# Ternary Weighted Function and Beurling Ternary Banach Algebral1ω(S)

Authors: Mehdi Dehghanian; Mohammad Sadegh Modarres Mosadegh
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

LetS be a ternary semigroup. In this paper, we introduce our notation and prove some elementary properties of a ternary weight function ω on S. Also, we make ternary weighted algebra l1ω(S) and show that l1ω(S) is a ternary Banach algebra.


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## Body

## 1. Introduction

The notion of an n-ary group was introduced by Dörnte [1] (inspired by E. Nöther) and is a natural generalization of the notion of a group and a ternary group considered by Certaine [2] and Kasner [3].In the first section, which have preliminary character, we review some basic definitions and properties related to ternary groups and semigroups (cf. also Belousov [4] and Rusakov [5]).Dudek [6], Feĭzullaev [7], Kim and Fred [8], and Lyapin [9] have also studied the properties of the ternary semigroups.The present paper may be described as an introduction to harmonic analysis on ternary semigroups. In Section2, we introduce our notation and prove some elementary properties of a ternary weight function. In Section 3, we make ternary weighted algebra l1ω(S) and show that l1ω(S) is a ternary Banach algebra.Definition 1.1.
A nonempty setG with one ternary operation []:G×G×G→G is called a ternary groupoid and denoted by (G,[]).
We say that(G,[]) is a ternary semigroup if the operation [] is associative, that is, if
(1.1)[[xyz]uv]=[x[yzu]v]=[xy[zuv]]
hold for all x,y,z,u,v∈G.Definition 1.2.
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)[xab]=[ayb]=[abz]=c.One can prove (post [10]) that elements x,y, and z are uniquely determined. Moreover, according to the suggestion of post [10], one can prove (cf. Dudek et al. [11]) that, in the above definition, under the assumption of the associativity, it suffices only to postulate the existence of a solution of [ayb]=c, or equivalently, of [xab]=[abz]=c.In a ternary group, the equation[xxz]=x has a unique solution which is denoted by z=x¯ and called the skew element to x (cf. Dörnte, [1]). As a consequence of results obtained in Dörnte [1], we have the following theorem.Theorem 1.3.
In any ternary group(G,[]) for all x,y,z∈G, the following identities take place:
(1.3)[xxx¯]=[xx¯x]=[x¯xx]=x,[yxx¯]=[yx¯x]=[xx¯y]=[x¯xy]=y,[xyz]¯=[z¯y¯x¯],x¯¯=x.Other properties of skew elements are described in Dudek [12] and I. Dudek and W. A. Dudek [13].Definition 1.4 (see [14]).
Let(G,[]) be a ternary group, -1 it’s inverse operation, and G be equipped with a topology O. Then, we say that (G,[],O) is a topological ternary group if and only if(i)
ternary operation[] is continuous in O, and(ii)
the 2-operation-1 is continuous in O.LetG be a ternary group and A any subset of G. We denote by A¯ denote set of all x¯ (skew element) such that x∈A, that is, (1.4)A¯={x¯:x∈A}.Definition 1.5.
A ternary Banach algebra is a complex Banach spaceA, equipped with a ternary product (x,y,z)→[xyz] of A3 into A, which is associative in the sense that [[xyz]uv]=[xy[zuv]]=[x[yzu]v], and satisfy ∥[xyz]∥≤∥x∥∥y∥∥z∥.LetA be a ternary Banach algebra and A1,A2, and A3 subsets of A. We define (1.5)[A1A2A3]={[a1a2a3]:a1∈A,a2∈A2,a3∈A3}.LetA and B be a ternary Banach algebra. A linear mapping ϕ:A→B is called to be a ternary homomorphism if ϕ[xyz]=[ϕ(x)ϕ(y)ϕ(z)].Definition 1.6.
LetS be a ternary semigroup, and let l1(S) denote the set of mappings f of S into ℂ such that
(1.6)∑s∈S|f(s)|<∞
with pointwise addition and scalar multiplication and with the norm
(1.7)‖f‖1=∑s∈S|f(s)|.LetS be a ternary semigroup; for f,g,h∈l1(S), we define (1.8)[f,g,h]*(x)=∑x=[rst]f(r)g(s)h(t)(x∈S),
and it is called ternary convolution product on l1(S).Theorem 1.7 (see [15]).
LetS be a ternary semigroup, then ternary convolution product on l1(S) is associative.Theorem 1.8 (see [15]).
LetS be a ternary semigroup, then (l1(S),[,,]*) under the usual norm is a ternary Banach algebra.


## 2. Ternary Weight Function on Ternary Semigroup and Ternary Group

Definition 2.1.
A ternary weight on ternary semigroup(S,[]) is a positive real function ω:S→ℝ+ such that
(2.1)ω([rst])≤ω(r)ω(s)ω(t)∀r,s,t∈S.Remark 2.2.
Ifω1 and ω2 are two ternary weight function, then ω1ω2 is a ternary weight function.Example 2.3.
Let(ℤ,[]) with [xyz]=x+y+z be a ternary group. For α>0, define ωα=(1+|n|)α. Then, ωα is a weight ternary function.Theorem 2.4.
LetK be a compact subset of a topological ternary group G, ω a ternary weight function on G, and the interior of {x:ω(x)<n} is nonempty for some n∈ℕ. Then, there exists a,b∈ℝ such that
(2.2)0<a≤ω(x)≤b
for all x∈K.Proof.
First we establish the existence ofb. To that end, for n∈ℕ, let
(2.3)Un:={x∈G:ω(x)<n}.
Clearly ∪n=1∞Un=G. Choose n∈ℕ such that (Un)∘≠∅(interior Un). Fix g∈(Un)∘, and let V=(Un)∘. Then V is an open neighborhood of the g, and, hence, by compactness of K, there exist y1,y2,…,ym∈K∪K¯ such that
(2.4)K∪K¯⊆[Vg¯y1]∪[Vg¯y2]∪⋯∪[Vg¯ym].
Now, define b>0 by
(2.5)b=nω(g¯)max⁡{ω(yi):1≤i≤m}.
If x∈K∪K¯, then x=[vg¯yi] for some v∈V and i∈{1,2,…,m}, and, hence,
(2.6)ω(x)=ω([vg¯yi])≤ω(v)ω(g¯)ω(yi)≤nω(g¯)ω(yi)≤b.
Thus, ω(x)≤b, for all x∈K∪K¯.
Next let(2.7)a=inf⁡{ω(x):x∈K},
and suppose that a=0. Then, there exists a sequence (xn)n in K such that ω(xn)→0. Since
(2.8)1≤ω(xn)ω(x¯n),
we must have ω(x¯n)→∞, which contradicts boundedness of ω on the compact set K¯.Proposition 2.5.
Letω be a ternary weight function on a ternary semigroup S such that {x∈S:ω(x)<ϵ} is finite for some ϵ>0. Then ω(x)≥1, for all x∈S.Proof.
Suppose thatω(x)<1 for some x∈S. Choose p∈ℕ such that ω(x3p)<ϵ. Then
(2.9)ω(x3p+3n+1)≤ω(x3p)ω(x3n)ω(x)<ω(x3p)ω(x3n)≤ω(x3p)ω(x)3n<ϵ.
Thus, {x3p+3n:n∈ℕ} is an infinite subset of {y∈S:ω(y)<ϵ} contrary to the hypothesis. Thus, the result follows.Corollary 2.6.
Letω be a ternary weight function on a compact topological ternary group G, and the interior of {x:ω(x)<n} is nonempty for some n∈ℕ. Then, ω(x)≥1 for all x∈G.Proof.
This follows trivially from Theorem2.4 and Proposition 2.5.LetC(S) be the set of all complex-valued continuous functions on S, Cb(S) the space of all bounded functions in C(S) under the supremum norm ∥·∥S and ω a continuous ternary weight function on S. We define the space of ternary weighted continuous functions C(S,ω) by (2.10)C(S,ω)={f∈C(S):fω∈Cb(S)}
with the norm given by (2.11)‖f‖ω=‖fω‖S.Define(2.12)Ω(x,y):=sup⁡{ω(z)ω([xyz]):z∈S},x,yf(z)=f([xyz])
for all x,y,z∈S.Lemma 2.7.
Letω be a continuous ternary weight function on a ternary semigroup S. Then, x,yf∈C(S,ω), for all f∈C(S,ω) and x∈S if and only if Ω(x,y) is finite.Proof.
To establish the necessary condition, we note thatω-1∈C(S,ω) and so if x∈S, we have x,yω-1∈C(S,ω). Hence,
(2.13)∞>‖x,yω-1‖=sup⁡{ω(z)ω([xyz]):z∈S}=Ω(x,y).
Conversely, suppose that Ω(x,y) is finite, for all x∈S, and let
(2.14)|x,yf(z)ω(z)|=|f([xyz])ω(z)|=|f([xyz])ω([xyz])|ω(z)ω([xyz])≤‖f‖ωΩ(x,y)
and so ∥x,yf∥ω<∞. Hence, x,yf∈C(S,ω).Corollary 2.8.
Letω be a continuous ternary weight function on a compact topological ternary group G, and the interior of {x:ω(x)<n} is nonempty for some n∈ℕ. Then, x,yf∈C(S,ω) for all f∈C(S,ω) and x∈G.Proof.
For allx and y in G, we have
(2.15)ω(z)ω([xyz])=ω([y¯x¯[xyz]])ω([xyz])≤ω(y¯)ω(x¯)ω([xyz])ω([xyz])=ω(y¯)ω(x¯),
and so Ω(x,y)≤ω(x¯)ω(y¯). By Lemma 2.7, our result follows.


## 3. Ternary Beurling Algebral1ω(S)

LetS be a ternary semigroup. In [15] introduce ternary Banach algebra l1(S). Now, we make ternary Beurling algebra l1ω(S) and show some elementary properties.Definition 3.1.
LetS be a ternary semigroup, let ω be a ternary weight on S, and let l1ω(S) denote the set of mappings f of S into ℂ such that
(3.1)∑s∈S|f(s)|ω(s)<∞,
with pointwise addition and scalar multiplication, with ternary convolution
(3.2)[f,g,h]*(x)=∑x=[rst]f(r)g(s)h(t)(x∈S),
and with the norm
(3.3)‖f‖1,ω=‖fω‖1=∑s∈S|f(s)|ω(s).Theorem 3.2.
LetS be a ternary semigroup and ω be a ternary weight function on S. Then, (l1ω(S),[,,]*,∥·∥1,ω) is a ternary Banach algebra.Proof.
By Theorem1.7, we need only to check that
(3.4)‖[f,g,h]*‖1,ω≤‖f‖1,ω‖g‖1,ω‖h‖1,ω,
for all f,g,h∈l1ω(S). This justifies the last inequality of the following calculation:
(3.5)‖[f,g,h]*‖1,ω=∑x∈S|[f,g,h]*(x)ω(x)|=∑x∈S|∑x=[rst]f(r)g(s)h(t)ω([rst])|≤∑x∈S∑x=[rst]|f(r)ω(r)‖g(s)ω(s)‖h(t)ω(t)|≤∑r|f(r)ω(r)|∑s|g(s)ω(s)|∑t|h(t)ω(t)|=‖f‖1,ω‖g‖1,ω‖h‖1,ω.The ternary algebral1ω(S) is called the ternary Beurling algebra on S associated with the ternary weight ω.Ifω(s)=1,(s∈S), we obtained l1ω(S)=l1(S).Ifω(s)≥1 for all s∈S, then l1ω(S) is a ternary subalgebra of l1(S), and if ω(s)≤1 for all s∈S, then l1(S) is a ternary subalgebra of l1ω(S).Proposition 3.3.
Letω and ω′ are ternary weights on S and ϕ:l1ω(S)→l1ω′(S) is a continuous nonzero ternary homomorphism. If [l1ω(S),f,g]* is norm dense in l1ω(S). Then, the norm closure of [l1ω′(S),ϕ(f),ϕ(g)]* contains the norm closure of [l1ω′(S),l1ω′(S),ϕ(h)]* for all h in l1ω(S).Proof.
Since[l1ω(S),f,g]* is norm dense in l1ω(S), we can find a sequence {λn} in l1ω(S) for which lim([λn,f,g]*)=h, with the limit taken in the norm topology. By the continuity of ϕ, this implies that
(3.6)lim⁡([ϕ(λn),ϕ(f),ϕ(g)]*)=ϕ(h).
Hence, ϕ(h), and, therefore, the norm closure of [l1ω′(S),l1ω′(S),ϕ(h)]* as well, belongs to the norm closure of [l1ω′(S),ϕ(f),ϕ(g)]*. This completes the proof of the proposition.Proposition 3.4.
Letω and ω′ are ternary weights on S, and ϕ:l1ω(S)→l1ω′(S) is a continuous nonzero ternary homomorphism. If [l1ω(S),l1ω(S),f]* is norm dense in l1ω(S). Then, the norm closure of [l1ω′(S),l1ω′(S),ϕ(f)]* contains the norm closure of [l1ω′(S),l1ω′(S),ϕ(g)]* for all g in l1ω(S).Proof.
Since[l1ω(S),l1ω(S),f]* is norm dense in l1ω(S), we can find sequences {hn} and {λn} in l1ω(S) for which lim([hn,λn,f]*)=g, with the limit taken in the norm topology. By the continuity of ϕ, this implies that
(3.7)lim⁡([ϕ(hn),ϕ(λn),ϕ(f)]*)=ϕ(g).
Hence,ϕ(g), and, therefore, the norm closure of [l1ω′(S),l1ω′(S),ϕ(g)]* as well, belongs to the norm closure of [l1ω′(S),l1ω′(S),ϕ(f)]*. This completes the proof of the proposition.Definition 3.5.
LetG be a ternary group, let ω be a ternary weight on G, and let l1ω(G) denote the set of mappings f of G into ℂ such that
(3.8)∑s∈G|f(s)|ω(s)<∞
with pointwise addition and scalar multiplication, with ternary convolution
(3.9)[f,g,h]*(x)=∑s,t∈Gf([xs¯t¯])g(t)h(s)(x∈G)
and with the norm
(3.10)‖f‖1,ω=‖fω‖1=∑s∈G|f(s)|ω(s).It is clear thatl1ω(G) is a ternary Banach algebra.Theorem 3.6.
LetG be a topological ternary group, ω a ternary weight function on G such that interior {x:ω(x)<n} not empty for some n∈ℕ. Then,(i)
every compactly supported function inl1(G) belongs to l1ω(G) and(ii)
For everyx,y∈G and f∈l1ω(G), y¯,x¯f∈l1ω(G) and ∥y¯,x¯f∥1,ω≤ω(x)ω(y)∥f∥1,ω.Proof.
(i) It is immediately sinceω is bounded on compact subsets of G by Theorem 2.4.
(ii) It follows simply from submultiplicativity ofω(3.11)‖y̅,x̅f‖1,ω=∑z∈G|f([y̅x̅z])|ω(z)=∑z∈G|f([y̅x̅z])|ω([y̅x̅z])ω(z)ω([y̅x̅z])≤ω(x)ω(y)∑z∈G|f([y̅x̅z])|ω([y̅x̅z])=ω(x)ω(y)‖f‖1,ω.LetG be a ternary group and ω be a ternary weight function, define (3.12)lpω(G)={f:fω∈lp(G)}
such that (3.13)‖f‖p,ω=‖fω‖p=(∑x∈G|f(x)|pω(x)p)1/p.
For p≥1, we have l1ω(G)⊂lpω(G) and ∥f∥p,ω≤∥f∥1,ω.Theorem 3.7.
LetG be a discrete uncountable ternary group. Then, there exists set A⊂G such that A=A¯ and lp(A)⊂lpω(G).Proof.
SinceG is uncountable, for some C>0, the set A={x:max(ω(x),ω(x¯))≤C} is also uncountable. Note that A=A¯. Now, if f∈lp(A), then
(3.14)‖f‖p,ωp=∑x∈A|f(x)ω(x)|p≤Cp∑x∈A‖f‖pp
so that lp(A)⊂lpω(G).



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