# On Amenability-Like Properties of a Class of Matrix Algebras

Authors: M. Rostami; S. F. Shariati; A. Sahami
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, we show that a matrix algebraℒℳIpA is a dual Banach algebra, where A is a dual Banach algebra and 1≤p≤2. We show that ℒℳIpℂ is Connes amenable if and only if I is finite, for every nonempty set I. Additionally, we prove that ℒℳIpℂ is always pseudo-Connes amenable, for 1≤p≤2. Also, Connes amenability and approximate Connes biprojectivity are investigated for generalized upper triangular matrix algebras. Finally, we show that UpIpA∗∗ is approximately biflat if and only if A∗∗ is approximately biflat and I is a singleton.


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## Body

## 1. Introduction

A special class of matrix algebras, so-calledℓ1-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 ℓ1-Munn algebras into arbitrary (not necessarily commutative) Banach algebras and improved results in [5]. He also discussed homomorphisms on Banach algebras with range in ℓ1-Munn algebras. Habibian et al., etal in [6], using a new method developed by Niels Grønbæk [7, 8], namely, Morita equivalence, studied some homological properties of ℓ1-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 ℓ1-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 ofℓ1-Munn algebras, Lashkarizadeh Bami and Naseri introduced the notion of ℓp-Munn algebras. Let I be a nonempty set and let A be a Banach algebra. We denote ℒℳIpA for the vector space of I×I matrices aij over A such that(1)aijp=∑i,j∈Iaijp1/p<∞.They proved thatℒℳIpA with matrix multiplication and ⋅p is a Banach algebra if 1≤p≤2 (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 p=1, it has been shown that ℒℳIpA is a dual Banach algebra, where A is a dual Banach algebra [14]. In this study, first, we generalize this fact, for 1<p≤2. 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, ℒℳIpℂ. In Section 4, we concentrate on the generalized upper triangular Banach algebras UpIpA, where 1≤p≤2. We study the approximate biflatness of the second dual UpIpA∗∗ which leads us to the conclusion that the nonempty set I must be a singleton.


## 2. Connes Amenability-Like Properties ofℒℳIpℂ

We recall some terminology from [13]. Let A be a Banach algebra, and let E be a Banach A-bimodule. An A-bimodule E is called dual if there exists a closed submodule E∗ of E∗ such that E=E∗∗. The Banach algebra A is called dual if it is dual as a Banach A-bimodule.It is worth noting that there is an isometric-isomorphismℒℳIpA≅ℓp0⊕j∈JAj as Banach spaces, where, for every j∈J, Aj=A and J=I×I.Theorem 1.
LetA be a dual Banach algebra with respect to predual A∗, and let I be a nonempty index set. Then, ℒℳIpA is a dual Banach algebra with respect to the predual ℓq0⊕i∈JAi∗, where 1<p≤2, 1/p+1/q=1, and for every i∈J, Ai∗=A∗, and J=I×I.Proof.
The proof is divided into two steps.Claim 1.
First, we show thatℓq0⊕i∈JAi∗∗≅ℓp0⊕i∈JAi.
To see this, letg=gαα∈J∈ℓp0⊕i∈JAi. Define ψg:ℓq0⊕i∈JAi∗⟶ℂ by ψgf=∑α∈Jgαfα, where f=fαα∈J∈ℓq0⊕i∈JAi∗. We have from Young’s inequality [15]:(2)ψgf≤∑α∈Jgαfα≤∑α∈Jfαgα≤1q∑α∈Jfαq+1p∑α∈Jgαp.
Moreover, applying the last inequality tofα/fq and gα/gp instead of fα and gα leads to(3)gp−1fq−1∑α∈Jfαgα≤1q+1p=1.
So,ψgf≤gpfq. It follows that ψg is bounded and(4)ψg≤gp.
We show that the following map is an isometric-isomorphism:(5)Θ:ℓp0⊕i∈JAi⟶ℓq0⊕i∈JAi∗∗,g↦ψg.
Obviously, the mapΘ is linear. Let ψ∈ℓq0⊕i∈JAi∗∗; we claim that there exists g=gαα∈J∈ℓp0⊕i∈JAi, such that Θg=ψ. For α∈J and λ∈A∗, let ξα,λ be the element of ℓq0⊕i∈JAi∗ defined by(6)ξα,λβ=λ,β=α0,otherwise.
Furthermore,ξα,λq=λ. Set gαλ:=ψξα,λ, for every α∈J and λ∈A∗. It may be observed that gα≤ψ, for every α∈J. So, each gα is a continuous linear functional on A∗ and drops in A. Take fαα∈J∈ℓq0⊕i∈JAi∗. Consider(7)χα=ψξα,fα¯ψξα,fα,ψξα,fα≠0,1,otherwise.
Hence, we see thatχα=1, for every α∈J. Let F be an arbitrary finite subset of J. We have(8)∑α∈Fgαfα=∑α∈Fψξα,fα=∑α∈Fχαψξα,fα=∑α∈Fψχαξα,fα=∑α∈Fψξα,χαfα=ψ∑α∈Fξα,χαfα≤ψfqq≤ψf=fαα∈J∈ℓq0⊕i∈JAi∗,fq≤1.
Therefore,∑α∈Fgα≤ψ. From (Proposition 6.11 of [16]), we deduce that(9)∑α∈Jgαp1/p≤ψ,showing that g∈ℓp0⊕i∈JAi. Let f=fαα∈J∈c000⊕i∈JAi∗. So, there exists a finite subset E of J such that, for any α∈J,E, fα=0. We have(10)Θgf=ψgf=∑α∈Jgαfα=∑α∈Eψξα,fα=ψ∑α∈Eξα,fα=ψf.
We hence see thatΘg=ψ on c000⊕i∈JAi∗. As c000⊕i∈JAi∗ is dense in ℓq0⊕i∈JAi∗, with respect to ⋅q, this shows that Θg=ψ on ℓq0⊕i∈JAi∗. Equations (4) and (9) imply that(11)gp≤ψ=Θg≤gp.
Thus, the mapΘ is an isometry, and using the open mapping theorem, we conclude ℓp0⊕i∈JAi≅ℓq0⊕i∈JAi∗∗.Claim 2.
ℓq0⊕i∈JAi∗ is a closed ℒℳIpA-submodule of ℓq0⊕i∈JAi∗.
Suppose thata=aαα∈J and f=fαα∈J are arbitrary elements in ℓp0⊕i∈JAi and ℓq0⊕i∈JAi∗, respectively. Since ℓq0⊕i∈JAi∗ is an ℒℳIpA-bimodule with dual actions, we have a⋅f∈ℓq0⊕i∈JAi∗. Moreover, for every b=bαα∈J in ℓp0⊕i∈JAi,(12)a⋅fb=fba=∑α∈Jfαcα,where cαα∈J=ci,ji,j∈I=ba with respect to the matrix multiplication in ℒℳIpA. We claim that a⋅f belongs to ℓq0⊕i∈JAi∗. Equation (12) implies that(13)a⋅fb=∑i,j∈Ici,j,fi,j=∑i,j∈I∑k∈Ibi,kak,j,fi,j=∑i,j∈I∑k∈Ibi,kak,j,fi,j=∑i,j∈I∑k∈Ibi,k,ak,j⋅fi,j.
Take a finite subsetF of I. Using Proposition 6.11 of [16], b∞<∞ and then(14)∑i,j∈F∑k∈Fbi,k,ak,j⋅fi,j≤∑i,j∈F∑k∈Fak,jfi,jbi,k≤b∞∑i,j∈F∑k∈Fak,jfi,j≤b∞∑i,j∈F∑k∈F1pak,jp+1qfi,jq≤b∞1papp+1qfqq.
So,∑i,j∈I∑k∈Ibi,k,ak,j⋅fi,j<∞. By applying rearrangement series in equation (13),(15)a⋅fb=∑i,j∈I∑k∈Ibi,j,aj,k⋅fi,k=∑i,j∈Ibi,j,∑k∈Iaj,k⋅fi,k.
Therefore,a⋅f=xαα∈J, where, for every α=i,j∈J, we have xi,j=∑k∈Iaj,k⋅fi,k which belongs to A∗. So, ℓq0⊕i∈JAi∗ is a closed ℒℳIpA-submodule of ℓq0⊕i∈JAi∗, 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,ℒℳIpℂ.LetA be a dual Banach algebra. A dual Banach A-bimodule E is normal if the left and right module actions of A on E are wk∗-wk∗ continuous. A dual Banach algebra A is called Connes amenable if, for every normal dual Banach A-bimodule E, every wk∗-wk∗ continuous derivation D:A⟶E is inner. For a given dual Banach algebra A and a Banach A-bimodule E, σwcE denotes the set of all elements x∈E such that the module maps,(16)A⟶E,a⟶a⋅x,x⋅a,are wk∗-wk continuous which is a closed submodule of E. It is known from Corollary 4.6 of [17] that π∗A∗⊆σwcA⊗^A∗ and then an A-bimodule morphism πσwc:σwcA⊗^A∗∗⟶A is induced by the product map πA:A⊗^A⟶A, where πAa⊗b=ab, for every a,b∈A. For further details, we refer to [13, 17].It was shown in [14] that ℓ1-Munn algebra ℒℳI1ℂ is Connes amenable if and only if I is finite.Theorem 2.
Let1<p≤2. Then, ℒℳIpℂ is Connes amenable if and only if I is finite.Proof.
IfℒℳIpℂ 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 I is finite.
Conversely, if the index setI is finite, then ℒℳIpℂ≅ℒℳI1ℂ 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 A is called Connes biprojective if there exists a bounded A-bimodule morphism ρ:A⟶σwcA⊗^A∗∗ such that πσwc°ρ=idA; for more details, see [18]. In the sequel, for a Banach space X, we write κX:X⟶X∗∗ for the canonical map given by κXxf=fx, for x∈X and f∈X∗.Theorem 3.
LetI be a nonempty index set and 1≤p≤2. Then, ℒℳIpℂ is Connes biprojective.Proof.
LetA=ℒℳIpℂ. Fixed k∈I. Take a finite subset F of I×I. For every a=ai,j∈A, define ρ:A⟶σwcA⊗^A∗∗ by(17)ρa=ι∗°κA⊗^Aa,where a=∑i,j∈Faijϵik⊗ϵkj∈A⊗^A and ι:σwcA⊗^A∗⟶A⊗^A∗ is the inclusion map. This latter exists because ap<∞. It can be observed that ρ is a bounded A-bimodule morphism and further(18)πσwc°ρa=πσwc°id∗°κA⊗^Aa=πσwcκA⊗^Aa|σwcA⊗^A∗=πA⊗^Aa=a.Lemma 1.
LetA be a unital dual Banach algebra, 1≤p≤2, and let I be a nonempty set. Then, ℒℳIpA has an approximate identity.Proof.
Assume thatℱ is the family of all finite subsets of I that is a directed set via inclusion. Consider the net eFF∈ℱ in ℒℳIpA by(19)eF=eij=eii=1A,∀i∈F,eij=0,otherwise.
We claim thateFF∈ℱ is an approximate identity for ℒℳIpA. Take an arbitrary element a=aij in ℒℳIpA. Since ap<∞, for every ϵ>0, there exists a finite subset E∈ℱ such that, for every F⊇E in ℱ,(20)eFa−a=∑i,j∈I∖Faijp1/p<ε.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, A is said to be pseudoamenable if there is a net mα⊆A⊗^A provided that(21)a⋅mα−mα⋅a⟶0,πAmαa⟶a,for all elements a∈A, where the module actions are given by a⋅b⊗c=ab⊗c and b⊗c⋅a=b⊗ca, for a,b,c∈A.Following Definition 4.3 in [21], we have the following definition.Definition 1.
A dual Banach algebraA is called pseudo-Connes amenable if there exists a net Mα in A⊗^A such that a⋅Mα−Mα⋅a⟶wk∗0 in σwcA⊗^A∗∗ and πσwcMαa⟶wk∗a in A.Proposition 1.
LetA be a Connes biprojective dual Banach algebra with an approximate identity. Then, A is pseudo-Connes amenable.Proof.
Leteαα∈J be an approximate identity for A, and let ρ:A⟶σwcA⊗^A∗∗ be a bounded A-bimodule morphism satisfying πσwc°ρ=idA. Define mα=ρeα. By Goldstein’s theorem, for each α∈J, there exists a net uβαβ∈Ψα in A⊗^A such that wk∗0limβuβα=mα in σwcA⊗^A∗∗. Furthermore, for every a∈A,(22)wk∗0limαwk∗0limβa⋅uβα−uβα⋅a=wk∗0lima⋅mα−mα⋅a=0inσwcA⊗^A∗∗.
Moreover, the duality ofA and wk∗-continuity of πσwc imply that(23)wk∗0limαwk∗0limβaπσwcuβα=wk∗0limαaπσwcmα=aa∈A.
PutΔ=J×∏α∈JΨα as a directed set with product ordering defined via(24)α,βγ≤Δα′,βγ′⇔α≤Jα′,βγ≤∏βγ′α,α′∈J,βγ,βγ′∈∏α∈JΨα,where βγ≤∏βγ′ means that βγ≤Ψγβγ′, for every γ∈J. Let δ=α,βγ∈Δ and uδ=uβαα. Applying iterated limit theorem (Page 69 of [22]),(25)wk∗0limδa⋅uδ−uδ⋅a=0inσwcA⊗^A∗∗,and also,(26)wk∗0limδaπσwcuδ=ainA.
Hence,uδΔ has the required properties and A is pseudo-Connes amenable.From Theorem3, Lemma 1, and Proposition 1, we deduce the following result.Corollary 1.
LetI be a nonempty index set and 1≤p≤2. Then, ℒℳIpℂ is pseudo-Connes amenable.


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

LetA be a Banach algebra, and let I be a totally ordered set. The generalized upper triangular Banach algebra, as a closed subalgebra of ℓp-Munn algebras, is denoted by(27)UpIpA=ai,j∈ℒℳIpA|ai,j=0for everyi>j,where 1≤p≤2. Some homological properties of this class of Banach algebras were investigated in [23]. In the case p=1 for the classes of dual Banach algebras, UpIpA is a dual Banach algebra, where A is a dual Banach algebra [9]. With a similar argument in Theorem 1, we generalize this fact, for 1<p≤2. The proof is similar, so we omit the proof.Theorem 4.
IfA is a dual Banach algebra with the predual A∗ and I is a totally ordered set, then UpIpA is a dual Banach algebra, where 1<p≤2.LetA be a dual Banach algebra. The set of all wk∗-continuous homomorphism from A onto ℂ is denoted by Δwk∗A.Theorem 5.
LetI be a totally ordered set, and let A be a dual Banach algebra with a left identity such that Δwk∗A≠∅ and 1≤p≤2. Then, UpIpA is Connes amenable if and only if I is singleton and A is Connes amenable.Proof.
IfUpIpA is Connes amenable, then it has a unit (Proposition 4.1 of [13]). It implies that I must be a finite set. Put I=i1,…,in. Consider the wk∗-continuous character Υ on UpIpA given by ai,ji,j∈I⟶φain,in, for every ai,ji,j∈I∈UpIpA, where φ∈Δwk∗A. According to Theorem 2.4 of [24], there exists an element m in UpIpA such that(28)am−Υam=0andΥm=φmin,in=1,a∈UpIpA.
Assume thatI>1. Choose an element,(29)a=0⋯0e:⋯::0⋯0e0⋯00,in UpIpA, where e is a left unit for the Banach algebra A. So, Υa=φ0=0. It follows that am=0. One can see that min,in=0, which is a contradiction with φmin,in=1. The converse is clear.Recently, approximate homological notion for dual Banach algebras have been introduced. A dual Banach algebraA is called approximately Connes biprojective if there exists a net ραα of continuous A-bimodule morphisms from A into σwcA⊗^A∗∗ such that πσwc°ραa⟶a, for every a∈A [25].Theorem 6.
LetA be a dual Banach algebra with a right identity which Δwk∗A≠∅, 1≤p≤2. Let I be a totally ordered set which has a smallest element i0. Then, UpIpA is approximately Connes biprojective if and only if A is approximately Connes biprojective and I is singleton.Proof.
Suppose thatUpIpA is approximately Connes biprojective. Since A has a right identity e, the Banach algebra UpIpA has a right approximate identity (Lemma 5.1 of [26]). Consider the wk∗-continuous character Υ on UpIpA defined via(30)Υaij=φai0,i0,for every ai,ji,j∈I∈UpIpA, where φ∈Δwk∗A. Applying Theorem 2.4 of [25], there exists an element u in UpIpA such that ua=Υau and Υu=φui0,i0=1, for every a∈UpIpA. By contrary, suppose that I has at least two elements. Choose an element a in UpIpA which i0,jth entry is e and others are zero, where j≠i0. Therefore, ua=0. By a simple computation, it may be observed that ui0,i0=0, which is a contradiction with Υu=φui0,i0=1. 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.etalin [27] generalized the concept of biflatness and created approximately biflat Banach algebras. A Banach algebra A is approximately biflat if there exists a net of A-bimodule morphisms ρα from A⊗^A∗ into A∗ such that ρα°πA∗⟶W∗OTidA∗, where W∗OT is denoted for the weak∗ operator topology and idA∗ is the identity map on A∗. For two Banach spaces X and Y, recall that Tα⟶W∗OTT in BX,Y∗ if Tαx⟶wk∗Tx in Y∗, for every x∈X, where BX,Y∗ the space of all bounded linear operators from X to Y∗. Bodaghi and Tanha, in [28], gave a modification of this notion and introduced module approximate biflat Banach algebras.Lemma 2.
LetA be a Banach algebra. If A is approximately biflat, then there exists a net ρα of bounded A-bimodule morphisms from A into A⊗^A∗∗ such that πA∗∗°ρα⟶W∗OTκA.Proof.
SinceA is approximately biflat, there exists a net of bounded A-bimodule morphisms ζα:A⊗^A∗⟶A∗ such that ζα°πA∗⟶W∗OTidA∗. Let ρα=ζα∗°κA. So, for every a∈A and f∈A∗, we have(31)limαπA∗∗°ραa,f=limαπA∗∗°ζα∗°κAa,f=limαζα∗°κAa,πA∗f=limακAa,ζα°πA∗f=κAa,f.
It follows thatπA∗∗°ραa⟶wk∗κAa, for every a∈A.A Banach algebraA which is the predual of a von Neumann algebra, and the identity of A∗ is multiplicative was considered in 1983 by To-Ming Lau [29]. This nice class of Banach algebras was called F-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 φ-amenability for a Banach algebra as a considerable generalization of Lau’s left amenability was defined by Kaniuth et al. etal.[30]. In fact, a Banach algebra A is called left φ-amenable if there exists an element m in A∗∗ such that a⋅m=φam and mφ=1, for every a∈A, where φ is a character on A. Many aspects of this notion have been studied in [31, 32]. In the sequel, we state the relation between left φ-amenability and approximate biflatness of Banach algebras.Theorem 7.
LetA be a Banach algebra with a left approximate identity and φ∈ΔA. If A∗∗ is approximately biflat, then A is left φ-amenable.Proof.
LetA∗∗ be an approximately biflat Banach algebra. Lemma 2 implies that there exists a net ρα of bounded A∗∗-bimodule morphisms from A∗∗ into A∗∗⊗^A∗∗∗∗ such that πA∗∗∗∗°ρα⟶W∗OTκA∗∗. Let K=kerφ. According to Lemma 1.7 of [33], there exists a bounded linear map ψ:A∗∗⊗^A∗∗⟶A⊗^A∗∗ such that, for every a,b∈A and m∈A∗∗⊗^A∗∗, we have(i)
ψa⊗b=a⊗b(ii)
ψm⋅a=ψm⋅a,a⋅ψm=ψa⋅m(iii)
πA∗∗ψm=πA∗∗m
Letξα:=idA⊗q∗∗∗∗°ψ∗∗°ρα°κA, where q:A⟶A/K is the quotient map. So, ξα is a net of bounded A-bimodule morphisms from A into A⊗^A/K∗∗∗∗. It may be observed that ξα|K=0. For this purpose, since A has a left approximate identity, AK¯=K. If y is an arbitrary element in K, then there are two nets xβ in A and yβ in K satisfying y=limβxβyβ. Moreover,(32)ξαy=idA⊗q∗∗∗∗°ψ∗∗°ρα°κAy=limβidA⊗q∗∗∗∗°ψ∗∗°ρα°κAxβyβ=limβidA⊗q∗∗∗∗ψ∗∗ρακAxβ⋅κAyβ=0.
Therefore,ξα can be dropped on A/K. That is, the maps ξ˜α:A/K⟶A⊗^A/K∗∗∗∗ given by ξ˜αa+K=ξαa are well defined. Let θα:=idA⊗φ⌣∗∗∗∗°ξ˜α:A/K⟶A∗∗∗∗, where φ˘ is the character on A/K induced by φ. Choose an element b0∈A such that φb0=1. Set nα=θαb0+K. Since ab0−φab0∈K,(33)a⋅nα=a⋅idA⊗φ⌣∗∗∗∗°ξ˜αb0+K=idA⊗φ⌣∗∗∗∗°ξ˜αab0+K=φaidA⊗φ⌣∗∗∗∗°ξ˜αb0+K=φanα.
We denoteφ˜ for the character on A∗∗ defined by φ˜F=Fφ, for every F∈A∗∗. We have(34)nαφ˜=φ˜,idA⊗φ⌣∗∗∗∗°ξ˜αb0+K=φ˜,idA⊗φ⌣∗∗∗∗°ξαb0=φ˜,idA⊗φ⌣∗∗∗∗°idA⊗q∗∗∗∗°ψ∗∗°ρα°κAb0=φ˜,πA∗∗∗∗°ψ∗∗°ρα°κAb0=φ˜,πA∗∗∗∗°ρα°κAb0.
Sincelimαφ˜,πA∗∗∗∗°ρα°κAb0=φ,b0=1, by replacing nα with nα/nαφ˜, there exists an element n in A∗∗∗∗ such that a⋅n=φan, for every a∈A and nφ˜=1. By Goldstine’s theorem, there is mβ⊆A∗∗ such that, for every a∈A, a⋅mβ−mβ⋅a⟶0 weakly in A∗∗ and φ˜mβ⟶1. Finally, for each finite subset F⊆A, say F=a1,…,ak, let(35)V=a1⋅x−φa1x,a2⋅x−φa2x,…,ak⋅x−φakx,φ˜x−1|x∈A∗∗,x≤n.
It is obvious thatV is a convex subset of ∏i=1kA∗∗⊕1ℂ, and by applying Mazur’s lemma,(36)0,0,…,0∈V¯wk=V¯⋅.
Therefore, there exists a bounded netvβ in A∗∗ such that a⋅vβ−φavβ⟶0, for every a∈A and φ˜vβ⟶1. The boundedness condition implies that wk∗-limit point of vβ exists in A∗∗. Let v=wk∗0limβvβ. It follows that a⋅v=φav and vφ=1. Hence, A is left φ-amenable.As an application of the above theorem, we characterize approximate biflatness of the second dual of generalized upper triangular matrix algebras.Theorem 8.
LetA be a Banach algebra with a right identity, ΔA≠∅, 1≤p≤2, and also, let I be a totally ordered set which has a smallest element i0. Then, UpIpA∗∗ is approximately biflat if and only if A∗∗ is approximately biflat and I is singleton.Proof.
We only need to prove the ‘if’ part, and the converse is clear. Suppose thatUpIpA∗∗ is approximately biflat. Consider the character Υ on UpIpA defined via Υai,j=φai0,i0, for every ai,ji,j∈I∈UpIpA, where φ∈ΔA. The right version of Theorem 7 and Lemma 5.1 of [26] imply that UpIpA is right Υ-amenable. Applying Theorem 1.4 of [30], there exist a bounded net mα in UpIpA such that(37)mαa−Υamα⟶0andΥmα=φmi0,i0α=1,a∈UpIpA.
Suppose towards a contradiction thatI>1. Consider the element(38)a=0e0⋯000⋯::::000⋯::::,in UpIpA, where e is the right identity for the Banach algebra A. So, Υa=φ0=0. Hence, we see that mαa⟶0. It is easy to see that mi0,i0α⟶0. The continuity of character φ implies that φmi0,i0α⟶0, which is a contradiction with φmi0,i0α=1.



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