# Bounded Approximate Identities in Ternary Banach Algebras

Authors: Madjid Eshaghi Gordji; Ali Jabbari; Gwang Hui Kim
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

LetA be a ternary Banach algebra. We prove that if A has a left-bounded approximating set, then A has a left-bounded approximate identity. Moreover, we show that if A has bounded left and right approximate identities, then A has a bounded approximate identity. Hence, we prove Altman’s Theorem and Dixon’s Theorem for ternary Banach algebras.


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## Body

## 1. Introduction

Ternary algebraic operations were considered in the 19th century by several mathematicians such as Cayley [1] who introduced the notion of cubic matrix which in turn was generalized by Kapranov et al. in  [2]. The comments on physical applications of ternary structures can be found in [3–7].A nonempty setG with a ternary operation [·,·,·]:G×G×G→G is called a ternary groupoid and denoted by (G,[·,·,·]). The ternary groupoid (G,[·,·,·]) is called a ternary semigroup if the operation [·,·,·] is associative, that is, if
(1.1)[[x,y,z],u,v]=[x,[y,z,u],v]=[x,y,[z,u,v]]holds for all x,y,z,u,v∈G. A ternary semigroup (G,[·,·,·])) is a ternary group if for all a,b,c∈G, there are x,y,z∈G such that(1.2)[x,a,b]=[a,y,b]=[a,b,z]=c, where  the elements x,y,z are uniquely determined (see [8]).A ternary Banach algebra is a complex Banach spaceA, equipped with a ternary product (x,y,z)↦[x,y,z] of A3 into A, which is associative in the sense that [[x,y,z],u,v]=[x,y,[z,u,v]]=[x,[y,z,u],v] and satisfies ∥[x,y,z]∥≤∥x∥∥y∥∥z∥. An element e∈A is an identity of A if x=[x,e,e]=[e,e,x] for all x∈A.Assume thatA is a ternary Banach algebra a bounded net (eα,fα) is a left-bounded approximate identity for A if limα[eα,fα,a]=a for all a∈A. Similarly, a bounded net (eα,fα) is a right-bounded approximate identity for A if limα[a,eα,fα]=a for all a∈A. Also, (eα,fα) is a middle-bounded approximate identity for A if limα[eα,a,fα]=a for all a∈A. A net (eα,fα) is a bounded approximate identity for A if (eα,fα) is a left-, right-,and middle-bounded approximate identity for A.For ternary Banach algebraA, a set U×V is said to be an approximating set for A (U and V are bounded subsets of A) if for every ϵ>0, and every a∈A, there exist u∈U,v∈V such that ∥[u,v,a]-a∥<ϵ,∥[u,a,v]-a∥<ϵ,∥[a,u,v]-a∥<ϵ.Existence of bounded approximating set for binary Banach algebras guarantees existing of bounded approximate identity (Altman’s Theorem [9, Proposition 2, page 58] or [10]) and also this notion generalized for commutative Fréchet algebras [11]. For normed algebra A with left-bounded approximate identity and right-bounded approximate identity, Dixon [12] proved that A has a bounded approximate identity [13, Proposition 2.9.3].In this paper, we prove Altman’s Theorem and Dixon’s Theorem for ternary Banach algebras. By“∘′′, we mean the quasiproduct between elements x,y of binary algebra A which are defined by x∘y=x+y-xy.


## 2. Main Results

We start our work with the following theorem which can be regarded as a version of Altman’s Theorem for ternary Banach algebras.Theorem 2.1.
LetA be a ternary Banach algebra and U,V be bounded subsets of A such that for given a∈A and ϵ>0 there are u∈U and v∈V, ∥[u,v,a]-a∥<ϵ. Then A possess a left-bounded approximate identity.Proof.
Letϵ>0, and set
(2.1)W=UV∘UV={(u1v1)∘(u2v2):u1,u2∈U,v1v2∈V}.
For proof of theorem, it is sufficient to show that for every finite subsetF⊂A, there exists w=uv∘st∈W such that ∥[uv∘st,a]-a∥<ϵ for every a∈F.
Step 1.
LetF={a} be singleton. Then, there are u∈U and v∈V such that ∥uv∥<M, and
(2.2)‖[u,v,a]-a‖<ϵ(M+1).
Letting w=uv∘uv, then
(2.3)‖[uv∘uv,a]-a‖=‖[u,v,[u,v,a]-a]-([u,v,a]-a)‖<ϵ.Step 2.
LetF={a1,a2}. There is a (u1,v1)∈U×V such that ∥[u1,v1,a1]-a1∥<ϵ/(1+M), and for [u1,v1,a2]-a2∈A there is a (u2,v2)∈U×V such that
(2.4)‖[u2,v2,[u1,v1,a2]-a2]-([u1,v1,a2]-a2)‖<ϵ.
Putw1=u1v1 and w2=u2v2. Then
(2.5)‖[w2∘w1,ai]-ai‖=‖[u2,v2,ai]+[u1,v1,ai]-[u2,v2,[u1,v1,ai]]-ai‖=‖[u2,v2,ai-[u1,v1,ai]]-(ai-[u1,v1,ai])‖<ϵ,
for i=1,2.Step 3.
Now, suppose that obtained results in Steps1 and 2 are true for i=1,2,…,n. Let F={a1,a2,…,an+1}, and let K=max{∥ai∥:i=1,…,n}. There exist w1∘w2∈W such that ∥ai-[w2∘w1,ai]∥<ϵ/3(M+1)2, for i=1,2,…,n, where w1 and w2 are defined as in Step 2. Also, we can choose α1=θ1η1 and α2=θ2η2 such that α1∘α2∈W,
(2.6)‖[α2∘α1,w2∘w1]-w2∘w1‖=‖[θ2,η2,w2∘w1-[θ1,η1,w2∘w1]]-(w2∘w1-[θ1,η1,w2∘w1])‖<ϵ3K,
and ∥[α2∘α1,an+1]-an+1∥<ϵ. Then for every j=1,2,…,n we have
(2.7)‖[α2∘α1,aj]-aj‖≤‖aj-[w2∘w1,aj]‖+‖[α2∘α1,aj]-[α2∘α1,[w2∘w1,aj]]‖+‖[α2∘α1,[w2∘w1,aj]]-[w2∘w1,aj]‖≤‖aj-[w2∘w1,aj]‖+‖α2∘α1‖‖aj-[w2∘w1,aj]‖+‖[α2∘α1,w2∘w1]-w2∘w1‖‖aj‖<ϵ. Let F(A) be the collection of all finite subsets of A and Λ=ℕ×ℕ×F(A). Then Λ is a direct set with the following partial order:
(2.8)(n1,m1,F1)≤(n2,m2,F2)iffF1⊆F2,n1≤n2,m1≤m2.
Now, we can choose a bounded approximate identity(eλ,fλ)λ∈Λ for A.Now, we prove Dixon’s Theorem for ternary Banach algebras. Hence, we prove that if a ternary Banach algebra has both left- and right-bounded approximate identities, then it has a bounded approximate identity.Theorem 2.2.
LetA be a ternary Banach algebra and (eα,fα) and (eβ,fβ) be bounded left and right approximate identities of A, respectively. Then A has a bounded approximate identity.Proof.
Consider(cα,β,dα,β)=(eαfα∘eβfβ,eβfβ∘eαfα). We claim that (cα,β,dα,β) is a bounded approximate identity for A. Boundedness of mentioned net is clear. Therefore, we have to show that (cα,β,dα,β) is a right, left, and middle approximate identity for A.
Step 1.
(cα,β,dα,β) is a left approximate identity. Because
(2.9)‖[eαfα∘eβfβ,eβfβ∘eαfα,a]-a‖=‖[eαfα,eβfβ∘eαfα,a]+[eβfβ,eβfβ∘eαfα,a]-[eαfαeβfβ,eβfβ∘eαfα,a]-a‖≤‖[eαfα,eαfα,a]-a‖+‖[eαfα,eβfβ,a]-[eαfαeβfβ,eαfα,a]‖+‖[eαfαeβfβ,eαfα,a]-[eβfβ,eαfα,a]‖+‖[eαfαeβfβ,eβfβ,a]-[eβfβ,eβfβ,a]‖+‖[eαfαeβfβ,eβfβeαfα,a]-[eβfβeβfβ,eαfα,a]‖≤‖[eα,fα,[eα,fα,a]]-[eα,fα,a]‖+‖[eα,fα,a]-a‖+‖eαfα‖‖eβfβ‖‖a-[eα,fα,a]‖+‖[eα,fα,[eβfβ,eαfα,a]]-[eβfβ,eαfα,a]‖+‖[eα,fα,[eβfβ,eβfβ,a]]-[eβfβ,eβfβ,a]‖+‖[eα,fα,[eβfβ,eβfβeαfα,a]]-[eβfβeβfβ,eαfα,a]‖≤5ϵMN+1+MNϵMN+1<ϵ,
where ∥eαfα∥≤∥eα∥∥fα∥≤M, and ∥eβfβ∥≤∥eβ∥∥fβ∥≤N.Step 2.
(cα,β,dα,β) is a right approximate identity because
(2.10)‖[a,eαfα∘eβfβ,eβfβ∘eαfα]-a‖=‖[a,eαfα,eβfβ∘eαfα]+[a,eβfβ,eβfβ∘eαfα]-[a,eαfαeβfβ,eβfβ∘eαfα]-a‖≤‖[a,eβfβ,eβfβ]-a‖+‖[a,eαfα,eβfβ]-[a,eαfαeβfβ,eβfβ]‖+‖[a,eαfαeβfβ,eαfα]-[a,eαfα,eαfα]‖+‖[a,eβfβ,eβfβeαfα]-[a,eβfβ,eαfα]‖+‖[a,eαfαeβfβ,eβfβeαfα]-[a,eαfα,eβfβeαfα]‖≤‖[[a,eβ,fβ],eβ,fβ]-[a,eβ,fβ]‖+‖[a,eβ,fβ]-a‖+‖[a,eα,fα]-[[a,eα,fα],eβ,fβ]‖‖eβfβ‖+‖[[a,eα,fα],eβ,fβ]-[a,eα,fα]‖‖eαfα‖+‖[[a,eβ,fβ],eβ,fβ]-[a,eβ,fβ]‖‖eαfα‖+‖[[a,eαfα,eβfβ],eβfβ]-[a,eαfα,eβfβ]‖‖eαfα‖≤2ϵMN+1+3MϵMN+1+NϵMN+1<ϵ.Step 3.
By the similar method, we show that the net(cα,β,dα,β) is a middle approximate identity:
(2.11)‖[eαfα∘eβfβ,a,eβfβ∘eαfα]-a‖=‖[eαfα,a,eβfβ∘eαfα]+[eβfβ,a,eβfβ∘eαfα]-[eαfαeβfβ,a,eβfβ∘eαfα]-a‖≤‖[eαfα,a,eβfβ]-a‖+‖[eαfα,a,eαfα]-[eαfα,a,eβfβeαfα]‖+‖[eαfαeβfβ,a,eβfβ]-[eβfβ,a,eβfβ]‖+‖[eαfαeβfβ,a,eαfα]-[eβfβ,a,eαfα]‖+‖[eαfαeβfβ,a,eβfβeαfα]-[eβfβ,a,eβfβeαfα]‖≤‖[eα,fα,[a,eβ,fβ]]-[a,eβ,fβ]‖+‖[a,eβ,fβ]-a‖+‖[eα,fα,a]-[[eα,fα,a],eβ,fβ]‖‖eαfα‖+‖[eα,fα,[eβfβ,a,eβfβ]]-[eβfβ,a,eβfβ]‖+‖[eα,fα,[eβfβ,a,eαfα]]-[eβfβ,a,eαfα]‖+‖[eα,fα,[eβfβ,a,eβfβeαfα]]-[eβfβ,a,eβfβeαfα]‖≤5ϵMN+1+MϵMN+1<ϵ.
This completes the proof of theorem.



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*Source: 386785-2012-03-05.xml*