Approximation of derivations and the superstability in random Banach ∗ algebras

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R E S E A R C H

Open Access

Approximation of derivations and the

superstability in random Banach

∗

-algebras

Reza Saadati

1*

_{and Choonkil Park}

2

*_{Correspondence:}

[email protected];[email protected] 1_{Department of Mathematics, Iran}

University of Science and Technology, Tehran, Iran Full list of author information is available at the end of the article

Abstract

We prove that approximations of derivations on random Banach∗-algebras are exactly derivations by using a ﬁxed point method. Furthermore, we show that approximations of quadratic∗-derivations on random Banach∗-algebras are exactly quadratic∗-derivations. We, moreover, prove that approximations of derivations on randomC∗-ternary algebras are exactly derivations by using a ﬁxed point method.

MSC: 46S50; 47H10; 26E60

Keywords: Derivation; Quadratic derivation; Superstability; Fixed point method; Random Banach∗-algebra; RandomC∗-ternary algebra

1 Introduction

Ulam [1] presented an eﬀective lecture at the University of Wisconsin in which he stated a number of essential unsolved problems, in the fall of 1940. The next question concerning the stability of homomorphisms was among those:

Assume thatΩ1is a group and suppose thatΩ2is a metric group with a metric(·,·). Let ξ > 0, is there η> 0 such that if a function ϕ :Ω1 →Ω2 satisﬁes the inequality (ϕ(uv),ϕ(u)ϕ(v)) <ηfor allu,v∈Ω1then there is a homomorphismΦ:Ω1→Ω2with (ϕ(u),Φ(u)) <ξfor allu∈Ω1?

When the answer is established, the functional equation for homomorphisms is stable. The ﬁrst mathematician who presented the result concerning the stability of functional equations was Hyers [2]. He intelligently answered Ulam’s question whenΩ1andΩ2are Banach spaces. Recently, Rassias [3] and others have obtained important results on stabil-ity and applied them to the investigations in the nonlinear sciences.

2 Preliminaries

Assume that+is the family of distribution functions, i.e., the family of all left-continuous functions G: [–∞,∞]→ [0, 1] such that G is increasing on [–∞,∞], G(0) = 0 and

G(+∞) = 1.D+⊆+contains each functionG∈+for which–G(+∞) = 1 and–g(x) is the left limit of the mapgatx, i.e.,–_g₍_x_{) =}_lim

t→x–g(t). In+, we haveH≤Fif and only

ifH(s)≤F(s) for allsinR(partially ordered). Note that the functionεudeﬁned by

εu(s) =

⎧ ⎨ ⎩

0, ifs≤u,

1, ifs>u,

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is an element of+ _and_ε

0 is the maximal element in this space. For more details see

[4–6].

Deﬁnition 2.1([6]) LetI= [0, 1]. A continuous triangular norm (brieﬂy, ct-norm) is a functionTfromItoIwith continuity property such that:

(a) T(θ,ϑ) =T(ϑ,θ)andT(θ,T(ϑ,ι)) =T(T(θ,ϑ),ι)for allθ,ϑ,ι∈I; (b) T(θ, 1) =θfor0≤θ≤1;

(c) T(θ,ϑ)≤T(ι,κ)wheneverθ≤ιandϑ≤κfor eachθ,ϑ,ι,κ∈I.

TP(θ,ϑ) =θ ϑ,TM(θ,ϑ) =min(θ,ϑ) andTL(θ,ϑ) =max(θ +ϑ– 1, 0) (the Lukasiewicz

t-norm) are some examples oft-norms. Also, we deﬁnen_j=1θj=Tn–1(θ1, . . . ,θn).

Deﬁnition 2.2([6]) Suppose thatTis act-norm,Vis a vector space and letμbe a map fromVtoD+_{. In this case, the ordered triple (}_V_,_μ_,_T_{) with the properties}

(RN1) μv(θ) =ε0(θ)for allθ> 0if and only ifv= 0;

(RN2) μαv(θ) =μv(_|α|θ )for allv∈V,α= 0;

(RN3) μu+v(θ+ϑ)≥T(μu(θ),μv(ϑ))for allu,v∈Vand allθ,ϑ≥0,

is said to be arandom normed space(in short, RN-space).

Let (V, · ) be a linear normed space. Then

μv(ϑ) = ϑ

ϑ+v

for allϑ> 0, deﬁnes a random norm, and the ordered triple (V,μ,TM) is an RN-space.

Deﬁnition 2.3 Assume that the following algebraic structure on an RN-space (V,μ,T) holds:

(RN-4) μuv(θ ϑ)≥T(μu(θ),μv(ϑ))for eachu,v∈Vand allθ,ϑ> 0, whereTis a

ct-norm.

Then (V,μ,T,T) is called arandom normed algebra.

Suppose that (V, · ) is a normed algebra. Then (V,μ,TM,TP) is a random normed

algebra, where

μv(ϑ) = ϑ

ϑ+v

for allϑ> 0 if and only if

uv ≤ vu+θu+ϑv (v,u∈V;θ,ϑ> 0).

For more details, see [7–22].

Deﬁnition 2.4 A random Banach ∗-algebra B is a random complex Banach algebra

(B,μ,T,T), together with an involution onBwhich is a mappingg→g∗ fromBinto

Bthat satisﬁes

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(ii) (ag+bh)∗=ag∗+bh∗; (iii) (gh)∗=h∗g∗forg,h∈B.

If, in addition,μg∗g(θ ϑ) =T(μg(θ),μg(ϑ)) forg∈Bandθ,ϑ> 0, thenBis called a

ran-domC∗-algebra.

Assume thatBis a random Banach∗-algebra. AderivationonBis a mappingδfromB toBsuch that:

δ(λg+h) =λδ(g) +δ(h), (2.1)

δ(gh) =δ(g)h+gδ(h) (2.2)

for allg,h∈Band allλ∈C. A derivationδis called a∗-derivation onBifδ(g∗) =δ(g)∗for allg∈B(see [23]).

Recall that

ω(u+v) =ω(u) +ω(v), (2.3)

ω(u+v) +ω(u–v) = 2ω(u) + 2ω(v), (2.4)

respectively, are Cauchy additive and Cauchy quadratic functional equations.

Firstly, Baker, Lawrence and Zorzitto [24] deﬁned the concept of superstability. Let (B,μ,T,T) be an RN algebra. The random norm is multiplicative ifμuv(θ ϑ) =T(μu(θ), μv(ϑ)) for allu,v∈Band allθ,ϑ> 0.

Suppose thatΓ =∅. A function:Γ ×Γ →[0,∞] is ageneralized metric(GM) onΓ

if

(1) (ρ,) = 0if and only ifρ=; (2) (ρ,) =(,ρ)for allρ,∈Γ;

(3) (ρ,)≤(ρ,σ) +(σ,)for allρ,,σ∈Γ.

Theorem 2.1 ([25,26]) Suppose that(Γ,)is a complete GM space and assume that the selfmappingΥ onΓ with Lipschitz constant0 <L< 1is strictly contractive.Then,for

∈Γ,either

Υn,Υn+1=∞

for each0≤n∈Z,or there exists n0∈Nsuch that

(1) (Υn_,_Υn+1_{) <}_∞,_∀n_≥_n 0;

(2) the sequence{Υn}tends toσ∗inΓ; (3) Υ(σ∗) =σ∗;

(4) Υ(σ∗) =σ∗and is unique inE={σ∈Γ|(Υn0_,_σ_{) <}∞} (5) (1 –L)(σ,σ∗)≤(σ,Υ σ)for allσ∈Γ.

3 Approximation of derivations on random Banach∗-algebras

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Theorem 3.1 Letψ1:B×B→D+andψ2:B→D+ be distribution functions.Assume

that f :B→Bis a mapping such that

μf(ξp+q)–ξf(p)–f(q)(t)≥ψ1(p,q,t), (3.1)

μf(pq)–pf(q)–f(p)q(t)≥ψ1(p,q,t), (3.2)

μf(p∗)–f(p)∗(t)≥ψ2(p,t), (3.3)

for allξ∈T,p,q∈Band t> 0.If there exist n∈Nand0 <L< 1such thatψ1(sp,sq,Lst) > ψ1(p,q,t),ψ1(sp,q,Lst) >ψ1(p,q,t),ψ1(p,sq,Lst) >ψ1(p,q,t)andψ2(sp,Lst) >ψ2(p,t)for

all p,q∈Band t> 0.Then f onBis a∗-derivation.

Proof Puttingp=qandξ= 1 in (3.1), we get

μf(2p)–2f(p)(t)≥ψ1(p,p,t) (3.4)

for allp∈Bandt> 0. By induction, we can prove that

μf(np)–nf(p)(t)≥ n–1

j=1

ψ1(jp,p,tj) (3.5)

for allp,q∈B,t> 0 andn≥2 where n–1_j=1 tj=t.

Deﬁne

Ψ(p,t) =

s–1

j=1

ψ1(jp,p,tj)

forp∈B,t> 0 ands≥2 where s–1_j=1tj=t. So

μf(sp)–sf(p)(t)≥Ψ(p,t). (3.6)

PutΓ ={g;g:B→B}. Deﬁne a function:Γ ×Γ →[0,∞] such that

(ϑ,υ) =infν> 0 :μϑ(p)–υ(p)(νt)≥Ψ(p,t),∀p∈B,t> 0

,

whereϑ,υ∈Γ. Miheţ and Radu [28] proved that (Γ,) is a complete GM space. Deﬁne a mappingH:Γ →Γ byH(ϑ)(p) =s–1_υ₍_sp_{). Put}

(ϑ,υ) =ν,

whereϑ,υ∈Γ. Then

μH(ϑ)(p)–H(υ)(p)(t) =μϑ(sp)–υ(sp)(st)≥Ψ

sp, s

αt

≥Ψ

p, t

Lα

.

So, forϑ,υ∈S, we have

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Now, we prove the derivation property ofh. In (3.2), we replacepbysn_p_,_q_by_sn_q_{, divide}

bys2n_{and get}

μf(snpsnq)

s2n –p f(snq)

sn – f(snp)

sn p (t)≥ψ1

snp,snq,s2nt≥ψ1

p,q, t

L2n

. (3.9)

In (3.9), lettingn→ ∞, we get

h(pq) =ph(q) +h(p)q (3.10)

for allp,q∈B. Sohis a∗-derivation onB. Now, in (3.2), replacingpbysnpand dividing bysn_{, we get}

μf(snpq)

sn –pf(q)– f(snp)

sn q (t)≥ψ1

snp,q,snt≥ψ1

p,q, t

Ln

for allp,q∈B,n∈Nandt> 0. Lettingn→ ∞, we get

h(pq) =pf(q) +h(p)q (3.11)

for allp,q∈B. Fixm∈N. From

pfsmq=hsmpq–h(p)smq

=smpf(q) (3.12)

for allp,q∈B, we havepf(q) =pf(s_smmq) for allp,q∈Bandm∈N. Lettingm→ ∞, we getpf(q) =ph(q). Puttingp=e, we geth(q) =f(q) for allq∈B. Hencef is a∗-derivation

onB.

4 Approximation of quadratic∗-derivations on random Banach∗-algebras Deﬁnition 4.1 Assume that a mappingδ:B→Bsatisﬁes

(1) δ(η+κ) +δ(η–κ) – 2δ(η) – 2δ(κ) = 0;

(2) δis quadratic homogeneous, that is,δ(λη) =λ2_δ₍_η₎_;

(3) δ(ηκ) =δ(η)κ2+η2δ(κ); (4) δ(η∗) =δ(η)∗;

for allη,κ∈Bandλ∈C. Then it is called a∗-quadratic derivation onB.

Theorem 4.2 Assume thatψ1:B×B→D+andψ2:B→D+are distribution functions.

Let f :B→Bbe a function such that

μf(p+q)+f(p–q)–2f(p)–2f(q)(t)≥ψ1(p,q,t), (4.1)

μ_f_(pq)–p2_f_(q)–f_(p)q2(t)≥ψ₁(p,q,t), (4.2)

μ_f₍_ξ_p)–_λ2_f(p)(t)≥ψ₂(p,t), (4.3)

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for allξ ∈C,p,q∈Band t> 0.If there exist s∈Nand0 <L< 1such thatψ1(2sp, 2sq,

which means that HonΓ, with Lipschitz constant Lis a strictly contractive mapping. Also, forp∈B, we have

μ(Hf)(p)–f(p)(t) =μ2–2s_f(2s_p)–f_(p)(t) =μ_f(2s₎₂2s_f_(p)

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which implies that(H(f),f)≤1/22s_{. Using Theorem}_2.1_{, we conclude that, in the set}

which implies thathis quadratic homogeneous.

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5 Derivations on randomC∗-ternary algebras

A complex random Banach space (B,μ,T,T), which has a ternary product (f,g,h)−→ [f,g,h] ofB3intoB, is a randomC∗-ternary algebra if (see [29]):

(1) [ξf +v,g,h] =ξ[f,g,h] + [v,g,h]for allξ∈C; (2) [f,ξg+v,h] =ξ[f,g,h] + [f,v,h]for allξ∈C; (3) [f,g,ξh+v] =ξ[f,g,h] + [f,g,v]for allξ∈C; (4) [f,g, [h,k,j]] = [f, [k,h,g],j] = [[f,g,h],k,j]; (5) [f,g,h] ≤ f · g · h;

(6) [f,f,f]=f3_;

forf,g,h,v,k,j∈B.

If (B,μ,T,T) has the unitesatisfyingf = [f,e,e] = [e,e,f] for allf ∈B, then the random

C∗-ternary algebra has unite. If forf ∈B, we have [e,f,e] =f∗, then∗is an involution on theC∗-ternary algebra. AC∗-ternary derivation is a mappingδ:B−→Bsuch that

δ[f,g,h]=δ(f),g,h+f,δ(g),h+f,g,δ(h),

δ(ξf+g) =ξ δ(f) +δ(g)

for allf,g,h∈Bandξ∈C. Recall thatδ([e,f,e]) = [e,δ(f),e] implies thatδis an involution.

Theorem 5.1 Assume thatBis a random C∗-ternary algebra which has the unit e.Suppose thatψ1:B2−→[0,∞)andψ2:B3−→[0,∞)are functions.Let f:B−→Bbe a mapping

such that

μf(ξp+q)–λf(p)–f(q)(t)≥ψ1(p,q,t), (5.1)

μf([p,q,r])–[f(p),q,r]–[p,f(q),r][p,q,f(r)](t)≥ψ2(p,q,r,t), (5.2)

μf([e,q,e])–[e,f(q),e](t)≥ψ2(e,q,e,t) (5.3)

for all λ ∈ C, p,q,r ∈ B and t > 0. Assume there exist s ∈ N and 0 < L< 1 such that ψ1(sip,sjq,s(i+j)L(i+j)t) >ψ1(p,q,t), ψ2(sip,sjq,skr,s(i+j+k)L(i+j+k)t) >ψ2(p,q,r,t)for all

p,q,r∈Band i,j,k= 0, 1.Then onB,f is a∗-derivation.

Proof Put

Ψ(p,t) =

s–1

j=1

ψ1(jp,p,tj)

forp∈Bandt> 0 where s–1_j=1tj=t. Then we have

μf(sp)–sf(p)(t)≥Ψ(p,t). (5.4)

We use similar method presented in the proof of Theorem3.1. LetΓ be the set of all mappingsr:B−→B. Deﬁne a function:Γ ×Γ −→[0,∞] by

(ζ,η) =infν> 0 :μζ(z)–η(z)(νs)≥Ψ(z,s)

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forζ,η∈Γ,z∈Bandt> 0. Miheţ and Radu [28] proved that (Γ,) is a complete GM

ThereforeHonΓ with Lipschitz constantLis a strictly contractive function. From (5.4), we have

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Now, in (5.2), we replaceqbysn_q_,_r_by_sn_r_{and divide by}_s2n_{. Letting}_n_{→ ∞}_{, we get}

The authors declare that they have no competing interests.

Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Iran University of Science and Technology, Tehran, Iran.}2_{Research Institute for Natural} Sciences, Hanyang University, Seoul, Republic of Korea.

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Received: 9 August 2018 Accepted: 9 November 2018 References

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