra, where is a dual Banach algebra and . We show that is Connes amenable if and only if is finite, for every nonempty set . Additionally, we prove that is always pseudo-Connes amenable, for . Also, Connes amenability and approximate Connes biprojectivity are investigated for generalized upper triangular matrix algebras. Finally, we show that is approximately biflat if and only if is approximately biflat and is a singleton.

1. Introduction

A special class of matrix algebras, so-called -Munn algebras, was introduced and investigated by Esslamzadeh [1] and is mainly applied in the study of certain semigroup algebras. Furthermore, he studied some cohomological properties of this type of matrix algebra. For more details, see [1]. Some other cohomological properties of this class of matrix algebras, such as approximate amenability and ultraamenability, have been discovered in [2, 3], respectively. Esslamzadeh, in [4], characterized homomorphisms from -Munn algebras into arbitrary (not necessarily commutative) Banach algebras and improved results in [5]. He also discussed homomorphisms on Banach algebras with range in -Munn algebras. Habibian et al., in [6], using a new method developed by Niels Grønbæk [7, 8], namely, Morita equivalence, studied some homological properties of -Munn algebras such as strict projectivity, biprojectivity, biflatness, and contractibility. In the end, they applied their results to semigroup algebras. Some homological properties of upper triangular Banach algebras as closed subalgebras of -Munn algebras have investigated in [9]. Upper triangular Banach algebras are a generalization of triangular Banach algebras, which were first studied in [10]. For more information on this class of Banach algebras, see [11].

As a generalization of -Munn algebras, Lashkarizadeh Bami and Naseri introduced the notion of -Munn algebras. Let be a nonempty set and let be a Banach algebra. We denote for the vector space of matrices over such that

They proved that with matrix multiplication and is a Banach algebra if (see Theorem 2.2 of [12]). Also, they obtained some algebraic properties of these Banach algebras.

The category of dual Banach algebras was introduced by Runde [13]. In the category of dual Banach algebras, for , it has been shown that is a dual Banach algebra, where is a dual Banach algebra [14]. In this study, first, we generalize this fact, for . Then, we investigate some modified concepts of amenability, such as Connes amenability, pseudo-Connes amenability, and also some homological notions such as Connes biprojectivity for this kind of dual Munn algebras, namely, . In Section 4, we concentrate on the generalized upper triangular Banach algebras , where . We study the approximate biflatness of the second dual which leads us to the conclusion that the nonempty set must be a singleton.

2. Connes Amenability-Like Properties of

We recall some terminology from [13]. Let be a Banach algebra, and let be a Banach -bimodule. An -bimodule is called dual if there exists a closed submodule of such that . The Banach algebra is called dual if it is dual as a Banach -bimodule.

It is worth noting that there is an isometric-isomorphism as Banach spaces, where, for every , and .

Theorem 1. Let be a dual Banach algebra with respect to predual , and let be a nonempty index set. Then, is a dual Banach algebra with respect to the predual , where , , and for every , , and .

Proof. The proof is divided into two steps.

Claim 1. First, we show that .
To see this, let . Define by , where . We have from Young’s inequality [15]:Moreover, applying the last inequality to and instead of and leads toSo, . It follows that is bounded andWe show that the following map is an isometric-isomorphism:Obviously, the map is linear. Let ; we claim that there exists , such that . For and , let be the element of defined byFurthermore, . Set , for every and . It may be observed that , for every . So, each is a continuous linear functional on and drops in . Take . ConsiderHence, we see that , for every . Let be an arbitrary finite subset of . We haveTherefore, . From (Proposition 6.11 of [16]), we deduce thatshowing that . Let . So, there exists a finite subset of such that, for any , . We haveWe hence see that on . As is dense in , with respect to , this shows that on . Equations (4) and (9) imply thatThus, the map is an isometry, and using the open mapping theorem, we conclude .

Claim 2. is a closed -submodule of .
Suppose that and are arbitrary elements in and , respectively. Since is an -bimodule with dual actions, we have . Moreover, for every in ,where with respect to the matrix multiplication in . We claim that belongs to . Equation (12) implies thatTake a finite subset of . Using Proposition 6.11 of [16], and thenSo, . By applying rearrangement series in equation (13),Therefore, , where, for every , we have which belongs to . So, is a closed -submodule of , as required.

In the sequel, we study some modified concepts of amenability such as Connes amenability and pseudo-Connes amenability and further some homological notions such as Connes biprojectivity and pseudo-Connes amenability for the specific types of dual Munn algebras, namely, .

Let be a dual Banach algebra. A dual Banach -bimodule is normal if the left and right module actions of on are - continuous. A dual Banach algebra is called Connes amenable if, for every normal dual Banach -bimodule , every - continuous derivation is inner. For a given dual Banach algebra and a Banach -bimodule , denotes the set of all elements such that the module maps,are - continuous which is a closed submodule of . It is known from Corollary 4.6 of [17] that and then an -bimodule morphism is induced by the product map , where , for every . For further details, we refer to [13, 17].

It was shown in [14] that -Munn algebra is Connes amenable if and only if is finite.

Theorem 2. Let . Then, is Connes amenable if and only if is finite.

Proof. If is Connes amenable, then it has a unit (Proposition 4.1 of [13]). It follows from Theorem 2.4 (i) of [12] that the index set is finite.
Conversely, if the index set is finite, then is Connes amenable [[14], Theorem 3.1].

We remark that, in [18], an analogue of biprojectivity in the category of dual Banach algebras, namely, Connes biprojectivity, was introduced that seems to be fit well with Connes amenability: a dual Banach algebra is called Connes biprojective if there exists a bounded -bimodule morphism such that ; for more details, see [18]. In the sequel, for a Banach space , we write for the canonical map given by , for and .

Theorem 3. Let be a nonempty index set and . Then, is Connes biprojective.

Proof. Let . Fixed . Take a finite subset of . For every , define bywhere and is the inclusion map. This latter exists because . It can be observed that is a bounded -bimodule morphism and further

Lemma 1. Let be a unital dual Banach algebra, , and let be a nonempty set. Then, has an approximate identity.

Proof. Assume that is the family of all finite subsets of that is a directed set via inclusion. Consider the net in byWe claim that is an approximate identity for . Take an arbitrary element in . Since , for every , there exists a finite subset such that, for every in ,

Ghahramani and Zhang, in [19], extended the classical notion of amenability introduced by Johnson [20] by dropping the requirement that the aforementioned net in approximate diagonal is bounded and introduced pseudoamenable Banach algebras. Indeed, is said to be pseudoamenable if there is a net provided thatfor all elements , where the module actions are given by and , for .

Following Definition 4.3 in [21], we have the following definition.

Definition 1. A dual Banach algebra is called pseudo-Connes amenable if there exists a net in such that in and in .

Proposition 1. Let be a Connes biprojective dual Banach algebra with an approximate identity. Then, is pseudo-Connes amenable.

Proof. Let be an approximate identity for , and let be a bounded -bimodule morphism satisfying . Define . By Goldstein’s theorem, for each there exists a net in such that in . Furthermore, for every ,Moreover, the duality of and -continuity of imply thatPut as a directed set with product ordering defined viawhere means that , for every . Let and . Applying iterated limit theorem (Page 69 of [22]),and also,Hence, has the required properties and is pseudo-Connes amenable.

From Theorem 3, Lemma 1, and Proposition 1, we deduce the following result.

Corollary 1. Let be a nonempty index set and . Then, is pseudo-Connes amenable.

3. Some Connes Homological Properties of Generalized Upper Triangular Matrix Algebras

Let be a Banach algebra, and let be a totally ordered set. The generalized upper triangular Banach algebra, as a closed subalgebra of -Munn algebras, is denoted bywhere . Some homological properties of this class of Banach algebras were investigated in [23]. In the case for the classes of dual Banach algebras, is a dual Banach algebra, where is a dual Banach algebra [9]. With a similar argument in Theorem 1, we generalize this fact, for . The proof is similar, so we omit the proof.

Theorem 4. If is a dual Banach algebra with the predual and is a totally ordered set, then is a dual Banach algebra, where .

Let be a dual Banach algebra. The set of all -continuous homomorphism from onto is denoted by .

Theorem 5. Let be a totally ordered set, and let be a dual Banach algebra with a left identity such that and . Then, is Connes amenable if and only if is singleton and is Connes amenable.

Proof. If is Connes amenable, then it has a unit (Proposition 4.1 of [13]). It implies that must be a finite set. Put . Consider the -continuous character on given by , for every , where . According to Theorem 2.4 of [24], there exists an element in such thatAssume that . Choose an element,in , where is a left unit for the Banach algebra . So, . It follows that . One can see that , which is a contradiction with . The converse is clear.

Recently, approximate homological notion for dual Banach algebras have been introduced. A dual Banach algebra is called approximately Connes biprojective if there exists a net of continuous -bimodule morphisms from into such that , for every [25].

Theorem 6. Let be a dual Banach algebra with a right identity which , . Let be a totally ordered set which has a smallest element . Then, is approximately Connes biprojective if and only if is approximately Connes biprojective and is singleton.

Proof. Suppose that is approximately Connes biprojective. Since has a right identity , the Banach algebra has a right approximate identity (Lemma 5.1 of [26]). Consider the -continuous character on defined viafor every , where . Applying Theorem 2.4 of [25], there exists an element in such that and , for every . By contrary, suppose that has at least two elements. Choose an element in which th entry is and others are zero, where . Therefore, . By a simple computation, it may be observed that , which is a contradiction with . The converse is clear.

4. Approximate Biflatness of Generalized Upper Triangular Matrix Algebras

In the spirit of the definitions of approximate amenability and pseudoamenability, Samei et al. in [27] generalized the concept of biflatness and created approximately biflat Banach algebras. A Banach algebra is approximately biflat if there exists a net of -bimodule morphisms from into such that , where is denoted for the weak∗ operator topology and is the identity map on . For two Banach spaces and , recall that in if in , for every , where the space of all bounded linear operators from to . Bodaghi and Tanha, in [28], gave a modification of this notion and introduced module approximate biflat Banach algebras.

Lemma 2. Let be a Banach algebra. If is approximately biflat, then there exists a net of bounded -bimodule morphisms from into such that .

Proof. Since is approximately biflat, there exists a net of bounded -bimodule morphisms such that . Let . So, for every and , we haveIt follows that , for every .

A Banach algebra which is the predual of a von Neumann algebra, and the identity of is multiplicative was considered in 1983 by To-Ming Lau [29]. This nice class of Banach algebras was called -algebras, includes the group algebra, measure algebra, and Fourier algebra of a locally compact group. In [29], the author investigated left amenability of these algebras. The notion of left -amenability for a Banach algebra as a considerable generalization of Lau’s left amenability was defined by Kaniuth et al. [30]. In fact, a Banach algebra is called left -amenable if there exists an element in such that and , for every , where is a character on . Many aspects of this notion have been studied in [31, 32]. In the sequel, we state the relation between left -amenability and approximate biflatness of Banach algebras.

Theorem 7. Let be a Banach algebra with a left approximate identity and . If is approximately biflat, then is left -amenable.

Proof. Let be an approximately biflat Banach algebra. Lemma 2 implies that there exists a net of bounded -bimodule morphisms from into such that . Let . According to Lemma 1.7 of [33], there exists a bounded linear map such that, for every and , we have(i)(ii)(iii)Let , where is the quotient map. So, is a net of bounded -bimodule morphisms from into . It may be observed that . For this purpose, since has a left approximate identity, . If is an arbitrary element in , then there are two nets in and in satisfying . Moreover,Therefore, can be dropped on . That is, the maps given by are well defined. Let , where is the character on induced by . Choose an element such that . Set . Since ,We denote for the character on defined by , for every . We haveSince , by replacing with , there exists an element in such that , for every and . By Goldstine’s theorem, there is such that, for every , weakly in and . Finally, for each finite subset , say , letIt is obvious that is a convex subset of , and by applying Mazur’s lemma,Therefore, there exists a bounded net in such that , for every and . The boundedness condition implies that -limit point of exists in . Let . It follows that and . Hence, is left -amenable.

As an application of the above theorem, we characterize approximate biflatness of the second dual of generalized upper triangular matrix algebras.

Theorem 8. Let be a Banach algebra with a right identity, , , and also, let be a totally ordered set which has a smallest element . Then, is approximately biflat if and only if is approximately biflat and is singleton.

Proof. We only need to prove the ‘if’ part, and the converse is clear. Suppose that is approximately biflat. Consider the character on defined via , for every , where . The right version of Theorem 7 and Lemma 5.1 of [26] imply that is right -amenable. Applying Theorem 1.4 of [30], there exist a bounded net in such thatSuppose towards a contradiction that . Consider the elementin , where is the right identity for the Banach algebra . So, . Hence, we see that . It is easy to see that . The continuity of character implies that , which is a contradiction with .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The third author is thankful to Ilam university for their support.

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