# On Pseudo-contractibility and Ultra Central Approximate Identity of Some Semigroup Algebras

Authors: Amir Sahami; Seyedeh Fatemeh Shariati
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we investigate the existence of an ultra central approximate identity for a Banach algebraA and its second dual A∗∗. Also, we prove that for a left cancellative regular semigroup S, ℓ1S∗∗ has an ultra central approximate identity if and only if S is a group. As an application, we show that for a left cancellative semigroup S, ℓ1S∗∗ is pseudo-contractible if and only if S is a finite group. We also study this property for φ-Lau product Banach algebras and the module extension Banach algebras.


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## Body

## 1. Introduction and Preliminaries

A Banach algebraA is amenable if there exists a bounded net mα in A⊗^A such that a⋅mα−mα⋅a⟶0 and πAmαa⟶a for each a∈A, where A⊗^A is the projective tensor product A with A and πA:A⊗^A⟶A is denoted for a bounded product morphism given by πAa⊗b=ab. It is well-known that every amenable Banach algebra has a bounded approximate identity πAmα. The existence of a bounded approximate identity may change the structure of a Banach algebra. For instance, the Fourier algebra AG with respect to a locally compact group G has a bounded approximate identity if and only if G is amenable (Leptin’s Theorem), see [1].Some generalized notions of amenability like pseudo-contractibility and pseudo-amenability are given by Ghahramani and Zhang. A Banach algebraA is called pseudo-contractible (pseudo-amenable) if there exists a not necessarily bounded net mα in A⊗^A such that a⋅mα=mα⋅aa⋅mα−mα⋅a⟶0 and πAmαa⟶a for each a∈A, respectively (see [2]). In fact, πAmα is a central approximate identity (approximate identity) with respect to the notion pseudo-contractibility (pseudo-amenability), respectively. We have to remind that for a locally compact group G, the Segal algebra SG has a central approximate identity if and only if G is a SIN group (see [3]).Recently, the first author Pourabbas gave a notion of Johnson pseudo-contractibility for Banach algebras. In fact, a Banach algebraA is Johnson pseudo-contractible if and only if there exists a not necessarily bounded net mα in A⊗^A∗∗ such that a⋅mα=mα⋅a and πA∗∗mαa⟶a for each a∈A (see [4]). It is easy to see that πA∗∗mα is a net in A∗∗ such that(1)aπA∗∗mα=πA∗∗mαa,πA∗∗mαa⟶a,a∈A.Motivated by these considerations in [5], the notion of ultra central approximate identity for Banach algebras is defined. In fact, a Banach algebra A has an ultra central approximate identity if there exists a net eα in A∗∗ such that aeα=eαa and eαa⟶a, for every a∈A. In [5], the existence of an ultra central approximate identity for the semigroup algebra ℓ1S, where S is a uniformly locally finite inverse semigroup, was characterized. They also showed that for the Brandt semigroup S=M0G;I over a nonempty set I, ℓ1S has an ultra central approximate identity if and only if I is finite.In this note, we study the existence of ultra central approximate identity for the second duals of certain semigroup algebras related to left cancellative semigroup. As an application, we show that for a left cancellative regular semigroupS, ℓ1S∗∗ is pseudo-contractible if and only if S is a finite group. Some Banach algebras which do not have an ultra central approximate identity are given. We also investigate the property of ultra central approximate identity for φ-Lau product Banach algebras and the module extension Banach algebras.


## 2. General Properties

LetA be a Banach algebra. Throughout this note, the symbol A∗∗ is denoted for the second duals of a Banach algebra A which is a Banach algebra with respect to the first Arens product. Jabbari et al. have introduced the notion of φ-inner amenability [6]. For a given Banach algebra A, the set of all nonzero multiplicative linear functionals on A is denoted by ΔA. Let Aφ=a∈A:φa=1, where φ∈ΔA. A Banach algebra A is called φ-inner amenable if there exists a bounded linear functional m on A∗ satisfying mφ=1 and mf⋅a=ma⋅f for every f∈A∗ and for every a∈Aφ. This notion is equivalent with the existence of a bounded net aα in A such that aaα−aαa⟶0 and φaα=1, for every a∈A [7], [Proposition 2.2].Theorem 1.
LetAbe a Banach algebra andφ∈ΔA. IfA∗∗has an ultra central approximate identity, thenAisφ-inner amenable.Proof.
Suppose thatA∗∗ has an ultra central approximate identity. Then, there exists a net eα in A∗∗∗∗ such that(2)aeα=eαa,aeα⟶a,a∈A∗∗.We denote φ˜ for unique extension of φ to A∗∗ is defined by φ˜λ=λφ for every λ∈A∗∗ and also φ˜˜ is denoted for the unique extension of φ˜ to A∗∗∗∗ is defined in the same way. Clearly φ˜˜∈ΔA∗∗∗∗ and φ˜∈ΔA∗∗. So φ˜˜eα⟶1 for every a∈A∗∗. Thus for a sufficient large α, φ˜˜eα stays away from zero. Replacing eα/φ˜˜eα with eα, we may assume that φ˜˜eα=1. By Goldstein’s theorem for every α, there exists a net mβαβ in A∗∗ such that wk∗‐limβmβα=eα in A∗∗∗∗ and mβα≤eα. So by equation (2), wk∗‐limβamβα−mβαa=0 in A∗∗∗∗ for every a∈A. Thus wk‐limβamβα−mβαa=0 in A∗∗ for every a∈A. Let F=a1,a2,…,an be an arbitrary finite subset of A. Set(3)V=a1b−ba1,a2b−ba2,…,anb−ban,φ˜b−1|b∈A∗∗,b≤eα≤∏i=1nAi⊕1ℂ,
where for everyi, Ai=A∗∗. It is easy to see that V is a convex set and 0,0,…,0 is a wk-limit point of V. Since 0,0,…,0∈V¯w=V¯⋅, there exists a bounded net nβ,Fαβ∈I in A∗∗ such that(4)anβ,Fα−nβ,Fαa<ε,φ˜nβ,Fα−1<ε,a∈F.
Consider the following order:(5)β,F,ε≤β1,F1,ε1⇔β≤Iβ1,F⊆F1,ε≥ε1.
So, the bounded netnβ,Fαβ,F,ε satisfies(6)anβ,Fα−nβ,Fαa⟶0,φ˜nβ,Fα−1⟶0,a∈A.
By Banach–Alaoglu’s theorem,nβ,Fαβ,F has a wk∗-limit point say N∈A∗∗. One can see that(7)aN=Na,φ˜N=1,a∈A.
It follows thatA is φ-inner amenable.
Using the similar argument to those in the proof of Theorem1, we obtain the following corollary:Corollary 1.
LetAbe a Banach algebra andφ∈ΔA. IfAhas an ultra central approximate identity, thenAisφ-inner amenable.Example 1.
LetA=ℂℂℂ⋯000⋯:⋯:⋯ℕ×ℕ. With the finite ℓ1-norm and matrix multiplication, A becomes a Banach algebra. We claim that A∗∗ does not have an ultra central approximate identity. Arguing by contradiction, we suppose that A∗∗ has an ultra central approximate identity. Define φ:A⟶ℂ by φai,ji,j∈ℕ=a11 for every ai,ji,j∈ℕ∈A. It is easy to see that φ∈ΔA. Thus, by Theorem 1, A is φ-inner amenable. By [6] and Theorem 1, there exists a bounded net nα=nα1nα2nα3⋯000⋯:⋯:⋯ℕ×ℕ in A such that(8)anα−nαa⟶0,φnα=nα1⟶1,for all a∈A. Put a1=010⋯000⋯:⋯:⋯ℕ×ℕ instead of a in (8), we have nα1⟶0 which is a contradiction. So, A∗∗ does not have an ultra central approximate identity.Example 2.
LetS be a left zero semigroup with S≥2, that is, a semigroup with st=s, for all s,t∈S. We claim that the semigroup algebra ℓ1S∗∗ does not have an ultra central approximate identity. Suppose conversely that ℓ1S∗∗ has an ultra central approximate identity. It is easy to see that fg=φSgf, where φS is denoted for the augmentation character on ℓ1S. Applying Corollary 1, follows that ℓ1S is φS-inner amenable. Then, by [6] and Theorem 1, there exists a bounded net nα in ℓ1S such that(9)fnα−nαf⟶0,ϕSnα⟶1,for all f∈ℓ1S. Since S≥2, set f=δs1 and f=δs2 and put f in (9), we have nα⟶δs1 and nα⟶δs2 which is a contradiction. Thus, ℓ1S does not have an ultra central approximate identity.
LetA and B be Banach algebras with φ∈ΔB. Consider the Cartesian product A×B. Equip A×B with the norm a,b=aA+bB and φ-Lau product, that is, the product which is defined by(10)a,bc,d=ac+φda+φbc,bd,a,b,c,d∈A.
Then,A×B becomes a Banach algebra which we denote it with A×φB. Note that A×φB∗∗ (with the first Arens product) is isometrically isomorphism with A∗∗×φB∗∗, and also for every m,n and a,b in A∗∗×φB∗∗, we have(11)m,na,b=ma+nϕa+bϕm,nb.
For more information about the Lau product and its generalization, see [8–10].Proposition 1.
LetAandBbe Banach algebras withφ∈ΔB. IfA×φBhas an ultra central approximate identity, thenBhas an ultra central approximate identity.Proof.
LetA×φB has an ultra central approximate identity. Then, there exists a net mα,nα in A∗∗×φB∗∗ such that(12)mα,nαa,b=a,bmα,nα,a,bmα,nα⟶a,b,a∈A,b∈B.Thus, we have(13)mαa+nαφa+φbmα,nαb=amα+nαϕa+φbmα,bnα⟶a,b,a∈A,b∈B.Therefore, nαb=bnα⟶b for all b∈B. So, B has an ultra central approximate identity.
LetA be a Banach algebra and X be a Banach A-bimodule. The module extension A⊕X is a Banach algebra with the following multiplication:(14)a,xb,y=ab,ay+bx,a,b∈A,x,y∈X,and the norm a,x=aA+xX. The Banach A-bimodule X is called commutative if ax=xa for every a∈A and x∈X. It is easy to see that A⊕X∗∗ can be identified with A∗∗⊕1X∗∗ (as a Banach space) and the first Arens product on A⊕X∗∗ is given by(15)m,λn,μ=mn,mμ+λn,m,n∈A∗∗,λ,μ∈X∗∗.
For more information about the module extension, see [11].Proposition 2.
LetAbe a Banach algebra andXbe a commutative BanachA-bimodule. If the module extensionA⊕Xhas an ultra central approximate identity, thenAhas an ultra central approximate identity.Proof.
Suppose that the module extensionA⊕X has an ultra central approximate identity. Then, there exists a net mα,λα in A⊕X∗∗ such that(16)mα,λαa,x=a,xmα,λα,mα,λαa,x⟶a,x,a∈A,x∈X.
It implies that for everya∈A and x∈X,(i)
mαa,mαx+λαa=amα,aλα+xmα,(ii)
mαa,mαx+λαa⟶a,x.
Somαa=amα and mαa⟶a. Thus, A has an ultra central approximate identity.


## 3. Applications for Certain Semigroup Algebras

We present some notions of the semigroup theory. The semigroupS is called left cancellative, if for every a,b,c in S,(17)ca=cb⇒a=b.The semigroupS is called regular semigroup, if for every s∈S, there exists s∗∈S such that s=ss∗s and s∗=s∗ss∗ [12].Theorem 2.
LetSbe a left cancellative regular semigroup. Then,ℓ1S∗∗has an ultra central approximate identity if and only ifSis a group.Proof.
Suppose thatℓ1S∗∗ has an ultra central approximate identity. Then, there exists a net eα in ℓ1S∗∗∗∗ such that aeα=eαa and eαa⟶a, for every a∈ℓ1S∗∗. By Goldstein’s theorem, there exists a net xβα in ℓ1S∗∗ such that wk∗‐limβxβα=eα in ℓ1S∗∗∗∗. Thus, wk∗‐limβaxβα−xβαa=0 in ℓ1S∗∗∗∗ for every a∈ℓ1S. So,(18)wk∗‐limαwk∗‐limβaxβα−xβαa=0,wk∗‐limαwk∗‐limβaxβα=a,a∈ℓ1S,in ℓ1S∗∗∗∗. Applying iterated limit theorem [13], we have a net Eγ in ℓ1S∗∗ such that(19)wk∗‐limγaEγ−Eγa=0,wk∗‐limγaEγ=a,a∈ℓ1S,in ℓ1S∗∗∗∗. It is easy to see that(20)wk‐limγaEγ−Eγa=0,wk‐limγaEγ=a,a∈ℓ1S,in ℓ1S∗∗. Let F=a1,a2,…,an be a finite subset of ℓ1S. Set(21)V=a1b−ba1,a2b−ba2,…,anb−ban,a1b−a1,a2b−a2,…,anb−an|=b∈ℓ1S∗∗.
It is easy to see thatV is a convex subset of ∏i=12nAi, where for every i, Ai=ℓ1S∗∗ and 0,0,…,0∈V¯w=V¯⋅. So, there exists a net (say again Eγ) in ℓ1S∗∗ which(22)⋅−limγaEγ−Eγa=0,⋅−limγaEγ=a,for every a∈ℓ1S. Using a similar arguments as above, we may suppose that Eγ is a net in ℓ1S such that(23)limγaEγ−Eγa=0,limγaEγ=a,for every a∈ℓ1S. So, ℓ1S has an approximate identity. It follows that δsEγ⟶δs. Then, there exists t∈S such that st=s. Suppose in contradiction that for every t∈S, st≠s. Assume that Eγ has a form ∑t∈Sbtγδt, where btγ∈ℂ for every t∈S. Let ε=1/2. Since δsEγ⟶δs, there exists γ0 such that ∑t∈Sbtγ0δst−δs≤1/2. Since st≠s, ∑t∈Sbtγ0+1≤1/2. So ∑t∈Sbtγ0≤−1/2, which is a contradiction. Thus, for every s0∈S, we have sts0=ss0. The left cancellative property of S implies that ts0=s0. Hence, t is a left identity for S. So,(24)δt=limγδtEγ=limγEγ.
For everys∈S by equation (24), limγδsEγ=δst. On the other hand, Eγ is an approximate identity for ℓ1S. So, limγδsEγ=δs. It follows that s=st, so t is the identity of S. Since S is a regular, for every s∈S, there exists s∗∈S such that(25)ss∗s=s=st,s∗ss∗=s∗=s∗t.
Again, the left cancellative property ofS implies that s∗s=ss∗=t. It deduces that S is a group.
For converse, ifS is a group, then ℓ1S has an identity. So ℓ1S∗∗ has an identity. It follows that ℓ1S∗∗ has an ultra central approximate identity.
A Banach algebraA is contractible (or super amenable) if for every bounded derivation D:A⟶X, there exists an element x0 in X such that(26)Da=a⋅x0−x0⋅aa∈A,for every Banach A− bimodule X, see [1]. It is easy to see that every contractible Banach algebra is amenable.Corollary 2.
LetSbe a left cancellative semigroup. Then,ℓ1S∗∗is pseudo-contractible if and only ifSis a finite group.Proof.
Letℓ1S∗∗ be a pseudo-contractible Banach algebra. So, we have a net mα in ℓ1S∗∗⊗^ℓ1S∗∗ such that a⋅mα=mα⋅a and πℓ1S∗∗mαa⟶a for all a∈ℓ1S∗∗. It is known that there is a bounded linear map χ:ℓ1S∗∗⊗^ℓ1S∗∗⟶ℓ1S⊗^ℓ1S∗∗ such that for each a,b∈ℓ1S and m∈ℓ1S∗∗⊗^ℓ1S∗∗, we have(i)
χa⊗b=a⊗b,(ii)
χm⋅a=χm⋅a, a⋅χm=χa⋅m,(iii)
πℓ1S∗∗χm=πℓ1S∗∗m,
[[14], Lemma 1.]. Therefore, the net ψmα gives that(27)a⋅χmα=χa⋅mα=χmα⋅a=χmα⋅a,πℓ1S∗∗χmαa=πℓ1S∗∗mαa=πℓ1S∗∗mαa⟶a,for all a∈ℓ1S. Then, we have πℓ1S∗∗χmα satisfying to be an ultra central approximate identity for ℓ1S. Following a similar method as in [15], Theorem 1, and Corollary 2, we can see that S is a regular semigroup. Thus, the previous theorem gives that S must be a group, say G. Then, ℓ1G∗∗ is pseudo-contractible. On the other hand, ℓ1G is an algebra with identity, so ℓ1G∗∗ possess an identity. Therefore [[2], Theorem 2.4] follows that ℓ1G∗∗ is contractible. Then, ℓ1G∗∗ is amenable. Now applying [[14], Theorem 1.3], G is a finite group.
Converse is clear.



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