Approximate 𝑛-Lie Homomorphisms and Jordan 𝑛-Lie Homomorphisms on 𝑛-Lie Algebras

Abstract

Using fixed point methods, we establish the stability of 𝑛-Lie homomorphisms and Jordan 𝑛-Lie homomorphisms on 𝑛-Lie algebras associated to the following generalized Jensen functional equation 𝜇𝑓(∑^𝑛_𝑖=1𝑥_𝑖/𝑛)+𝜇∑^𝑛_𝑗=2𝑓(∑^𝑛_{𝑖=1,𝑖≠𝑗}𝑥_𝑖−(𝑛−1)𝑥_𝑗/𝑛)=𝑓(𝜇𝑥₁)(𝑛≥2).

1. Introduction

Let 𝑛 be a natural number greater or equal to 3. The notion of an 𝑛-Lie algebra was introduced by Filippov in 1985 [1]. The Lie product is taken between 𝑛 elements of the algebra instead of two. This new bracket is 𝑛-linear, antisymmetric and satisfies a generalization of the Jacobi identity. For 𝑛=3 this product is a special case of the Nambu bracket, well known in physics, which was introduced by Nambu [2] in 1973, as a generalization of the Poisson bracket in Hamiltonian mechanics.

An 𝑛-Lie algebra is a natural generalization of a Lie algebra. Namely, a vector space 𝑉 together with a multilinear, antisymmetric 𝑛-ary operation []:Λ^𝑛𝑉→𝑉 is called an 𝑛-Lie algebra, 𝑛≥3, if the 𝑛-ary bracket is a derivation with respect to itself, that is,𝑥₁,…,𝑥_𝑛,𝑥_𝑛+1,…,𝑥_2𝑛−1=^𝑛_𝑖=1𝑥₁,…,𝑥_𝑖−1𝑥_𝑖,𝑥_𝑛+1,…,𝑥_2𝑛−1,…,𝑥_𝑛, (1.1) where 𝑥₁,𝑥₂,…,𝑥_2𝑛−1∈𝑉. Equation (1.1) is called the generalized Jacobi identity. The meaning of this identity is similar to that of the usual Jacobi identity for a Lie algebra (which is a 2-Lie algebra).

In [1] and several subsequent papers, [3–5] a structure theory of finite-dimensional 𝑛-Lie algebras over a field 𝔽 of characteristic 0 was developed.

𝑛-ary algebras have been considered in physics in the context of Nambu mechanics [2, 6] and, recently (for 𝑛=3), in the search for the effective action of coincident 𝑀2-branes in 𝑀-theory initiated by the Bagger-Lambert-Gustavsson (BLG) model [7, 8] (further references on the physical applications of 𝑛-ary algebras are given in [9]).

From now on, we only consider 𝑛-Lie algebras over the field of complex numbers. An 𝑛-Lie algebra 𝐴 is a normed 𝑛-Lie algebra if there exists a norm ‖‖ on 𝐴 such that ‖[𝑥₁,𝑥₂,…,𝑥_𝑛]‖≤‖𝑥₁‖‖𝑥₂‖⋯‖𝑥_𝑛‖ for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝐴. A normed 𝑛-Lie algebra 𝐴 is called a Banach 𝑛-Lie algebra, if (𝐴,‖‖) is a Banach space.

Let (𝐴,[]_𝐴) and (𝐵,[]_𝐵) be two Banach 𝑛-Lie algebras. A ℂ-linear mapping 𝐻∶(𝐴,[]_𝐴)→(𝐵,[]_𝐵) is called an 𝑛-Lie homomorphism if𝐻𝑥₁𝑥₂⋯𝑥_𝑛𝐴=𝐻𝑥₁𝐻𝑥₂⋯𝐻𝑥_𝑛𝐵 (1.2) for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝐴. A ℂ-linear mapping 𝐻∶(𝐴,[]_𝐴)→(𝐵,[]_𝐵) is called a Jordan 𝑛-Lie homomorphism if𝐻[𝑥𝑥⋯𝑥]_𝐴=[𝐻(𝑥)𝐻(𝑥)⋯𝐻(𝑥)]_𝐵 (1.3) for all 𝑥∈𝐴.

The study of stability problems had been formulated by Ulam [10] during a talk in 1940. Under what condition does there exist a homomorphism near an approximate homomorphism? In the following year, Hyers [11] answered affirmatively the question of Ulam for Banach spaces, which states that if 𝜀>0 and 𝑓∶𝑋→𝑌 is a map with 𝑋 a normed space, 𝑌 a Banach spaces such that𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜀 (1.4) for all 𝑥,𝑦∈𝑋, then there exists a unique additive map 𝑇∶𝑋→𝑌 such that𝑓(𝑥)−𝑇(𝑥)≤𝜀 (1.5) for all 𝑥∈𝑋. A generalized version of the theorem of Hyers for approximately linear mappings was presented by Rassias [12] in 1978 by considering the case when inequality (1.4) is unbounded. Due to that fact, the additive functional equation 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦) is said to have the generalized Hyers-Ulam-Rassias stability property. A large list of references concerning the stability of functional equations can be found in [13–32].

In 1982–1994, Rassias (see [26–28]) solved the Ulam problem for different mappings and for many Euler-Lagrange type quadratic mappings, by involving a product of different powers of norms. In addition, Rassias considered the mixed product sum of powers of norms control function. For more details see [33–57].

In 2003 Cădariu and Radu applied the fixed-point method to the investigation of the Jensen functional equation [58]. They could present a short and a simple proof (different of the “direct method”, initiated by Hyers in 1941) for the generalized Hyers-Ulam stability of Jensen functional equation [58] and for quadratic functional equation.

Park and Rassias [59] proved the stability of homomorphisms in 𝐶^∗-algebras and Lie 𝐶^∗-algebras and also of derivations on 𝐶^∗-algebras and Lie 𝐶^∗-algebras for the Jensen-type functional equation𝜇𝑓𝑥+𝑦2+𝜇𝑓𝑥−𝑦2−𝑓𝜇𝑥=0 (1.6) for all 𝜇∈𝕋¹∶={𝜆∈ℂ;|𝜆|=1}.

In this paper, by using the fixed-point methods, we establish the stability of 𝑛-Lie homomorphisms and Jordan 𝑛-Lie homomorphisms on 𝑛-Lie Banach algebras associated to the following generalized Jensen type functional equation:∑𝜇𝑓^𝑛𝑖=1𝑥_𝑖𝑛+𝜇^𝑛_𝑗=2𝑓∑^𝑛_{𝑖=1,𝑖≠𝑗}𝑥_𝑖−(𝑛−1)𝑥_𝑗𝑛−𝑓𝜇𝑥₁=0 (1.7) for all 𝜇∈(𝕋¹_1/𝑛𝑜∶={𝑒^𝑖𝜃;0≤𝜃≤2𝜋/𝑛_𝑜}∪{1}), where 𝑛≥2.

Throughout this paper, assume that (𝐴,[]_𝐴),(𝐵,[]_𝐵) are two 𝑛-Lie Banach algebras.

2. Main Results

Before proceeding to the main results, we recall a fundamental result in fixed point theory.

Theorem 2.1 (see [60]). Let (Ω,𝑑) be a complete generalized metric space, and let 𝑇∶Ω→Ω be a strictly contractive function with Lipschitz constant 𝐿. Then for each given 𝑥∈Ω, either 𝑑𝑇^𝑚𝑥,𝑇^𝑚+1𝑥=∞∀𝑚≥0, (2.1) or other exists a natural number 𝑚₀ such that(i)𝑑(𝑇^𝑚𝑥,𝑇^𝑚+1𝑥)<∞ for all 𝑚≥𝑚₀;(ii)the sequence {𝑇^𝑚𝑥} is convergent to a fixed point 𝑦^∗ of 𝑇;(iii)𝑦^∗ is the unique fixed point of 𝑇 in Λ={𝑦∈Ω∶𝑑(𝑇^𝑚0𝑥,𝑦)<∞};(iv)𝑑(𝑦,𝑦^∗)≤(1/(1−𝐿))𝑑(𝑦,𝑇𝑦) for all 𝑦∈Λ.

We start our work with the main theorem of the our paper.

Theorem 2.2. Let 𝑛₀∈ℕ be a fixed positive integer number. Let 𝑓∶𝐴→𝐵 be a function for which there exists a function 𝜙∶𝐴^𝑛→[0,∞) such that ∑𝜇𝑓^𝑛𝑖=1𝑥_𝑖𝑛+𝜇^𝑛_𝑗=2𝑓∑^𝑛_{𝑖=1,𝑖≠𝑗}𝑥_𝑖−(𝑛−1)𝑥_𝑗𝑛−𝑓(𝜇𝑥₁)_𝐵≤𝜙𝑥₁,𝑥₂,…,𝑥_𝑛 (2.2) for all 𝜇∈(T¹_1/𝑛𝑜∶={𝑒^𝑖𝜃;0≤𝜃≤2𝜋/𝑛_𝑜}∪{1}) and all 𝑥₁,…,𝑥_𝑛∈𝐴, and that 𝑓𝑥₁𝑥₂⋯𝑥_𝑛𝐴−𝑓𝑥₁𝑓𝑥₂⋯𝑓𝑥_𝑛𝐵𝐵≤𝜙𝑥₁,𝑥₂,…,𝑥_𝑛 (2.3) for all 𝑥₁,…,𝑥_𝑛∈𝐴. If there exists an 𝐿<1 such that 𝜙𝑥₁,𝑥₂,…,𝑥_𝑛𝑥≤𝑛𝐿𝜙¹𝑛,𝑥₂𝑛𝑥,…,^𝑛𝑛 (2.4) for all 𝑥₁,…,𝑥_𝑛∈𝐴, then there exists a unique 𝑛-Lie homomorphism 𝐻∶𝐴→𝐵 such that 𝑓(𝑥)(𝑥≤𝐿−𝐻)𝜙()1−𝐿𝑥,0,0,…,0 (2.5) for all 𝑥∈𝐴.

Proof. Let Ω be the set of all functions from 𝐴 into 𝐵 and let 𝑑𝑔,ℎ∶=inf𝐶∈ℝ⁺∶𝑔(𝑥)−ℎ(𝑥)_𝐵≤𝐶𝜙(𝑥,0,…,0),∀𝑥∈𝐴. (2.6) It is easy to show that (Ω,𝑑) is a generalized complete metric space [61].
Now we define the mapping 𝐽∶Ω→Ω by 𝐽(ℎ)(𝑥)=(1/𝑛)ℎ(𝑛𝑥) for all 𝑥∈𝐴.
Note that for all 𝑔,ℎ∈Ω, 𝑑𝑔(𝑥)(𝑥()⟹1𝑔,ℎ<𝐶⟹−ℎ)≤𝐶𝜙𝑥,0,…,0,∀𝑥∈𝐴,𝑛𝑔()−1𝑛𝑥𝑛ℎ(≤1𝑛𝑥)|𝑛|^ℓ()⟹1𝐶𝜙𝑛𝑥,0,…,0,∀𝑥∈𝐴,𝑛𝑔()−1𝑛𝑥𝑛ℎ(()𝐽𝑔(ℎ)𝑛𝑥)≤𝐿𝐶𝜙𝑥,0,…,0,∀𝑥∈𝐴,⟹𝑑,𝐽≤𝐿𝐶. (2.7) Hence we see that 𝑑𝐽𝑔,𝐽(ℎ)≤𝐿𝑑𝑔,ℎ (2.8) for all 𝑔,ℎ∈Ω. It follows from (2.4) that lim_𝑚→∞1𝑛^𝑚𝜙𝑛^𝑚𝑥₁,𝑛^𝑚𝑥₂,…,𝑛^𝑚𝑥_𝑛=0 (2.9) for all 𝑥₁,…,𝑥_𝑛∈𝐴. Putting 𝜇=1,𝑥₁=𝑥, and 𝑥_𝑗=0 (𝑗=2,…,𝑛) in (2.2), we obtain 𝑥𝑛𝑓𝑛−𝑓(𝑥)_𝐵≤𝜙(𝑥,0,…,0) (2.10) for all 𝑥∈𝐴. Thus by using (2.4), we obtain that 1𝑛𝑓(𝑛𝑥)−𝑓(𝑥)_𝐵≤1𝑛𝜙(𝑛𝑥,0,…,0)≤𝐿𝜙(𝑥,0,…,0) (2.11) for all 𝑥∈𝐴, that is, 𝑑𝑓,𝐽𝑓≤𝐿<∞. (2.12) By Theorem 2.1, 𝐽 has a unique fixed point in the set 𝑋₁∶={ℎ∈Ω∶𝑑(𝑓,ℎ)<∞}. Let 𝐻 be the fixed point of 𝐽. 𝐻 is the unique mapping with 𝐻(𝑛𝑥)=𝑛𝐻(𝑥) (2.13) for all 𝑥∈𝐴, such that there exists 𝐶∈(0,∞) satisfying 𝑓(𝑥)−𝐻(𝑥)_𝐵≤𝐶𝜙(𝑥,0,…,0) (2.14) for all 𝑥∈𝐴. On the other hand we have lim_𝑚→∞𝑑(𝐽^𝑚(𝑓),𝐻)=0, so lim_𝑚→∞1𝑛^𝑚𝑓𝑛^𝑚𝑥=𝐻(𝑥) (2.15) for all 𝑥∈𝐴. Also by Theorem 2.1, we have 𝑑≤1𝑓,𝐻𝑑𝑓.1−𝐿𝑓,𝐽 (2.16) It follows from (2.12) and (2.16) that 𝑑≤𝐿𝑓,𝐻.1−𝐿 (2.17) This implies the inequality (2.5). By (2.21), we have 𝐻(𝑥₁𝑥₂⋯𝑥_𝑛𝐴)−[𝐻(𝑥₁)𝐻(𝑥₂)𝐻(𝑥₃)⋯𝐻(𝑥_𝑛)]_𝐵𝐵=lim_𝑚→∞1𝑛^𝑛𝑚𝐻𝑛^𝑚𝑥₁𝑛^𝑚𝑥₂⋯𝑛^𝑚𝑥_𝑛𝐴−1𝑛^𝑛𝑚𝐻𝑛^𝑚𝑥₁𝐻𝑛^𝑚𝑥₂𝐻𝑛^𝑚𝑥₃⋯𝐻𝑛^𝑚𝑥_𝑛𝐵≤lim_𝑚→∞1𝑛^𝑛𝑚𝜙𝑛^𝑚𝑥₁,𝑛^𝑚𝑥₂,…,𝑛^𝑚𝑥_𝑛=0 (2.18) for all 𝑥₁,…,𝑥_𝑛∈𝐴. Hence 𝐻𝑥₁𝑥₂⋯𝑥_𝑛𝐴=𝐻𝑥₁𝐻𝑥₂𝐻𝑥₃⋯𝐻𝑥_𝑛𝐵 (2.19) for all 𝑥₁,…,𝑥_𝑛∈𝐴.
On the other hand, it follows from (2.2), (2.9), and (2.15) that 𝐻∑^𝑛_𝑖=1𝑥_𝑖𝑛+^𝑛_𝑗=2𝐻∑^𝑛_{𝑖=1,𝑖≠𝑗}𝑥_𝑖−(𝑛−1)𝑥_𝑗𝑛−𝐻𝑥_1𝐵=lim_𝑚→∞1𝑛^𝑚𝑓𝑛^𝑚−1𝑛_𝑖=1𝑥_𝑖+^𝑛_𝑗=2𝑓𝑛^𝑚−1𝑛_{𝑖=1,𝑖≠𝑗}𝑥_𝑖−(𝑛−1)𝑥_𝑗−𝑓(𝑛^𝑚𝑥₁)_𝐵≤lim_𝑚→∞1𝑛^𝑚𝜙𝑛^𝑚𝑥₁,𝑛^𝑚𝑥₂,…,𝑛^𝑚𝑥_𝑛=0 (2.20) for all 𝑥₁,…,𝑥_𝑛∈𝐴. Then 𝐻∑^𝑛_𝑖=1𝑥_𝑖𝑛+^𝑛_𝑗=2𝐻∑^𝑛_{𝑖=1,𝑖≠𝑗}𝑥_𝑖−(𝑛−1)𝑥_𝑗𝑛=𝐻𝑥₁ (2.21) for all 𝑥₁,…,𝑥_𝑛∈𝐴. Putting 𝑠₁=∑^𝑛_𝑖=1𝑥_𝑖/𝑛 and 𝑠_𝑗=∑^𝑛_{𝑖=1,𝑖≠𝑗}𝑥_𝑖−(𝑛−1)𝑥_𝑗/𝑛 (𝑗=2,3,…,𝑛) in (2.21), we obtain 𝐻^𝑛_𝑗=1𝑠_𝑗=^𝑛_𝑗=1𝐻𝑠_𝑗 (2.22) for all 𝑠₁,…,𝑠_𝑛∈𝐴. Setting 𝑠_𝑗=0(𝑗=3,4,…,𝑛) in (2.22) to get 𝐻𝑠₁+𝑠₂=𝐻𝑠₁+𝐻𝑠₂ (2.23) hence 𝐻 is cauchy additive. Letting 𝑥_𝑖=𝑥 for all 𝑖=1,2,…,𝑛 in (2.2), we obtain 𝜇𝑓(𝑥)−𝑓𝜇𝑥_𝐵≤𝜙(𝑥,𝑥,…,𝑥) (2.24) for all 𝑥∈𝐴. It follows that 𝐻𝜇𝑥−𝜇𝐻(𝑥)=lim_𝑚→∞1𝑛^𝑚𝑓𝜇𝑛^𝑚𝑥−𝜇𝑓𝑛^𝑚𝑥_𝐵≤lim_𝑚→∞1𝑛^𝑚𝜙𝑛^𝑚𝑥,𝑛^𝑚𝑥,…,𝑛^𝑚𝑥=0 (2.25) for all 𝜇∈𝕋¹_1/𝑛𝑜, and all 𝑥∈𝐴. One can show that the mapping 𝐻∶𝐴→𝐵 is ℂ-linear. Hence, 𝐻∶𝐴→𝐵 is an 𝑛-Lie homomorphism satisfying (2.5), as desired.

Corollary 2.3. Let 𝜃 and 𝑝 be nonnegative real numbers such that 𝑝<1. Suppose that a function 𝑓∶𝐴→𝐵 satisfies ∑𝜇𝑓^𝑛𝑖=1𝑥_𝑖𝑛+𝜇^𝑛_𝑗=2𝑓∑^𝑛_{𝑖=1,𝑖≠𝑗}𝑥_𝑖−(𝑛−1)𝑥_𝑗𝑛−𝑓𝜇𝑥_1𝐵≤𝜃^𝑛_𝑖=1𝑥_𝑖^𝑝_𝐴 (2.26) for all 𝜇∈𝕋¹ and all 𝑥₁,…,𝑥_𝑛∈𝐴 and 𝑓𝑥₁𝑥₂⋯𝑥_𝑛𝐴−𝑓𝑥₁𝑓𝑥₂⋯𝑓𝑥_𝑛𝐵𝐵≤𝜃^𝑛_𝑖=1𝑥_𝑖^𝑝_𝐴 (2.27) for all 𝑥₁,…,𝑥_𝑛∈𝐴. Then there exists a unique 𝑛-Lie homomorphism 𝐻∶𝐴→𝐵 such that 𝑓(𝑥)−𝐻(𝑥)_𝐵≤2^𝑝ℓ(2−2^𝑝)𝜃‖𝑥‖^𝑝_𝐴 (2.28) for all 𝑥∈𝐴.

Proof. Put 𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛)∶=𝜃∑^𝑛_𝑖=1(‖𝑥_𝑖‖^𝑝_𝐴) for all 𝑥₁,…,𝑥_𝑛∈𝐴 in Theorem 2.2. Then (2.9) holds for 𝑝<1, and (2.28) holds when 𝐿=2^(𝑝−1).

Theorem 2.4. Let 𝑛₀∈ℕ be a fixed positive integer number. Let 𝑓∶𝐴→𝐵 be a function for which there exists a function 𝜙∶𝐴^𝑛→[0,∞) such that ∑𝜇𝑓^𝑛𝑖=1𝑥_𝑖𝑛+𝜇^𝑛_𝑗=2𝑓∑^𝑛_{𝑖=1,𝑖≠𝑗}𝑥_𝑖−(𝑛−1)𝑥_𝑗𝑛−𝑓𝜇𝑥_1𝐵≤𝜙𝑥₁,𝑥₂,…,𝑥_𝑛 (2.29) for all 𝜇∈(𝕋¹_1/𝑛𝑜∶={𝑒^𝑖𝜃;0≤𝜃≤2𝜋/𝑛_𝑜}∪{1}) and all 𝑥₁,…,𝑥_𝑛∈𝐴, and that 𝑓[𝑥𝑥⋯𝑥]_𝐴−𝑓(𝑥)𝑓(𝑥)⋯𝑓(𝑥)_𝐵𝐵≤𝜙(𝑥,𝑥,…,𝑥) (2.30) for all 𝑥∈𝐴. If there exists an 𝐿<1 such that 𝜙𝑥₁,𝑥₂,…,𝑥_𝑛𝑥≤𝑛𝐿𝜙¹𝑛,𝑥₂𝑛𝑥,…,^𝑛𝑛 (2.31) for all 𝑥₁,…,𝑥_𝑛∈𝐴, then there exists a unique Jordan 𝑛-Lie homomorphism 𝐻∶𝐴→𝐵 such that 𝑓(𝑥)(𝑥≤𝐿−𝐻)𝜙()1−𝐿𝑥,0,0,…,0 (2.32) for all 𝑥∈𝐴.

Proof. By the same reasoning as the proof of Theorem 2.2, we can define the mapping 𝐻(𝑥)=lim_𝑚→∞1𝑛^𝑚ℓ𝑓𝑛^𝑚ℓ𝑥 (2.33) for all 𝑥∈𝐴. Moreover, we can show that 𝐻 is ℂ-linear. The inequality (2.30) follows that 𝐻([𝑥𝑥⋯𝑥]_𝐴)−[𝐻(𝑥)𝐻(𝑥)⋯𝐻(𝑥)]_𝐵𝐵=lim_𝑚→∞1𝑛^𝑛𝑚𝐻𝑛^𝑚𝑥⋯𝑛^𝑚𝑥_𝐴−1𝑛^𝑛𝑚𝐻𝑛^𝑚𝑥𝐻𝑛^𝑚𝑥⋯𝐻𝑛^𝑚𝑥_𝐵𝐵≤lim_𝑚→∞1𝑛^𝑛𝑚𝜙𝑛^𝑚𝑥,𝑛^𝑚𝑥,…,𝑛^𝑚𝑥=0 (2.34) for all 𝑥∈𝐴. So 𝐻[𝑥𝑥⋯𝑥]_𝐴=[𝐻(𝑥)𝐻(𝑥)⋯𝐻(𝑥)]_𝐵 (2.35) for all 𝑥∈𝐴. Hence 𝐻∶𝐴→𝐵 is a Jordan 𝑛-Lie homomorphism satisfying (2.32).

Corollary 2.5. Let 𝜃 and 𝑝 be nonnegative real numbers such that 𝑝<1. Suppose that a function 𝑓∶𝐴→𝐵 satisfies ∑𝜇𝑓^𝑛𝑖=1𝑥_𝑖𝑛+𝜇^𝑛_𝑗=2𝑓∑^𝑛_{𝑖=1,𝑖≠𝑗}𝑥_𝑖−(𝑛−1)𝑥_𝑗𝑛−𝑓(𝜇𝑥₁)_𝐵≤𝜃^𝑛_𝑖=1𝑥_𝑖^𝑝_𝐴 (2.36) for all 𝜇∈𝕋¹ and all 𝑥₁,…,𝑥_𝑛∈𝐴 and 𝑓[𝑥𝑥⋯𝑥]_𝐴−𝑓(𝑥)𝑓(𝑥)⋯𝑓(𝑥)_𝐵𝐵≤𝑛𝜃‖𝑥‖^𝑝_𝐴 (2.37) for all 𝑥∈𝐴. Then there exists a unique Jordan 𝑛-Lie homomorphism 𝐻∶𝐴→𝐵 such that 𝑓(𝑥)−𝐻(𝑥)_𝐵≤2^𝑝ℓ(2−2^𝑝)𝜃‖𝑥‖^𝑝_𝐴 (2.38) for all 𝑥∈𝐴.

Proof. It follows by Theorem 2.4 by putting 𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛)∶=𝜃∑^𝑛_𝑖=1(‖𝑥_𝑖‖^𝑝_𝐴) for all 𝑥₁,…,𝑥_𝑛∈𝐴 and 𝐿=2^(𝑝−1).