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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 206165, 9 pages
http://dx.doi.org/10.1155/2011/206165
Research Article

Ternary Weighted Function and Beurling Ternary Banach Algebra π‘™πœ”πŸ(𝑆)

Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran

Received 26 July 2011; Accepted 29 August 2011

Academic Editor: GabrielΒ Turinici

Copyright Β© 2011 Mehdi Dehghanian and Mohammad Sadegh Modarres Mosadegh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑆 be a ternary semigroup. In this paper, we introduce our notation and prove some elementary properties of a ternary weight function πœ” on 𝑆. Also, we make ternary weighted algebra π‘™πœ”1(𝑆) and show that π‘™πœ”1(𝑆) is a ternary Banach algebra.

1. Introduction

The notion of an n-ary group was introduced by DΓΆrnte [1] (inspired by E. NΓΆther) 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], FeΔ­zullaev [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 Section 2, we introduce our notation and prove some elementary properties of a ternary weight function. In Section 3, we make ternary weighted algebra π‘™πœ”1(𝑆) and show that π‘™πœ”1(𝑆) is a ternary Banach algebra.

Definition 1.1. A nonempty set 𝐺 with one ternary operation []βˆΆπΊΓ—πΊΓ—πΊβ†’πΊ 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 ]]=[π‘₯[]𝑣]=[[[[π‘₯𝑦𝑧𝑒𝑣𝑦𝑧𝑒π‘₯𝑦𝑧𝑒𝑣]](1.1) hold for all π‘₯,𝑦,𝑧,𝑒,π‘£βˆˆπΊ.

Definition 1.2. A ternary semigroup (𝐺,[]) is a ternary group if for all π‘Ž,𝑏,π‘βˆˆπΊ, there are π‘₯,𝑦,π‘§βˆˆπΊ such that []=[]=[]π‘₯π‘Žπ‘π‘Žπ‘¦π‘π‘Žπ‘π‘§=𝑐.(1.2)

One can prove (post [10]) that elements π‘₯,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 [π‘Žπ‘¦π‘]=𝑐, or equivalently, of [π‘₯π‘Žπ‘]=[π‘Žπ‘π‘§]=𝑐.

In a ternary group, the equation [π‘₯π‘₯𝑧]=π‘₯ has a unique solution which is denoted by 𝑧=π‘₯ and called the skew element to π‘₯ (cf. DΓΆrnte, [1]). As a consequence of results obtained in DΓΆrnte [1], we have the following theorem.

Theorem 1.3. In any ternary group (𝐺,[]) for all π‘₯,𝑦,π‘§βˆˆπΊ, the following identities take place: ξ€Ίπ‘₯π‘₯π‘₯ξ€»=ξ€Ίπ‘₯ξ€»=ξ€Ίπ‘₯π‘₯ξ€»ξ€Ίπ‘₯π‘₯π‘₯=π‘₯,𝑦π‘₯π‘₯ξ€»=𝑦=ξ€Ίπ‘₯π‘₯π‘₯ξ€»=ξ€Ίπ‘₯𝑦π‘₯π‘₯𝑦=𝑦,[]=ξ€Ίπ‘₯𝑦𝑧𝑧𝑦π‘₯ξ€»,π‘₯=π‘₯.(1.3)

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, βˆ’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 βˆ’1 is continuous in 𝑂.

Let 𝐺 be a ternary group and 𝐴 any subset of 𝐺. We denote by 𝐴 denote set of all π‘₯ (skew element) such that π‘₯∈𝐴, that is, 𝐴=ξ€Ύ.π‘₯∢π‘₯∈𝐴(1.4)

Definition 1.5. A ternary Banach algebra is a complex Banach space 𝐴, equipped with a ternary product (π‘₯,𝑦,𝑧)β†’[π‘₯𝑦𝑧] of 𝐴3 into 𝐴, which is associative in the sense that [[π‘₯𝑦𝑧]𝑒𝑣]=[π‘₯𝑦[𝑧𝑒𝑣]]=[π‘₯[𝑦𝑧u]𝑣], and satisfy β€–[π‘₯𝑦𝑧]‖≀‖π‘₯‖‖𝑦‖‖𝑧‖.

Let 𝐴 be a ternary Banach algebra and 𝐴1,𝐴2, and 𝐴3 subsets of 𝐴. We define 𝐴1𝐴2𝐴3ξ€»=π‘Žξ€½ξ€Ί1π‘Ž2π‘Ž3ξ€»βˆΆπ‘Ž1∈𝐴,π‘Ž2∈𝐴2,π‘Ž3∈𝐴3ξ€Ύ.(1.5)

Let 𝐴 and 𝐡 be a ternary Banach algebra. A linear mapping πœ™βˆΆπ΄β†’π΅ is called to be a ternary homomorphism if πœ™[π‘₯𝑦𝑧]=[πœ™(π‘₯)πœ™(𝑦)πœ™(𝑧)].

Definition 1.6. Let 𝑆 be a ternary semigroup, and let 𝑙1(𝑆) denote the set of mappings 𝑓 of 𝑆 into β„‚ such that ξ“π‘ βˆˆπ‘†||||𝑓(𝑠)<∞(1.6) with pointwise addition and scalar multiplication and with the norm ‖𝑓‖1=ξ“π‘ βˆˆπ‘†||||.𝑓(𝑠)(1.7)

Let 𝑆 be a ternary semigroup; for 𝑓,𝑔,β„Žβˆˆπ‘™1(𝑆), we define []𝑓,𝑔,β„Žβˆ—(π‘₯)=[]π‘₯=π‘Ÿπ‘ π‘‘π‘“(π‘Ÿ)𝑔(𝑠)β„Ž(𝑑)(π‘₯βˆˆπ‘†),(1.8) and it is called ternary convolution product on 𝑙1(𝑆).

Theorem 1.7 (see [15]). Let 𝑆 be a ternary semigroup, then ternary convolution product on 𝑙1(𝑆) is associative.

Theorem 1.8 (see [15]). Let 𝑆 be a ternary semigroup, then (𝑙1(𝑆),[,,]βˆ—) 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 πœ”βˆΆπ‘†β†’β„+ such that []πœ”(π‘Ÿπ‘ π‘‘)β‰€πœ”(π‘Ÿ)πœ”(𝑠)πœ”(𝑑)βˆ€π‘Ÿ,𝑠,π‘‘βˆˆπ‘†.(2.1)

Remark 2.2. If πœ”1 and πœ”2 are two ternary weight function, then πœ”1πœ”2 is a ternary weight function.

Example 2.3. Let (β„€,[]) with [π‘₯𝑦𝑧]=π‘₯+𝑦+𝑧 be a ternary group. For 𝛼>0, define πœ”π›Ό=(1+|𝑛|)𝛼. Then, πœ”π›Ό is a weight ternary function.

Theorem 2.4. Let 𝐾 be a compact subset of a topological ternary group 𝐺, πœ” a ternary weight function on 𝐺, and the interior of {π‘₯βˆΆπœ”(π‘₯)<𝑛} is nonempty for some π‘›βˆˆβ„•. Then, there exists π‘Ž,π‘βˆˆβ„ such that 0<π‘Žβ‰€πœ”(π‘₯)≀𝑏(2.2) for all π‘₯∈𝐾.

Proof. First we establish the existence of 𝑏. To that end, for π‘›βˆˆβ„•, let π‘ˆπ‘›βˆΆ={π‘₯βˆˆπΊβˆΆπœ”(π‘₯)<𝑛}.(2.3) Clearly βˆͺβˆžπ‘›=1π‘ˆπ‘›=𝐺. Choose π‘›βˆˆβ„• such that (π‘ˆπ‘›)βˆ˜β‰ βˆ…(interior π‘ˆπ‘›). Fix π‘”βˆˆ(π‘ˆπ‘›)∘, and let 𝑉=(π‘ˆπ‘›)∘. Then 𝑉 is an open neighborhood of the 𝑔, and, hence, by compactness of 𝐾, there exist 𝑦1,𝑦2,…,π‘¦π‘šβˆˆπΎβˆͺ𝐾 such that 𝐾βˆͺξ€Ίπ‘‰πΎβŠ†π‘”π‘¦1ξ€»βˆͺ𝑉𝑔𝑦2𝑉βˆͺβ‹―βˆͺπ‘”π‘¦π‘šξ€».(2.4) Now, define 𝑏>0 by 𝑏=π‘›πœ”π‘”ξ€Έξ€½πœ”ξ€·π‘¦max𝑖.∢1β‰€π‘–β‰€π‘š(2.5) If π‘₯∈𝐾βˆͺ𝐾, then π‘₯=[𝑣𝑔𝑦𝑖] for some π‘£βˆˆπ‘‰ and π‘–βˆˆ{1,2,…,π‘š}, and, hence, πœ”π‘£(π‘₯)=πœ”ξ€·ξ€Ίπ‘”π‘¦π‘–ξ€·ξ€»ξ€Έβ‰€πœ”(𝑣)πœ”π‘”ξ€Έπœ”ξ€·π‘¦π‘–ξ€Έξ€·β‰€π‘›πœ”π‘”ξ€Έπœ”ξ€·π‘¦π‘–ξ€Έβ‰€π‘.(2.6) Thus, πœ”(π‘₯)≀𝑏, for all π‘₯∈𝐾βˆͺ𝐾.
Next let π‘Ž=inf{πœ”(π‘₯)∢π‘₯∈𝐾},(2.7) and suppose that π‘Ž=0. Then, there exists a sequence (π‘₯𝑛)𝑛 in 𝐾 such that πœ”(π‘₯𝑛)β†’0. Since ξ€·π‘₯1β‰€πœ”π‘›ξ€Έπœ”ξ€·π‘₯𝑛,(2.8) we must have πœ”(π‘₯𝑛)β†’βˆž, which contradicts boundedness of πœ” on the compact set 𝐾.

Proposition 2.5. Let πœ” be a ternary weight function on a ternary semigroup 𝑆 such that {π‘₯βˆˆπ‘†βˆΆπœ”(π‘₯)<πœ–} is finite for some πœ–>0. Then πœ”(π‘₯)β‰₯1, for all π‘₯βˆˆπ‘†.

Proof. Suppose that πœ”(π‘₯)<1 for some π‘₯βˆˆπ‘†. Choose π‘βˆˆβ„• such that πœ”(π‘₯3𝑝)<πœ–. Then πœ”ξ€·π‘₯3𝑝+3𝑛+1ξ€Έξ€·π‘₯β‰€πœ”3π‘ξ€Έπœ”ξ€·π‘₯3π‘›ξ€Έπœ”ξ€·π‘₯(π‘₯)<πœ”3π‘ξ€Έπœ”ξ€·π‘₯3𝑛π‘₯β‰€πœ”3π‘ξ€Έπœ”(π‘₯)3𝑛<πœ–.(2.9) Thus, {π‘₯3𝑝+3π‘›βˆΆπ‘›βˆˆβ„•} is an infinite subset of {π‘¦βˆˆπ‘†βˆΆπœ”(𝑦)<πœ–} contrary to the hypothesis. Thus, the result follows.

Corollary 2.6. Let πœ” be a ternary weight function on a compact topological ternary group 𝐺, and the interior of {π‘₯βˆΆπœ”(π‘₯)<𝑛} is nonempty for some π‘›βˆˆβ„•. Then, πœ”(π‘₯)β‰₯1 for all π‘₯∈𝐺.

Proof. This follows trivially from Theorem 2.4 and Proposition 2.5.

Let 𝐢(𝑆) be the set of all complex-valued continuous functions on 𝑆, 𝐢𝑏(𝑆) the space of all bounded functions in 𝐢(𝑆) under the supremum norm ‖⋅‖𝑆 and πœ” a continuous ternary weight function on 𝑆. We define the space of ternary weighted continuous functions 𝐢(𝑆,πœ”) by 𝐢(𝑆,πœ”)=π‘“βˆˆπΆ(𝑆)βˆΆπ‘“πœ”βˆˆπΆπ‘ξ€Ύ(𝑆)(2.10) with the norm given by β€–π‘“β€–πœ”=β€–π‘“πœ”β€–π‘†.(2.11)

Defineξ‚»Ξ©(π‘₯,𝑦)∢=supπœ”(𝑧)[])ξ‚Ό,πœ”(π‘₯π‘¦π‘§βˆΆπ‘§βˆˆπ‘†π‘₯,𝑦𝑓[])(𝑧)=𝑓(π‘₯𝑦𝑧(2.12) for all π‘₯,𝑦,π‘§βˆˆπ‘†.

Lemma 2.7. Let πœ” be a continuous ternary weight function on a ternary semigroup 𝑆. Then, π‘₯,π‘¦π‘“βˆˆπΆ(𝑆,πœ”), for all π‘“βˆˆπΆ(𝑆,πœ”) and π‘₯βˆˆπ‘† if and only if Ξ©(π‘₯,𝑦) is finite.

Proof. To establish the necessary condition, we note that πœ”βˆ’1∈𝐢(𝑆,πœ”) and so if π‘₯βˆˆπ‘†, we have π‘₯,π‘¦πœ”βˆ’1∈𝐢(𝑆,πœ”). Hence, β€–β€–βˆž>π‘₯,π‘¦πœ”βˆ’1β€–β€–ξ‚»=supπœ”(𝑧)[])ξ‚Όπœ”(π‘₯π‘¦π‘§βˆΆπ‘§βˆˆπ‘†=Ξ©(π‘₯,𝑦).(2.13) Conversely, suppose that Ξ©(π‘₯,𝑦) is finite, for all π‘₯βˆˆπ‘†, and let ||π‘₯,𝑦||=||[]||=||[][])||𝑓(𝑧)πœ”(𝑧)𝑓(π‘₯𝑦𝑧)πœ”(𝑧)𝑓(π‘₯𝑦𝑧)πœ”(π‘₯π‘¦π‘§πœ”(𝑧)[])πœ”(π‘₯π‘¦π‘§β‰€β€–π‘“β€–πœ”Ξ©(π‘₯,𝑦)(2.14) and so β€–π‘₯,π‘¦π‘“β€–πœ”<∞. Hence, π‘₯,π‘¦π‘“βˆˆπΆ(𝑆,πœ”).

Corollary 2.8. Let πœ” be a continuous ternary weight function on a compact topological ternary group 𝐺, and the interior of {π‘₯βˆΆπœ”(π‘₯)<𝑛} is nonempty for some π‘›βˆˆβ„•. Then, π‘₯,π‘¦π‘“βˆˆπΆ(𝑆,πœ”) for all π‘“βˆˆπΆ(𝑆,πœ”) and π‘₯∈𝐺.

Proof. For all π‘₯ and 𝑦 in 𝐺, we have πœ”(𝑧)[])=πœ”πœ”(π‘₯𝑦𝑧𝑦π‘₯[]π‘₯𝑦𝑧[])β‰€πœ”ξ€·πœ”(π‘₯π‘¦π‘§π‘¦ξ€Έπœ”ξ€·π‘₯ξ€Έπœ”([])π‘₯𝑦𝑧[])ξ€·πœ”(π‘₯𝑦𝑧=πœ”π‘¦ξ€Έπœ”ξ€·π‘₯ξ€Έ,(2.15) and so Ξ©(π‘₯,𝑦)β‰€πœ”(π‘₯)πœ”(𝑦). By Lemma 2.7, our result follows.

3. Ternary Beurling Algebra π‘™πœ”1(𝑆)

Let 𝑆 be a ternary semigroup. In [15] introduce ternary Banach algebra 𝑙1(𝑆). Now, we make ternary Beurling algebra π‘™πœ”1(𝑆) and show some elementary properties.

Definition 3.1. Let 𝑆 be a ternary semigroup, let πœ” be a ternary weight on 𝑆, and let π‘™πœ”1(𝑆) denote the set of mappings 𝑓 of 𝑆 into β„‚ such that ξ“π‘ βˆˆπ‘†||||𝑓(𝑠)πœ”(𝑠)<∞,(3.1) with pointwise addition and scalar multiplication, with ternary convolution []𝑓,𝑔,β„Žβˆ—(π‘₯)=[]π‘₯=π‘Ÿπ‘ π‘‘π‘“(π‘Ÿ)𝑔(𝑠)β„Ž(𝑑)(π‘₯βˆˆπ‘†),(3.2) and with the norm ‖𝑓‖1,πœ”=β€–π‘“πœ”β€–1=ξ“π‘ βˆˆπ‘†||||𝑓(𝑠)πœ”(𝑠).(3.3)

Theorem 3.2. Let 𝑆 be a ternary semigroup and πœ” be a ternary weight function on 𝑆. Then, (π‘™πœ”1(𝑆),[,,]βˆ—,β€–β‹…β€–1,πœ”) is a ternary Banach algebra.

Proof. By Theorem 1.7, we need only to check that β€–β€–[]𝑓,𝑔,β„Žβˆ—β€–β€–1,πœ”β‰€β€–π‘“β€–1,πœ”β€–π‘”β€–1,πœ”β€–β„Žβ€–1,πœ”,(3.4) for all 𝑓,𝑔,β„Žβˆˆπ‘™πœ”1(𝑆). This justifies the last inequality of the following calculation: β€–β€–[𝑓,𝑔,β„Ž]βˆ—β€–β€–1,πœ”=π‘₯βˆˆπ‘†||[]𝑓,𝑔,β„Žβˆ—(||=π‘₯)πœ”(π‘₯)π‘₯βˆˆπ‘†|||||[]π‘₯=π‘Ÿπ‘ π‘‘[])|||||≀𝑓(π‘Ÿ)𝑔(𝑠)β„Ž(𝑑)πœ”(π‘Ÿπ‘ π‘‘π‘₯βˆˆπ‘†ξ“π‘₯=[π‘Ÿπ‘ π‘‘]||||≀𝑓(π‘Ÿ)πœ”(π‘Ÿ)‖𝑔(𝑠)πœ”(𝑠)β€–β„Ž(𝑑)πœ”(𝑑)π‘Ÿ||||𝑓(π‘Ÿ)πœ”(π‘Ÿ)𝑠||||𝑔(𝑠)πœ”(𝑠)𝑑||||β„Ž(𝑑)πœ”(𝑑)=‖𝑓‖1,πœ”β€–π‘”β€–1,πœ”β€–β„Žβ€–1,πœ”.(3.5)

The ternary algebra π‘™πœ”1(𝑆) is called the ternary Beurling algebra on 𝑆 associated with the ternary weight πœ”.

If πœ”(𝑠)=1,(π‘ βˆˆπ‘†), we obtained π‘™πœ”1(𝑆)=𝑙1(𝑆).

If πœ”(𝑠)β‰₯1 for all π‘ βˆˆπ‘†, then π‘™πœ”1(𝑆) is a ternary subalgebra of 𝑙1(𝑆), and if πœ”(𝑠)≀1 for all π‘ βˆˆπ‘†, then 𝑙1(𝑆) is a ternary subalgebra of π‘™πœ”1(𝑆).

Proposition 3.3. Let πœ” and πœ”ξ…ž are ternary weights on 𝑆 and πœ™βˆΆπ‘™πœ”1(𝑆)β†’π‘™πœ”β€²1(𝑆) is a continuous nonzero ternary homomorphism. If [π‘™πœ”1(𝑆),𝑓,𝑔]βˆ— is norm dense in π‘™πœ”1(𝑆). Then, the norm closure of [π‘™πœ”β€²1(𝑆),πœ™(𝑓),πœ™(𝑔)]βˆ— contains the norm closure of [π‘™πœ”β€²1(𝑆),π‘™πœ”β€²1(𝑆),πœ™(β„Ž)]βˆ— for all β„Ž in π‘™πœ”1(𝑆).

Proof. Since [π‘™πœ”1(𝑆),𝑓,𝑔]βˆ— is norm dense in π‘™πœ”1(𝑆), we can find a sequence {πœ†π‘›} in π‘™πœ”1(𝑆) for which lim([πœ†π‘›,𝑓,𝑔]βˆ—)=β„Ž, with the limit taken in the norm topology. By the continuity of πœ™, this implies that πœ™ξ€·πœ†lim𝑛,πœ™(𝑓),πœ™(𝑔)βˆ—ξ€Έ=πœ™(β„Ž).(3.6) Hence, πœ™(β„Ž), and, therefore, the norm closure of [π‘™πœ”β€²1(𝑆),π‘™πœ”β€²1(𝑆),πœ™(β„Ž)]βˆ— as well, belongs to the norm closure of [π‘™πœ”β€²1(𝑆),πœ™(𝑓),πœ™(𝑔)]βˆ—. This completes the proof of the proposition.

Proposition 3.4. Let πœ” and πœ”ξ…ž are ternary weights on 𝑆, and πœ™βˆΆπ‘™πœ”1(𝑆)β†’π‘™πœ”β€²1(𝑆) is a continuous nonzero ternary homomorphism. If [π‘™πœ”1(𝑆),π‘™πœ”1(𝑆),𝑓]βˆ— is norm dense in π‘™πœ”1(𝑆). Then, the norm closure of [π‘™πœ”β€²1(𝑆),π‘™πœ”β€²1(𝑆),πœ™(𝑓)]βˆ— contains the norm closure of [π‘™πœ”β€²1(𝑆),π‘™πœ”β€²1(𝑆),πœ™(𝑔)]βˆ— for all 𝑔 in π‘™πœ”1(𝑆).

Proof. Since [π‘™πœ”1(𝑆),π‘™πœ”1(𝑆),𝑓]βˆ— is norm dense in π‘™πœ”1(𝑆), we can find sequences {β„Žπ‘›} and {πœ†π‘›} in π‘™πœ”1(𝑆) for which lim([β„Žπ‘›,πœ†π‘›,𝑓]βˆ—)=𝑔, with the limit taken in the norm topology. By the continuity of πœ™, this implies that πœ™ξ€·β„Žlimξ€·ξ€Ίπ‘›ξ€Έξ€·πœ†,πœ™π‘›ξ€Έξ€»,πœ™(𝑓)βˆ—ξ€Έ=πœ™(𝑔).(3.7)
Hence, πœ™(𝑔), and, therefore, the norm closure of [π‘™πœ”β€²1(𝑆),π‘™πœ”β€²1(𝑆),πœ™(𝑔)]βˆ— as well, belongs to the norm closure of [π‘™πœ”β€²1(𝑆),π‘™πœ”β€²1(𝑆),πœ™(𝑓)]βˆ—. This completes the proof of the proposition.

Definition 3.5. Let 𝐺 be a ternary group, let πœ” be a ternary weight on 𝐺, and let π‘™πœ”1(𝐺) denote the set of mappings 𝑓 of 𝐺 into β„‚ such that ξ“π‘ βˆˆπΊ||||𝑓(𝑠)πœ”(𝑠)<∞(3.8) with pointwise addition and scalar multiplication, with ternary convolution []𝑓,𝑔,β„Žβˆ—(π‘₯)=𝑠,π‘‘βˆˆπΊπ‘“π‘₯𝑠𝑑𝑔(𝑑)β„Ž(𝑠)(π‘₯∈𝐺)(3.9) and with the norm ‖𝑓‖1,πœ”=β€–π‘“πœ”β€–1=ξ“π‘ βˆˆπΊ||||𝑓(𝑠)πœ”(𝑠).(3.10)

It is clear that π‘™πœ”1(𝐺) is a ternary Banach algebra.

Theorem 3.6. Let 𝐺 be a topological ternary group, πœ” a ternary weight function on 𝐺 such that interior {π‘₯βˆΆπœ”(π‘₯)<𝑛} not empty for some π‘›βˆˆβ„•. Then,(i)every compactly supported function in 𝑙1(𝐺) belongs to π‘™πœ”1(𝐺) and(ii)For every π‘₯,π‘¦βˆˆπΊ and π‘“βˆˆπ‘™πœ”1(𝐺), 𝑦,π‘₯π‘“βˆˆπ‘™πœ”1(𝐺) and ‖𝑦,π‘₯𝑓‖1,πœ”β‰€πœ”(π‘₯)πœ”(𝑦)‖𝑓‖1,πœ”.

Proof. (i) It is immediately since πœ” is bounded on compact subsets of 𝐺 by Theorem 2.4.
(ii) It follows simply from submultiplicativity of πœ”β€–β€–π‘¦,π‘₯𝑓‖‖1,πœ”=ξ“π‘§βˆˆπΊ||𝑓𝑦||=π‘₯π‘§ξ€»ξ€Έπœ”(𝑧)π‘§βˆˆπΊ||𝑓𝑦||πœ”π‘₯𝑧𝑦π‘₯π‘§ξ€»ξ€Έπœ”(𝑧)πœ”ξ€·ξ€Ίπ‘¦ξ“π‘₯π‘§ξ€»ξ€Έβ‰€πœ”(π‘₯)πœ”(𝑦)π‘§βˆˆπΊ||𝑓𝑦||πœ”π‘₯𝑧𝑦π‘₯𝑧=πœ”(π‘₯)πœ”(𝑦)‖𝑓‖1,πœ”.(3.11)

Let 𝐺 be a ternary group and πœ” be a ternary weight function, define π‘™πœ”π‘ξ€½(𝐺)=π‘“βˆΆπ‘“πœ”βˆˆπ‘™π‘ξ€Ύ(𝐺)(3.12) such that ‖𝑓‖𝑝,πœ”=β€–π‘“πœ”β€–π‘=π‘₯∈𝐺||||𝑓(π‘₯)π‘πœ”(π‘₯)𝑝ξƒͺ1/𝑝.(3.13) For 𝑝β‰₯1, we have π‘™πœ”1(𝐺)βŠ‚π‘™πœ”π‘(𝐺) and ‖𝑓‖𝑝,πœ”β‰€β€–π‘“β€–1,πœ”.

Theorem 3.7. Let 𝐺 be a discrete uncountable ternary group. Then, there exists set π΄βŠ‚πΊ such that 𝐴=𝐴 and 𝑙𝑝(𝐴)βŠ‚π‘™πœ”π‘(𝐺).

Proof. Since 𝐺 is uncountable, for some 𝐢>0, the set 𝐴={π‘₯∢max(πœ”(π‘₯),πœ”(π‘₯))≀𝐢} is also uncountable. Note that 𝐴=𝐴. Now, if π‘“βˆˆπ‘™π‘(𝐴), then ‖𝑓‖𝑝𝑝,πœ”=π‘₯∈𝐴||||𝑓(π‘₯)πœ”(π‘₯)𝑝≀𝐢𝑝π‘₯βˆˆπ΄β€–π‘“β€–π‘π‘(3.14) so that 𝑙𝑝(𝐴)βŠ‚π‘™πœ”π‘(𝐺).

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