Ternary Weighted Function and Beurling Ternary Banach Algebra 𝑙^𝜔𝟏(𝑆)

Abstract

Let 𝑆 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 𝑙^𝜔1(𝑆) and show that 𝑙^𝜔1(𝑆) is a ternary Banach algebra.

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 Section 2, we introduce our notation and prove some elementary properties of a ternary weight function. In Section 3, we make ternary weighted algebra 𝑙^𝜔1(𝑆) and show that 𝑙^𝜔1(𝑆) is a ternary Banach algebra.

Definition 1.1. A nonempty set 𝐺 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 [[𝑥𝑦𝑧]𝑢𝑣]=[𝑥[𝑦𝑧𝑢]𝑣]=[𝑥𝑦[𝑧𝑢𝑣]](1.1) 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 [𝑥𝑎𝑏]=[𝑎𝑦𝑏]=[𝑎𝑏𝑧]=𝑐.(1.2)

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: 𝑦𝑥𝑥𝑥𝑥=𝑥𝑥𝑥=𝑥𝑥𝑥=𝑥,𝑥=𝑦𝑥𝑥=𝑥𝑥𝑦=𝑥𝑥𝑦=𝑦,[𝑥𝑦𝑧]=𝑧𝑦𝑥,𝑥=𝑥.(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, −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.

Let 𝐺 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, 𝐴=𝑥∶𝑥∈𝐴.(1.4)

Definition 1.5. A ternary Banach algebra is a complex Banach space 𝐴, 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‖.

Let 𝐴 be a ternary Banach algebra and 𝐴₁,𝐴₂, and 𝐴₃ subsets of 𝐴. We define 𝐴₁𝐴₂𝐴₃=𝑎₁𝑎₂𝑎₃∶𝑎₁∈𝐴,𝑎₂∈𝐴₂,𝑎₃∈𝐴₃.(1.5)

Let 𝐴 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. Let 𝑆 be a ternary semigroup, and let 𝑙₁(𝑆) denote the set of mappings 𝑓 of 𝑆 into ℂ such that _𝑠∈𝑆||𝑓(𝑠)||<∞(1.6) with pointwise addition and scalar multiplication and with the norm ‖𝑓‖₁=_𝑠∈𝑆||𝑓(𝑠)||.(1.7)

Let 𝑆 be a ternary semigroup; for 𝑓,𝑔,ℎ∈𝑙₁(𝑆), we define [𝑓,𝑔,ℎ]_∗(𝑥)=_{[]𝑥=𝑟𝑠𝑡}𝑓(𝑟)𝑔(𝑠)ℎ(𝑡)(𝑥∈𝑆),(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),[,,]_∗) 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 𝜔([𝑟𝑠𝑡])≤𝜔(𝑟)𝜔(𝑠)𝜔(𝑡)∀𝑟,𝑠,𝑡∈𝑆.(2.1)

Remark 2.2. If 𝜔₁ 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 𝜔_𝛼=(1+|𝑛|)^𝛼. Then, 𝜔_𝛼 is a weight ternary function.

Theorem 2.4. Let 𝐾 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 0<𝑎≤𝜔(𝑥)≤𝑏(2.2) for all 𝑥∈𝐾.

Proof. First we establish the existence of 𝑏. To that end, for 𝑛∈ℕ, let 𝑈_𝑛∶={𝑥∈𝐺∶𝜔(𝑥)<𝑛}.(2.3) Clearly ∪^∞𝑛=1𝑈_𝑛=𝐺. Choose 𝑛∈ℕ such that (𝑈_𝑛)^∘≠∅(interior 𝑈_𝑛). Fix 𝑔∈(𝑈_𝑛)^∘, and let 𝑉=(𝑈_𝑛)^∘. Then 𝑉 is an open neighborhood of the 𝑔, and, hence, by compactness of 𝐾, there exist 𝑦₁,𝑦₂,…,𝑦_𝑚∈𝐾∪𝐾 such that 𝐾∪𝐾⊆𝑉𝑔𝑦₁∪𝑉𝑔𝑦₂∪⋯∪𝑉𝑔𝑦_𝑚.(2.4) Now, define 𝑏>0 by 𝑏=𝑛𝜔𝑔max𝜔𝑦_𝑖∶1≤𝑖≤𝑚.(2.5) If 𝑥∈𝐾∪𝐾, then 𝑥=[𝑣𝑔𝑦_𝑖] for some 𝑣∈𝑉 and 𝑖∈{1,2,…,𝑚}, and, hence, 𝜔(𝑥)=𝜔𝑣𝑔𝑦_𝑖≤𝜔(𝑣)𝜔𝑔𝜔𝑦_𝑖≤𝑛𝜔𝑔𝜔𝑦_𝑖≤𝑏.(2.6) Thus, 𝜔(𝑥)≤𝑏, for all 𝑥∈𝐾∪𝐾.
Next let 𝑎=inf{𝜔(𝑥)∶𝑥∈𝐾},(2.7) and suppose that 𝑎=0. Then, there exists a sequence (𝑥_𝑛)_𝑛 in 𝐾 such that 𝜔(𝑥_𝑛)→0. Since 1≤𝜔𝑥_𝑛𝜔𝑥_𝑛,(2.8) we must have 𝜔(𝑥_𝑛)→∞, which contradicts boundedness of 𝜔 on the compact set 𝐾.

Proposition 2.5. Let 𝜔 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 that 𝜔(𝑥)<1 for some 𝑥∈𝑆. Choose 𝑝∈ℕ such that 𝜔(𝑥^3𝑝)<𝜖. Then 𝜔𝑥^{3𝑝+3𝑛+1}≤𝜔𝑥^3𝑝𝜔𝑥^3𝑛𝜔(𝑥)<𝜔𝑥^3𝑝𝜔𝑥^3𝑛≤𝜔𝑥^3𝑝𝜔(𝑥)^3𝑛<𝜖.(2.9) Thus, {𝑥^3𝑝+3𝑛∶𝑛∈ℕ} is an infinite subset of {𝑦∈𝑆∶𝜔(𝑦)<𝜖} contrary to the hypothesis. Thus, the result follows.

Corollary 2.6. Let 𝜔 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 Theorem 2.4 and Proposition 2.5.

Let 𝐶(𝑆) be the set of all complex-valued continuous functions on 𝑆, 𝐶_𝑏(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 𝐶(𝑆,𝜔)=𝑓∈𝐶(𝑆)∶𝑓𝜔∈𝐶_𝑏(𝑆)(2.10) with the norm given by ‖𝑓‖_𝜔=‖𝑓𝜔‖_𝑆.(2.11)

DefineΩ(𝑥,𝑦)∶=sup𝜔(𝑧)𝜔([𝑥𝑦𝑧])∶𝑧∈𝑆_𝑥,𝑦,𝑓(𝑧)=𝑓([𝑥𝑦𝑧])(2.12) for all 𝑥,𝑦,𝑧∈𝑆.

Lemma 2.7. Let 𝜔 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 that 𝜔⁻¹∈𝐶(𝑆,𝜔) and so if 𝑥∈𝑆, we have _𝑥,𝑦𝜔⁻¹∈𝐶(𝑆,𝜔). Hence, ∞>‖‖_𝑥,𝑦𝜔⁻¹‖‖=sup𝜔(𝑧)𝜔([𝑥𝑦𝑧])∶𝑧∈𝑆=Ω(𝑥,𝑦).(2.13) Conversely, suppose that Ω(𝑥,𝑦) is finite, for all 𝑥∈𝑆, and let ||_𝑥,𝑦𝑓(𝑧)𝜔(𝑧)||=||𝑓([𝑥𝑦𝑧])𝜔(𝑧)||=||𝑓([𝑥𝑦𝑧])𝜔([𝑥𝑦𝑧])||𝜔(𝑧)≤‖𝑓‖_𝜔𝜔([𝑥𝑦𝑧])Ω(𝑥,𝑦)(2.14) and so ‖_𝑥,𝑦𝑓‖_𝜔<∞. Hence, _𝑥,𝑦𝑓∈𝐶(𝑆,𝜔).

Corollary 2.8. Let 𝜔 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 all 𝑥 and 𝑦 in 𝐺, we have 𝜔(𝑧)𝜔([𝑥𝑦𝑧])=𝜔𝑦𝑥[𝑥𝑦𝑧]𝜔([𝑥𝑦𝑧])≤𝜔𝑦𝜔𝑥𝜔([𝑥𝑦𝑧])𝜔([𝑥𝑦𝑧])=𝜔𝑦𝜔𝑥,(2.15) and so Ω(𝑥,𝑦)≤𝜔(𝑥)𝜔(𝑦). By Lemma 2.7, our result follows.

3. Ternary Beurling Algebra 𝑙^𝜔1(𝑆)

Let 𝑆 be a ternary semigroup. In [15] introduce ternary Banach algebra 𝑙₁(𝑆). Now, we make ternary Beurling algebra 𝑙^𝜔1(𝑆) and show some elementary properties.

Definition 3.1. Let 𝑆 be a ternary semigroup, let 𝜔 be a ternary weight on 𝑆, and let 𝑙^𝜔1(𝑆) denote the set of mappings 𝑓 of 𝑆 into ℂ such that _𝑠∈𝑆||𝑓(𝑠)||𝜔(𝑠)<∞,(3.1) with pointwise addition and scalar multiplication, with ternary convolution [𝑓,𝑔,ℎ]_∗(𝑥)=_{[]𝑥=𝑟𝑠𝑡}𝑓(𝑟)𝑔(𝑠)ℎ(𝑡)(𝑥∈𝑆),(3.2) and with the norm ‖𝑓‖_1,𝜔=‖𝑓𝜔‖₁=_𝑠∈𝑆||𝑓(𝑠)||𝜔(𝑠).(3.3)

Theorem 3.2. Let 𝑆 be a ternary semigroup and 𝜔 be a ternary weight function on 𝑆. Then, (𝑙^𝜔1(𝑆),[,,]_∗,‖⋅‖_1,𝜔) is a ternary Banach algebra.

Proof. By Theorem 1.7, we need only to check that ‖‖[𝑓,𝑔,ℎ]_∗‖‖_1,𝜔≤‖𝑓‖_1,𝜔‖𝑔‖_1,𝜔‖ℎ‖_1,𝜔,(3.4) for all 𝑓,𝑔,ℎ∈𝑙^𝜔1(𝑆). This justifies the last inequality of the following calculation: ‖‖[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‖||𝑓_1,𝜔(𝑟)𝜔(𝑟)||_𝑠||𝑔(𝑠)𝜔(𝑠)||_𝑡||ℎ(𝑡)𝜔(𝑡)||‖𝑔‖_1,𝜔‖ℎ‖_1,𝜔.(3.5)

The ternary algebra 𝑙^𝜔1(𝑆) is called the ternary Beurling algebra on 𝑆 associated with the ternary weight 𝜔.

If 𝜔(𝑠)=1,(𝑠∈𝑆), we obtained 𝑙^𝜔1(𝑆)=𝑙₁(𝑆).

If 𝜔(𝑠)≥1 for all 𝑠∈𝑆, then 𝑙^𝜔1(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 𝑙^𝜔1(𝑆).

Proposition 3.3. Let 𝜔 and 𝜔^ are ternary weights on 𝑆 and 𝜙∶𝑙^𝜔1(𝑆)→𝑙^𝜔′₁(𝑆) is a continuous nonzero ternary homomorphism. If [𝑙^𝜔1(𝑆),𝑓,𝑔]_∗ is norm dense in 𝑙^𝜔1(𝑆). Then, the norm closure of [𝑙^𝜔′₁(𝑆),𝜙(𝑓),𝜙(𝑔)]_∗ contains the norm closure of [𝑙^𝜔′₁(𝑆),𝑙^𝜔′₁(𝑆),𝜙(ℎ)]_∗ for all ℎ in 𝑙^𝜔1(𝑆).

Proof. Since [𝑙^𝜔1(𝑆),𝑓,𝑔]_∗ is norm dense in 𝑙^𝜔1(𝑆), we can find a sequence {𝜆_𝑛} in 𝑙^𝜔1(𝑆) for which lim([𝜆_𝑛,𝑓,𝑔]_∗)=ℎ, with the limit taken in the norm topology. By the continuity of 𝜙, this implies that lim𝜙𝜆_𝑛,𝜙(𝑓),𝜙(𝑔)_∗=𝜙(ℎ).(3.6) Hence, 𝜙(ℎ), and, therefore, the norm closure of [𝑙^𝜔′₁(𝑆),𝑙^𝜔′₁(𝑆),𝜙(ℎ)]_∗ as well, belongs to the norm closure of [𝑙^𝜔′₁(𝑆),𝜙(𝑓),𝜙(𝑔)]_∗. This completes the proof of the proposition.

Proposition 3.4. Let 𝜔 and 𝜔^ are ternary weights on 𝑆, and 𝜙∶𝑙^𝜔1(𝑆)→𝑙^𝜔′₁(𝑆) is a continuous nonzero ternary homomorphism. If [𝑙^𝜔1(𝑆),𝑙^𝜔1(𝑆),𝑓]_∗ is norm dense in 𝑙^𝜔1(𝑆). Then, the norm closure of [𝑙^𝜔′₁(𝑆),𝑙^𝜔′₁(𝑆),𝜙(𝑓)]_∗ contains the norm closure of [𝑙^𝜔′₁(𝑆),𝑙^𝜔′₁(𝑆),𝜙(𝑔)]_∗ for all 𝑔 in 𝑙^𝜔1(𝑆).

Proof. Since [𝑙^𝜔1(𝑆),𝑙^𝜔1(𝑆),𝑓]_∗ is norm dense in 𝑙^𝜔1(𝑆), we can find sequences {ℎ_𝑛} and {𝜆_𝑛} in 𝑙^𝜔1(𝑆) for which lim([ℎ_𝑛,𝜆_𝑛,𝑓]_∗)=𝑔, with the limit taken in the norm topology. By the continuity of 𝜙, this implies that lim𝜙ℎ_𝑛,𝜙𝜆_𝑛,𝜙(𝑓)_∗=𝜙(𝑔).(3.7)
Hence, 𝜙(𝑔), and, therefore, the norm closure of [𝑙^𝜔′₁(𝑆),𝑙^𝜔′₁(𝑆),𝜙(𝑔)]_∗ as well, belongs to the norm closure of [𝑙^𝜔′₁(𝑆),𝑙^𝜔′₁(𝑆),𝜙(𝑓)]_∗. This completes the proof of the proposition.

Definition 3.5. Let 𝐺 be a ternary group, let 𝜔 be a ternary weight on 𝐺, and let 𝑙^𝜔1(𝐺) denote the set of mappings 𝑓 of 𝐺 into ℂ such that _𝑠_∈𝐺||𝑓(𝑠)||𝜔(𝑠)<∞(3.8) with pointwise addition and scalar multiplication, with ternary convolution [𝑓,𝑔,ℎ]_∗(𝑥)=_{𝑠,𝑡∈𝐺}𝑓𝑥𝑠𝑡𝑔(𝑡)ℎ(𝑠)(𝑥∈𝐺)(3.9) and with the norm ‖𝑓‖_1,𝜔=‖𝑓𝜔‖₁=_𝑠∈𝐺||𝑓(𝑠)||𝜔(𝑠).(3.10)

It is clear that 𝑙^𝜔1(𝐺) is a ternary Banach algebra.

Theorem 3.6. Let 𝐺 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 in 𝑙₁(𝐺) belongs to 𝑙^𝜔1(𝐺) and(ii)For every 𝑥,𝑦∈𝐺 and 𝑓∈𝑙^𝜔1(𝐺), _𝑦,𝑥𝑓∈𝑙^𝜔1(𝐺) and ‖_𝑦,𝑥𝑓‖_1,𝜔≤𝜔(𝑥)𝜔(𝑦)‖𝑓‖_1,𝜔.

Proof. (i) It is immediately since 𝜔 is bounded on compact subsets of 𝐺 by Theorem 2.4.
(ii) It follows simply from submultiplicativity of 𝜔‖‖_𝑦,𝑥𝑓‖‖_1,𝜔==_𝑧∈𝐺||𝑓𝑦𝑥𝑧||𝜔(𝑧)_𝑧∈𝐺||𝑓𝑦𝑥𝑧||𝜔𝑦𝑥𝑧𝜔(𝑧)≤𝜔(𝑥)𝜔(𝑦)_𝑧∈𝐺𝜔𝑦𝑥𝑧=𝜔(𝑥)𝜔(𝑦)‖𝑓‖||𝑓_1,𝜔𝑦𝑥𝑧||𝜔𝑦𝑥𝑧.(3.11)

Let 𝐺 be a ternary group and 𝜔 be a ternary weight function, define 𝑙^𝜔𝑝(𝐺)=𝑓∶𝑓𝜔∈𝑙_𝑝(𝐺)(3.12) such that ‖𝑓‖_𝑝,𝜔=‖𝑓𝜔‖_𝑝=_𝑥∈𝐺||𝑓(𝑥)||^𝑝𝜔(𝑥)^𝑝^1/𝑝.(3.13) For 𝑝≥1, we have 𝑙^𝜔1(𝐺)⊂𝑙^𝜔𝑝(𝐺) and ‖𝑓‖_𝑝,𝜔≤‖𝑓‖_1,𝜔.

Theorem 3.7. Let 𝐺 be a discrete uncountable ternary group. Then, there exists set 𝐴⊂𝐺 such that 𝐴=𝐴 and 𝑙_𝑝(𝐴)⊂𝑙^𝜔𝑝(𝐺).

Proof. Since 𝐺 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∈𝑙_𝑝(𝐴), then ‖𝑓‖^{𝑝𝑝,𝜔}=_𝑥∈𝐴||𝑓(𝑥)𝜔(𝑥)||^𝑝≤𝐶^𝑝_𝑥_∈𝐴‖𝑓‖^𝑝𝑝(3.14) so that 𝑙_𝑝(𝐴)⊂𝑙^𝜔𝑝(𝐺).