# Approximate Biprojectivity ofℓ1-Munn Banach Algebras

Authors: G. Zarei; A. Pourabbas; M. Rostami; A. Sahami
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the present paper, we study the approximate biprojectivity and weak approximate biprojectivity ofℓ1-Munn Banach algebras when the related sandwich matrix is regular over InvA. In fact, we show that a ℓ1-Munn Banach algebra with the regular sandwich matrix over InvA is approximately biprojective (weak approximately biprojective) if and only if A 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 InvA is invertible. Finally, we apply our results to Rees semigroup algebras.


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## Body

## 1. Introduction

Biprojectivity is a significant notion in homology theory of Banach algebras which was introduced by Helemskii [1]. Indeed, a Banach algebra A is called biprojective if there exists a bounded A-bimodule morphism ρ:A⟶A⊗^A satisfying πA∘ρa=a for every a∈A, where πA:A⊗^A⟶A is the product morphism given by πAa⊗b=ab for every a,b∈A. 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 A is called approximately biprojective if there exists a net ρα of continuous A-bimodule morphisms from A into A⊗^A, satisfying πA∘ραa⟶a for every a∈A. Aghababa introduced a weaker notion than approximately biprojective called weak approximately biprojective [4]. A Banach algebra A is weak approximately biprojective if there exists a net of approximate A-bimodule maps ρα from A into A⊗^A such that(1)πA∘ραa⟶a,a∈A.Note that a netρα of linear maps from A into A⊗^A is called approximate A-bimodule, if for every a,b∈A, we have(2)ραab−ραa⋅b⟶0,ραab−a⋅ραb⟶0.Module amenable-like properties of Banach algebras which are the generalizations of the classical case have been introduced and studied in Refs. [5, 6].Theℓ1-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 InvA. They showed that the ℓ1-Munn Banach algebra with the regular sandwich matrix over InvA is biprojective if and only if the unital Banach algebra A is biprojective [9].Forrest and Marcoux studied the class of2×2-upper triangular Banach algebras [10]. Also, they investigated some notions of amenability and homological properties of triangular Banach algebras [11].In Section2, we show that the ℓ1-Munn Banach algebra with the regular sandwich matrix over InvA is approximately biprojective (weak approximately biprojective) if and only if the unital Banach algebra A is approximately biprojective (weak approximately biprojective) and this is equivalent to pseudo contractibility (pseudo amenability) of A, respectively. In Section 3, we show that the upper triangular Banach algebra, UPI,P,A, with an invertible sandwich matrix over InvA and totally ordered (finite) set with the smallest element is weak approximately biprojective if and only if I is singleton and the unital Banach algebra A is weak approximately biprojective, respectively. In the last section, as the application of these results, we expose some examples.


## 2. Approximate Biprojectivity ofℓ1-Munn Algebras

For a unital Banach algebraA, arbitrary index sets I and Λ, and a nonzero Λ×I The sandwich matrix P over A with suppλi:λ∈Λ,i∈I≤1, and we regard the Banach algebra of all I×Λ-matrices with elements in A equipped with the ℓ1-norm and the product as(3)M∘N=MPN.This Banach algebra which is calledℓ1-Munn Banach algebra is denoted by A:=ℳA,P,I,Λ. It is worthwhile to mention that the component-wise product M∘N is given by(4)miλ⋅njμ=mpλjniμ,pλj≠0,0,pλj=0,where M,N∈A, miλ∈M,njμ∈N, and the notation miλ means the i,λ-th entry of M is m. Throughout this work, we use the notations of Ref. [9]. Consider those as follows:(i)
We denote byAiλ the set of all I×Λ matrices such that i,λ-th entry is nonzero and zero elsewhere(ii)
We denote byAi the set of all I×Λ matrices such that for every λ∈Λ, i,λ-th entry is nonzero and zero elsewhere(iii)
We denote byAλ the set of all I×Λ matrices such that for every i∈I, i,λ-th entry is nonzero and zero elsewhereBy (4), the linear map A⟶iAλ, defined by a↦apλi−1iλ, is an isomorphism if pλi≠0, and Aiλ⋅Ajλ=0 if pλj=0. Therefore, A admits a relatively nice decomposition to its subalgebras as A=ℓ1−⊕i∈Iλ∈ΛAiλ. Suppose that InvA is the set of all invertible elements of A. The matrix P is called a regular sandwich matrix over InvA if every row and every column of P contains at least one nonzero entry. So for every i∈I, there is λ∈Λ such that pλi is a nonzero invertible element in A, and we obtain an idempotent in Ai, namely, ei=pλi−1iλ. It is an easy verification that by (4), for every i∈I, ei is a left identity for Ai. Similarly, for each λ∈Λ, eλ=pλi−1iλ is a right identity of Aλ.Note that for a Banach algebraA, the projective tensor product A⊗^A is a Banach A-bimodule with(5)a⋅b⊗c=ab⊗c,b⊗c⋅a=b⊗ca,a,b,c∈A.Habibian et al. [9] studied biprojectivity of A:=ℳA,P,I,Λ, and in this paper, we aim to investigate approximate biprojectivity of A. For this implication, we need the following definition.A Banach algebraA is pseudo contractible if there is a net mα⊆A⊗^A satisfying a⋅mα=mα⋅a and πAmαa⟶a for every a∈A [12].Throughout this section, we assume that A:=ℳA,P,I,Λ is the ℓ1-Munn Banach algebra, where P is a regular sandwich matrix over InvA.Theorem 1.
The following statements are equivalent:(i)
A is approximately biprojective(ii)
A is approximately biprojective(iii)
A is pseudo contractibleProof.
iii⇒i Suppose that A is pseudo contractible and there exists mα⊆A⊗^A, which satisfies c⋅mα=mα⋅c and πAmαc⟶c for every c∈A. Since mα⊆A⊗^A, there exist nαt and ναt in A such that mα=∑t=1∞nαt⊗ναt. Fix an idempotent pλi−1iλ. Define(6)ρα:A⟶A⊗^Abyραajμ=∑t=1∞anαtpλi−1jλ⊗ναtiμ,where ajμ∈A. We show that ρα is a A-bimodule morphism. For ajμ and bkγ in A, we have(7)ραbkγajμ=ραbpγjakμ=∑t=1∞bpγjanαtpλi−1kλ⊗ναtiμ=∑t=1∞bkγ⋅anαtpλi−1jλ⊗ναtiμ=bkγ⋅∑t=1∞anαtpλi−1jλ⊗ναtiμ=bkγ·ραajμ.
Note thatc⋅mα=mα⋅c for every c∈A, and therefore, ∑t=1∞cnαt⊗ναt=∑t=1∞nαt⊗ναtc. Applying this fact, we show that for every α, ρα is a right A-bimodule morphism. Indeed, we get(8)ραajμbkγ=ραapμkbjγ=∑t=1∞apμkbnαtpλi−1jλ⊗ναtiγ=∑t=1∞apji−1jj⋅pμkbnαtpλi−1iλ⊗ναtiγ=∑t=1∞apji−1jj⋅nαtpλi−1iλ⊗ναtpμkbiγ=∑t=1∞anαtpλi−1jλ⊗ναtpμkbiγ=∑t=1∞anαtpλi−1jλ⊗ναtiμ⋅bkγ=ραajμ⋅bkγ.
Also, forajμ∈A, we have(9)πA∘ραajμ=πA∑t=1∞anαtpλi−1jλ⊗ναtiμ=∑t=1∞πAanαtpλi−1jλ⊗ναtiμ=∑t=1∞anαtpλi−1jλναtiμ=∑t=1∞anαtpλi−1pλiναtjμ=∑t=1∞anαtναtjμ=a∑t=1∞nαtναtjμ=a∑t=1∞πAnαt⊗ναtjμ=aπA∑t=1∞nαt⊗ναtjμ=aπAmαjμ⟶ajμ.
SoπA∘ραajμ⟶ajμ.
i⇒iii Let ρα:A⟶A⊗^A be a net of not necessarily bounded A-bimodule morphism as required in the definition of approximately biprojective. Consider their restriction ρα:eAe⟶eA⊗^Ae for each α and regard an idempotent e=pλi−1iλ. For each α, one obtains(10)ραeae=eae⋅ραe=ραe⋅eae,a∈A,where we have taken into account the fact that ρα is a A-bimodule morphism, and e is an idempotent. In turn, we get(11)ραe∈eA⊗^Ae=⊕μ∈ΛAiμ⊗^⊕j∈IAjλ,=⊕μ∈Λj∈IAiμ⊗^Ajλ.
Therefore,(12)ραe=∑μ,jbiμα⊗cjλα,with j∈I,μ∈Λ. It follows that for all j∈I,μ∈Λ, we get(13)eae⋅biμα⊗cjλα=biμα⊗cjλα⋅eae.
For eachj∈I,μ∈Λ, choose fjμ∈Ajλ and ejμ∈Aiμ such that(14)fjμ=rljpμj−1λ,pμj≠0,0,pμj=0,ejμ=rlipλi−1μ,pμj≠0,0,pμj=0.
Consider the projective tensor product of Banach algebras with the following multiplication:(15)a⊗b⋅a′⊗b′=aa′⊗b′b,and define(16)mα=∑μ,jbiμα⊗cjλα⋅fjμ⊗ejμ.
We have(17)biμα⊗cjλα⋅fjμ⊗ejμ=biμαfjμ⊗ejμcjλα=bαpμjpμj−1iλ⊗pλi−1pμjcαiλ=biλα⊗pλi−1pμjcαiλ∈Aiλ⊗^Aiλ.
Somα∈eAe⊗^eAe. By (13), for a∈A,eae⋅mα=mα⋅eae, and by (17), we get(18)πeAemαeae=πeAeραeeae⟶eae.Then, eAe≅A is pseudo contractible.
ii⇔iii It is proved in ([12] Proposition 3.8).
Now, we recall a definition which we shall use it. A Banach algebraA is pseudo amenable if there is a net mα⊆A⊗^A satisfying a⋅mα−mα⋅a⟶0 and πAmαa⟶a for every a∈A [12].Theorem 2.
The following statements are equivalent:(i)
A is weak approximately biprojective(ii)
A is weak approximately biprojective(iii)
A is pseudo amenableProof.
iii⇒i Suppose that A is pseudo amenable. Then there exists mα⊆A⊗^A which satisfies a⋅mα−mα⋅a⟶0 and πAmαa⟶a. Since mα⊆A⊗^A, there exist nαt and ναt⊆A such that mα=∑t=1∞nαt⊗ναt. Define ρα:A⟶A⊗^A by(19)ραajμ=∑t=1∞anαtpλi−1jλ⊗ναtiμ,where ajμ∈A. For ajμ,bkγ∈A, we have(20)ραajμbkγ−ajμ⋅ραbkγ=ραapμkbjλ−ajμ⋅ραbkγ=∑t=1∞apμkbnαtpλi−1jλ⊗ναtiγ−ajμ⋅∑t=1∞bnαtpλi−1kλ⊗ναtiγ=∑t=1∞apμkbnαtpλi−1jλ⊗ναtiγ−∑t=1∞apμkbnαtpλi−1jλ⊗ναtiγ=∑t=1∞apμkbnαtpλi−1−apμkbnαtpλi−1jλ⊗ναtiγ=0,and(21)ραajμbkγ−ραajμ⋅bkγ=ραapμkbjγ−ραajμ⋅bkγ=∑t=1∞apμkbnαtpλi−1jλ⊗ναtiγ−∑t=1∞anαtpλi−1jλ⊗ναtiγ⋅bkγ=∑t=1∞apμkbnαtpλi−1jλ⊗ναtiγ−∑t=1∞anαtpλi−1jλ⊗ναtpμkbiγ=apλi−1jλ⋅∑t=1∞pμkbnαtpλi−1iλ⊗ναtpλi−1iλ−∑t=1∞nαtpλi−1iλ⊗ναtpμkbpλi−1iλ⋅pλiiγ=apλi−1jλ⋅pμkbpλi−1iλ⋅∑t=1∞nαtpλi−1iλ⊗ναtpλi−1iλ−∑t=1∞nαtpλi−1iλ⊗ναtpλi−1iλ⋅pμkbpλi−1iλ⋅pλiiγ=apλi−1jλ⋅pμkbpλi−1iλ⋅mα−mα⋅pμkbpλi−1iλ⋅pλiiγ⟶0.
Thus, for eachα, ρα is an approximate A-bimodule map. And also, we get(22)πA∘ραajμ=πA∑t=1∞anαtpλi−1jλ⊗ναtiμ=∑t=1∞πAanαtpλi−1jλ⊗ναtiμ=∑t=1∞anαtpλi−1jλναtiμ=∑t=1∞anαtpλi−1pλiναtjμ=∑t=1∞anαtναtjμ=a∑t=1∞nαtναtjμ=a∑t=1∞πAnαt⊗ναtjμ=aπA∑t=1∞nαt⊗ναtjμ=aπAmαjμ⟶ajμ.So A is weak approximately biprojective.
i⇒iii Suppose that A is weak approximately biprojective and fix an idempotent e=pλi−1iλ. Let ρα:A⟶A⊗^A be approximate A-bimodule map such that πA∘ραa⟶a for every a∈A. Since A=⊕i,λAiλ, we have(23)ρα:⊕i,λAiλ⟶⊕i,λAiλ⊗^⊕i,λAiλ=⊕i,μ⊕j,λAiμ⊗^Ajλ,and for each α denote their restrictions Aiλ⟶⊕i,μ⊕j,λAiμ⊗^Ajλ by ρα. By definition of ρα, for aiλ∈Aiλ, we have(24)ραaiλe−aiλ⋅ραe⟶0,ραeaiλ−ραe⋅aiλ⟶0.
Therefore,(25)aiλ⋅ραe−ραe⋅aiλ⟶0,but(26)ραe∈⊕i,μ⊕j,λAiμ⊗^Ajλ,so(27)ραe=∑i,μ∑j,λbiμα⊗cjλα.Then, for i,j∈I,μ,λ∈Λ and a∈A, we get(28)eae⋅biμα⊗cjλα−biμα⊗cjλα⋅eae⟶0.
For everyi,j∈I,μ,λ∈Λ, choose fjμ∈Ajλ and ejμ∈Aiμ such that(29)fjμ=rljpμj−1λ,pμj≠0,0,pμj=0,ejμ=rlipλi−1μ,pμj≠0,0,pμj=0.
Now consider the projective tensor product of Banach algebras with the following multiplication:(30)a⊗b⋅a′⊗b′=aa′⊗b′b,and take(31)mα=∑i,μ∑j,λbiμα⊗cjλα⋅fjμ⊗ejμ.
By easy calculation, we havemα⊆eAe⊗^eAe, and by (28), we have(32)eae⋅mα−mα⋅eae⟶0,a∈A,also(33)πeAebiμα⊗cjλα⋅fjμ⊗ejμ=πeAebiμαfjμ⊗ejμcjλα=πeAebiμα⊗cjλα,then by (33), we get(34)πeAemαeae=πeAeραeeae⟶eae.Therefore, eAe≅A is pseudo amenable.
ii⇔iii It is proved in [4] Corollary 3.6.


## 3. Applications to Upper Triangular Banach Algebras

LetA be a unital Banach algebra, I be an arbitrary index set, and let P be a sandwich matrix, which is a nonzero I×I-upper triangular matrix over A with suppλi:λ,i∈I≤1. The vector space of all I×I-upper triangular matrices M=ai,λi,λ∈I over A with ℓ1-norm M1=∑i,λ∈Iai,λ is such that M1<∞, and with the product, we get(35)M∘N=MPN,for M,N∈UPI,P,A is a Banach algebra, which is called upper triangular Banach algebra and is denoted by UPI,P,A. It should be noted that the components of matrix M∘N is given by(36)aiλ⋅bjμ=rlibpλjaμ,pλj≠0,0,pλj=0.For the identityI×I-matrix P over A, the Banach algebra UPI,P,A is denoted by UPI,A. Taking Aiλ=UPi,λ,P,A, the set of all I×I matrices in the sense that i,λ-entry is nonzero and zero elsewhere. Since A≅Aiλ for each i,λ∈I, then UPI,P,A=⊕i,λ∈IAiλ is decomposition of UPI,P,A into ℓ1-direct sum of subalgebras.Suppose thatInvA is the set of all invertible elements of A. When we say P is an invertible sandwich matrix over InvA, we mean an I×I-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 caseℓ1-Munn Banach algebra ℳA,P,I,Λ proved by Esslamzadeh [7].Lemma 1.
Suppose thatI is finite and V is an invertible I×I-upper triangular matrix over InvA. Then UPI,P,A and UPI,VP,A and also UPI,PV,A are isometrical isomorphisms.Proof.
LetV=vi,ji,j∈I be as in assumption. Define the map φ:UPI,P,A⟶UPI,VP,A by φa=aV−1, where V−1=wi,ji,j∈I is the inverse of V. It is easy to see that φ is a linear bijection. For a=ai,ji,j∈I∈UPI,P,A, we have(37)φa=aV−1=∑i,j∈I∑k∈Iai,kwk,j≤∑i,j∈I∑k∈Iai,kwk,j≤aV−1,and by open mapping theorem, φ−1 is bounded; thus, it is an isomorphism. Similarly for the other case, consider isomorphism φ:UPI,P,A⟶UPI,PV,A by φa=V−1a.
Note that ifI is finite, then by previous Lemma, we have UPI,P,A≅UPI,P−1P,A≅UPI,A. Also, since UPI,A is unital, UPI,P,A is unital too.Lemma 2.
LetA be a Banach algebra with an (a left/right) identity, I be a totally ordered set, and let P be an invertible sandwich matrix over InvA. Then UPI,P,A has an (a left/right) approximate identity, respectively.Proof.
Letℱ be the set of all finite subsets of I. Suppose that ε>0 and a=ai,ji,j∈I∈UPI,P,A are arbitrary. Then there existsF∈ℱ such that ∑i,j∈I/Fai,j<ε. Define eF=ei,ji,j∈I by ei,i=pi,i−1,pi,i≠0,0,pi,i=0, whenever i∈F, zero elsewhere and also define EF=Ei,ji,j∈I by Ei,i=eA,pi,i≠0,0,pi,i=0. whenever i∈F and zero elsewhere, where eA is the (left/right) identity of A and pi,i∈P. Thus, we get(38)eF∘a−a=eFPa−a=∑k∈Fei,kpk,ji,j∈Ia−a=EFa−a=∑i,j∈I/Fai,j≤∑i,j∈I/Fai,j<ε,and similarly, we get(39)a∘eF−a=aPeF−a=aEF−a=∑i,j∈I/Fai,j≤∑i,j∈I/Fai,j<ε,so UPI,P,A has an (a left/right) approximate identity.
We remind that a Banach algebraA is called approximate right φ-amenable if there exists a not necessarily bounded net aα in A such that aαa−φaaα⟶0 and φaα⟶1 for every a∈A and φ∈ΔA, where ΔA is the character space of A. The left case is defined similarly. For more information, see Refs. [13, 14].Theorem 3.
LetA be a unital Banach algebra, I be a totally ordered set with a smallest element i0 and φ∈ΔA, and let P be an invertible sandwich matrix over InvA such that the i0,i0-th entry of P is nonzero. Then UPI,P,A is approximately biprojective if and only if I is singleton and A is approximately biprojective.Proof.
Suppose thatUPI,P,A is approximately biprojective. Let i0 be the smallest element of I. Define(40)ψφ:UPI,P,A⟶ℂ,ψφai,ji,j∈I=φai0,i0pi0,i0,where pi0,i0 is the i0,i0-th entry of P. Note that pi0,i0∈InvA, then we have φpi0,i0≠0, and, therefore, ψφ is nonzero. Also, for ai,ji,j∈I,bi,ji,j∈I∈UPI,P,A, consider i0,i0-th entry of ai,ji,j∈I∘bi,ji,j∈I=ai,ji,j∈IPbi,ji,j∈I that is equal to(41)∑t∈I∑k∈Iai0,kpk,ti0,tbt,i0=∑k∈Iai0,kpk,i0bi0,i0=ai0,i0pi0,i0bi0,i0,so(42)ψφai,ji,j∈I∘bi,ji,j∈I=φai0,i0pi0,i0bi0,i0pi0,i0=φai0,i0pi0,i0φbi0,i0pi0,i0=ψφai,ji,j∈Iψφbi,ji,j∈I.
Thus,ψφ∈ΔUPI,P,A. Since A is unital by Lemma 2, UPI,P,A has a right approximate identity and by [15] Theorem 3.9, we can conclude that UPI,P,A is approximately right ψφ-amenable. Define a closed ideal in UPI,P,A by(43)J=ai,ji,j∈I∈UPI,P,A:ai,j=0,i≠i0.
We haveψφ|J≠0. Now by [16] Proposition 5.1, J is approximately right ψφ-amenable. So there exists a net mα⊆J such that(44)mα∘g−ψφgmα⟶0,ψφmα⟶1g∈J.
We go toward a contradiction and assume thatI has at least two elements. Set(45)g=0eA⋯eA⋯00⋯0⋯⋮⋮⋮⋮⋮00⋯0⋯⋮⋮⋮⋮⋮,where eA is the identity of A . For every α, mα has the form mi0α⋯miα⋯0⋯0⋯⋮⋮⋮⋮0⋯0⋯⋮⋮⋮⋮, where miα∈A for every i∈I. As every row and every column in P contains exactly one nonzero entry, by setting mα and g in (44), we conclude that mi0αpi0,i0⟶0, where pi0,i0 is nonzero element of i0-th row of P. Since φ is continuous, we have φmi0α⟶0, and therefore ψφmα⟶0 gives a contradiction. So I is singleton.Proposition 1.
LetA be a weak approximately biprojective Banach algebra and φ∈ΔA. If there exists an element a0∈A in the sense that aa0=a0a for every a∈A and φa0=1, then A is approximately right φ-amenable.Proof.
Suppose thatρα:A⟶A⊗^A is a net of approximate A-bimodule maps such that πA∘ραa⟶a for every a∈A. Define(46)T:A⊗^A⟶A,Ta⊗b=φab.
It is readily seen thatT is a bounded linear map and(47)Ta⋅x=φaTx,Tx⋅a=Txa,φ∘Tx=φ∘πAx,for every a∈A and x∈A⊗^A. Let a0 be as in assumption. Set mα=ραa0, then we have(48)φaTmα−Tmαa=Ta⋅mα−mα⋅a=Ta⋅ραa0−ραa0⋅a=Ta⋅ραa0−ραaa0+ραa0a−ραa0⋅a⟶0,and(49)φ∘Tmα=φ∘πAmα=φ∘πA∘ραa0⟶φa0=1.
SoA is approximately right φ-amenable.Proposition 2.
Suppose thatA is a unital Banach algebra, φ∈ΔA and P is an invertible sandwich matrix over InvA. If I is a finite set which has at least two elements, then UPI,P,A is not weak approximately biprojective.Proof.
Assume a contradiction thatUPI,P,A is weak approximately biprojective and I=i0,i1,…,in is a finite set. So UPI,P,A has an identity. Without loss of generality, we may assume that pi0,i0≠0. Define(50)ψφ:UPI,P,A⟶ℂψφai,ji,j∈I=φai0,i0pi0,i0,and then, ψφ∈ΔUPI,P,A. By Proposition 1, UPI,P,A is approximately right ψφ-amenable. Define a closed ideal in UPI,P,A by(51)J=ai,ji,j∈I∈UPI,P,A:ai,j=0,i≠i0.
We haveψφ|J≠0. Now by [16] Proposition 5.1, J is approximately right ψφ-amenable. So there exists a net mα⊆J such that(52)mα∘g−ψφgmα⟶0,ψφmα⟶1,g∈J.
A similar argument as in the proof of Theorem3, we obtain a contradiction and I must be singleton.Theorem 4.
LetA be a unital Banach algebra, φ∈ΔA and let I be finite. If P is an invertible sandwich matrix over InvA, then UPI,P,A is weak approximately biprojective if and only if I is singleton and A is weak approximately biprojective.Proof.
By Proposition2 the proof is trivial.Corollary 1.
LetA be a unital Banach algebra, φ∈ΔA and let I be finite. Then UPI,A is weak approximately biprojective if and only if I is singleton and A 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. LetG be a group, I and Λ be index sets, and G0=G∪0 be the group with zero arising from G by adjunction of a zero element. Let P=piλ be a Λ×I regular sandwich matrix over G0, so each row and each column of P contains at least one nonzero entry. The associated Rees semigroup is defined by(53)S=giλ:g∈G,i∈I,λ∈Λ∪θ,where giλ denotes the element of MI×ΛG0 with g in the i,λ-th place and 0 elsewhere, and θ is the zero matrix. The binary operation on S is defined by(54)giλ∘hjμ=gpλjhiμ,ifpλj≠0,θ,ifpλj=0,and(55)giλ∘θ=θ∘giλ=θ∘θ=θ.This semigroup is denoted byM0G0,P,I,Λ and called Rees matrix semigroup over G0. Further, by [7]Proposition 5.6, we get(56)ℓ1Sℂδθ=ℳℓ1G,P,I,Λ,where the zero of G0 is identified with the zero of the ℓ1-Munn algebra ℳℓ1G,P,I,Λ, and P is considered as a matrix over ℓ1G. Furthermore, the multiplication in ℓ1S satisfies the following equations for all F,G∈ℓ1G and λ,μ∈Λ,i,j∈I:(57)Fiλ⋆Gjμ=F⋆δpλj⋆Giμ,ifpλj≠0,∑s,t∈SFsGtδθ,ifpλj=0,and(58)F⋆δθ=δθ⋆F=∑s∈SFsδθ.It is worthwhile to mention that ifΛ and I are singleton sets and the sandwich matrix P is the identity matrix, with the group identity on the diagonal of P, then ℓ1S/ℂδθ is exactly equal to the traditional group algebra ℓ1G.As a direct consequence of Theorems1 and 2, we conclude the following results.Theorem 5.
LetG be a group and S=ℳ0G0,P,I,Λ be a Rees matrix semigroup with the invertible Sandwich matrix P. Then, we get those as follows:(i)
ℓ1S/ℂδθ is approximately biprojective if and only if G is finite(ii)
ℓ1S/ℂδθ is weak approximately biprojective if and only if G is amenableRemark 1.
We note that ifP is the identity matrix and Λ=I, then ℳ0G0,P,I,Λ is denoted by ℳ0G0,Λ and is called the Brandt semigroup over G0 with index set Λ. In this case, ℓ1S/ℂδ0=MΛℓ1G. In special case, if G is the trivial group G=e, then ℓ1S/ℂδ0=MΛℂ. These facts imply the following results:(i)
LetG be a group, Λ be a nonempty set, and let S=M0G0,Λ be the Brandt semigroup over G0 with index set Λ. Then ℓ1S/ℂδ0 is approximately biprojective if and only if G is finite.(ii)
LetΛ be a nonempty set. Then MΛℂ is (weak) approximately biprojective.



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*Source: 2112167-2022-11-14.xml*