derivations on C -ternary algebras

R E S E A R C H Open Access

C ^* -ternary -derivations on C ^* -ternary algebras

Mehdi Dehghanian¹, Seyed Mohammad Sadegh Modarres Mosadegh², Choonkil Park³and Dong Yun Shin^4*

*Correspondence: dyshin@uos.ac.kr

4Department of Mathematics, University of Seoul, Seoul, 130-743, Korea

Full list of author information is available at the end of the article

Abstract

In this paper, we prove the Hyers-Ulam stability ofC^*-ternary 3-derivations and of C^*-ternary 3-homomorphisms for the functional equation

f(x₁+x₂,y₁+y₂,z₁+z₂) =

1≤i,j,k≤2

f(x_i,y_j,z_k)

inC^*-ternary algebras.

MSC: Primary 17A40; 39B52; 46Lxx; 46K70; 46L05; 46B99

Keywords: Hyers-Ulam stability; 3-additive mapping;C^*-ternary algebra;C^*-ternary 3-derivation;C^*-ternary 3-homomorphism

1 Introduction and preliminaries

Ternary algebraic operations were considered in the nineteenth century by several math- ematicians such as Cayley [] who introduced the notion of a cubic matrix, which in turn was generalized by Kapranov, Gelfand and Zelevinskii []. The simplest example of such non-trivial ternary operation is given by the following composition rule:

{a,b,c}ijk=

≤l,m,n≤N

a_nilb_ljmc_mkn (i,j,k= , , . . . ,N).

Ternary structures and their generalization, the so-calledn-ary structures, raise certain hopes in view of their applications in physics. Some significant physical applications are as follows (see [, ]).

() The algebra of nonions generated by two matrices

⎛

⎜⎝

  

  

  

⎞

⎟⎠ and

⎛

⎜⎝

  

  ω

ω^  

⎞

⎟⎠ ω=e^πi

was introduced by Sylvester as a ternary analog of Hamiltons quaternions (cf.[]).

() The quark model inspired a particular brand of ternary algebraic systems. The so- called Nambu mechanics is based on such structures (see []).

©2013 Dehghanian et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation (cf.[, , ]).

A C^*-ternary algebra is a complex Banach spaceA, equipped with a ternary product (x,y,z)→[x,y,z] ofA^intoA, which isC-linear in the outer variables, conjugateC-linear in the middle variable, and associative in the sense that [x,y, [z,w,v]] = [x, [w,z,y],v] = [[x,y,z],w,v], and satisfies[x,y,z] ≤ x · y · zand[x,x,x]=x^(see []). Every left HilbertC^*-module is aC^*-ternary algebra via the ternary product [x,y,z] :=x,yz.

If aC^*-ternary algebra (A, [·,·,·]) has an identity,i.e., an elemente∈Asuch thatx= [x,e,e] = [e,e,x] for allx∈A, then it is routine to verify thatA, endowed withx◦y:= [x,e,y]

andx^*:= [e,x,e], is a unitalC^*-algebra. Conversely, if (A,◦) is a unitalC^*-algebra, then [x,y,z] :=x◦y^*◦zmakesAinto aC^*-ternary algebra.

Throughout this paper, assume thatC^*-ternary algebrasAandBare induced by unital C^*-algebras with unitseande, respectively.

A C-linear mappingH :A→Bis called aC^*-ternary homomorphismifH([x,y,z]) = [H(x),H(y),H(z)] for allx,y,z∈A. If, in addition, the mappingH is bijective, then the mappingH:A→Bis called aC^*-ternary algebra isomorphism. AC-linear mappingδ: A→Ais called aC^*-ternary derivationif

δ [x,y,z]

=

δ(x),y,z +

x,δ(y),z +

x,y,δ(z)

for allx,y,z∈A(see []).

In , Ulam [] gave a talk before the Mathematics Club of the University of Wiscon- sin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms:

We are given a group G and a metric group G with metricρ(·,·). Given> , does there exist aδ> such that if f :G→G satisfiesρ(f(xy),f(x)f(y)) <δ for all x,y∈G, then a homomorphism h:G→G exists withρ(f(x),h(x)) <for all x∈G?

In , Hyers [] gave the first partial solution to Ulam’s question for the case of ap- proximate additive mappings under the assumption thatG andG are Banach spaces.

Then, Aoki [] and Bourgin [] considered the stability problem with unbounded Cauchy differences. In , Rassias [] generalized the theorem of Hyers [] by con- sidering the stability problem with unbounded Cauchy differences. In , Gajda [], following the same approach as that by Rassias [], gave an affirmative solution to this question forp> . It was shown by Gajda [] as well as by Rassias and Šemrl [], that one cannot prove a Rassias-type theorem whenp= . Gˇavruta [] obtained the gener- alized result of the Rassias theorem which allows the Cauchy difference to be controlled by a general unbounded function. During the last two decades, a number of articles and research monographs have been published on various generalizations and applications of the Hyers-Ulam stability to a number of functional equations and mappings, for exam- ple, Cauchy-Jensen mappings,k-additive mappings, invariant means, multiplicative map- pings, bounded nth differences, convex functions, generalized orthogonality mappings, Euler-Lagrange functional equations, differential equations, and Navier-Stokes equations.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results, containing ternary homo- morphisms and ternary derivations, concerning this problem (see [–]).

Let X and Y be complex vector spaces. A mappingf :X×X×X→Y is called a

-additive mappingiff is additive for each variable, and a mappingf :X×X×X→Yis called a -C-linear mappingiff isC-linear for each variable.

A -C-linear mappingH:A×A×A→Bis called aC^*-ternary-homomorphismif it satisfies

H [x,y,z], [x,y,z], [x,y,z]

=

H(x,x,x),H(y,y,y),H(z,z,z)

for allx,y,z,x,y,z,x,y,z∈A.

For a given mappingf :A^→B, we define

Dλ,μ,νf(x,x,y,y,z,z)

:=f(λx_+λx_,μy_+μy_,νz_+νz_) –λμν

≤i,j,k≤

f(x_i,y_j,z_k)

for allλ,μ,ν∈S^:={λ∈C:|λ|= }and allx,x,y,y,z,z∈A.

Bae and Park [] proved the Hyers-Ulam stability of -homomorphisms inC^*-ternary algebras for the functional equation

Dλ,μ,νf(x,x,y,y,z,z) = .

Lemma . [] Let X and Y be complex vector spaces, and let f :X×X×X→Y be a-additive mapping such that f(λx,μy,νz) =λμνf(x,y,z)for allλ,μ,ν∈S^ and all x,y,z∈X.Then f is-C-linear.

Theorem .[] Let p,q,r∈(,∞)with p+q+r< andθ∈(,∞),and let f:A^→B be a mapping such that

Dλ,μ,νf(x,x,y,y,z,z)

≤θ·max

x,xp

·max

y,yq

·max

z,zr

()

and

f [x_,y_,z_], [x_,y_,z_], [x_,y_,z_] –

f(x_,x_,x_),f(y_,y_,y_),f(z_,z_,z_)

≤θ 

i=

xi^p· yi^q· zi^r ()

for all λ,μ,ν ∈S^ and all x,x,x,y,y,y,z,z,z ∈A. Then there exists a unique C^*-ternary-homomorphism H:A^→B such that

f(x,y,z) –H(x,y,z)≤ θ

^– ^p+q+rx^p· y^q· z^r ()

for all x,y,z∈A.

2 C^*-ternary3-homomorphisms inC^*-ternary algebras

Theorem . Let p,q,r∈(,∞) with p+q+r<  andθ ∈(,∞), and let f :A^ → B be a mapping satisfying () and (). If there exists an (x,y,z) ∈ A^ such that lim_n→∞^nf(ⁿx, ⁿy, ⁿz) =e,then the mapping f is a C^*-ternary-homomorphism.

Proof By Theorem ., there exists a uniqueC^*-ternary -homomorphismH:A^→B satisfying (). Note that

H(x,y,z) := lim

n→∞



ⁿf ⁿx, ⁿy, ⁿz for allx,y,z∈A. By the assumption, we get that

H(x_,y_,z_) = lim

n→∞



ⁿf ⁿx_, ⁿy_, ⁿz_

=e.

It follows from () that

H(x_,x,x),H(y_,y,y),H(z_,z,z) –

H(x_,x,x),H(y_,y,y),f(z_,z,z)

=H [x,y,z], [x,y,z], [x,y,z] –

H(x,x,x),H(y,y,y),f(z,z,z)

= lim

n→∞



^nf ⁿx, ⁿy,z

,

ⁿx, ⁿy,z

,

ⁿx, ⁿy,z

–

f ⁿx, ⁿx, ⁿx

,f ⁿy, ⁿy, ⁿy

,f(z,z,z)

≤ lim

n→∞

θ^n(p+q)

^n 

i=

x_i^p· y_i^q· z_i^r= 

for allx,x,x,y,y,y,z,z,z∈A. So, H(x,x,x),H(y,y,y),H(z,z,z)

=

H(x,x,x),H(y,y,y),f(z,z,z)

for allx,x,x,y,y,y,z,z,z∈A. Lettingx=y=x,x=y=yandx=y=zin the last equality, we getf(z,z,z) =H(z,z,z) for allz,z,z∈A. Therefore, the map-

pingf is aC^*-ternary -homomorphism.

Theorem . Let pi,qi,ri∈(,∞) (i= , , )such that pi= or qi= or ri= for some

≤i≤andθ,η∈(,∞),and let f:A^→B be a mapping such that Dλ,μ,νf(x,x,y,y,z,z)

≤θ x^p· x^p· y^q· y^q

+y^q· y^q· z^r· z^r+x^p· x^p· z^r· z^r

()

and

f [x,y,z], [x,y,z], [x,y,z] –

f(x,x,x),f(y,y,y),f(z,z,z)

≤ηx^p· x^p· x^p· y^q· y^q· y^q· z^r· z^r· z^r ()

for allλ,μ,ν∈S^and all x,x,x,y,y,y,z,z,z∈A.Then the mapping f :A^→B is a C^*-ternary-homomorphism.

Proof Lettingxi=yj=zk=  (i,j,k= , ) in (), we getf(, , ) = . Puttingλ=μ=ν= , x=  andyj=zk=  (j,k= , ) in (), we havef(x_, , ) =  for allx∈A. Similarly, we getf(,y, ) =f(, ,z) =  for ally,z∈A. Settingλ=μ=ν= ,x= ,y=  andz= z_= , we havef(x_,y_, ) =  for allx_,y_∈A. Similarly, we getf(x_, ,z_) =f(,y_,z_) =  for allx,y,z∈A. Now lettingλ=μ=ν=  andy=z=  in (), we have

f(x+x,y,z) =f(x,y,z) +f(x,y,z)

for allx,x,y,z∈A.

Similarly, one can show that the other equations hold. So,f is -additive.

Lettingx=y=z=  in (), we getf(λx,μy,νz) =λμνf(x,y,z) for allλ,μ,ν∈S^ and allx,y,z∈A. So, by Lemma ., the mappingf is -C-linear.

Without any loss of generality, we may suppose thatp_= .

Letp< . It follows from () that f [x,y,z], [x,y,z], [x,y,z]

–

f(x,x,x),f(y,y,y),f(z,z,z)

= lim

n→∞



ⁿf ⁿx_,y_,z_

, [x_,y_,z_], [x_,y_,z_] –

f ⁿx_,x_,x_

,f(y_,y_,y_),f(z_,z_,z_)

≤η lim

n→∞

^np

ⁿ

× x_^p· x_^p· x_^p· y_^q· y_^q· y_^q· z_^r· z_^r· z_^r

= 

for allx,x,x,y,y,y,z,z,z∈A.

Letp> . It follows from () that f [x,y,z], [x,y,z], [x,y,z]

–

f(x,x,x),f(y,y,y),f(z,z,z)

= lim

n→∞ⁿ f



ⁿx,y,z

, [x,y,z], [x,y,z]

–

f 

ⁿx,x,x

,f(y,y,y),f(z,z,z)

≤η lim

n→∞

ⁿ

^np

× x_^p· x_^p· x_^p· y_^q· y_^q· y_^q· z_^r· z_^r· z_^r

= 

for allx,x,x,y,y,y,z,z,z∈A. Therefore,

f [x,y,z], [x,y,z], [x,y,z]

=

f(x,x,x),f(y,y,y),f(z,z,z)

for allx,x,x,y,y,y,z,z,z∈A. So, the mappingf :A^→Bis aC^*-ternary -homo-

morphism.

Theorem . Letϕ:A^→[,∞)andψ:A^→[,∞)be functions such that

ϕ(x_, . . . ,x_) = 

if xi= for some≤i≤and



ⁿψ x, . . . , ⁿxi, . . . ,x

=  or ⁿψ

x, . . . , 

ⁿxi, . . . ,x

= .

Suppose that f :A^→B is a mapping satisfying Dλ,μ,νf(x,x,y,y,z,z)≤ϕ(x,x,y,y,z,z)

and

f [x_,y_,z_], [x_,y_,z_], [x_,y_,z_] –

f(x_,x_,x_),f(y_,y_,y_),f(z_,z_,z_)

≤ψ(x,x,x,y,y,y,z,z,z)

for all λ,μ,ν ∈ S^ and all x_,x_,x_,y_,y_,y_,z_,z_,z_ ∈ A. Then the mapping f is a C^*-ternary-homomorphism.

Proof The proof is similar to the proof of Theorem ..

Corollary . Let pi,qi,ri∈(,∞) (i= , , )such that pi= or qi= or ri= for some

≤i≤andθ,η∈(,∞),and let f:A^→B be a mapping such that

Dλ,μ,νf(x,x,y,y,z,z)≤θx^p· x^p· y^q· y^q· z^r· z^r

and

f [x_,y_,z_], [x_,y_,z_], [x_,y_,z_] –

f(x_,x_,x_),f(y_,y_,y_),f(z_,z_,z_)

≤ηx^p· x^p· x^p· y^q· y^q· y^q· z^r· z^r· z^r

for allλ,μ,ν∈S^and all x_,x_,x_,y_,y_,y_,z_,z_,z_∈A.Then the mapping f :A^→B is a C^*-ternary-homomorphism.

3 C^*-ternary3-derivations onC^*-ternary algebras

Definition . A -C-linear mappingD:A^→Ais called aC^*-ternary-derivationif it satisfies

D [x,y,z], [x,y,z], [x,y,z]

=

D(x,x,x), y,y^*_,y

, z,z^*_,z

+

x,x^*_,x

,D(y,y,y), z,z^*_,z

+

x,x^*_,x

, y,y^*_,y

,D(z,z,z)

for allx,x,x,y,y,y,z,z,z∈A.

Theorem . Let p,q,r∈(,∞)with p+q+r< andθ∈(,∞),and let f :A^→A be a mapping such that

Dλ,μ,νf(x,x,y,y,z,z)

≤θ·max

x,xp·max

y,yq·max

z,zr

()

and

f [x_,y_,z_], [x_,y_,z_], [x_,y_,z_] –

f(x_,x_,x_),

y_,y^*_,y_ ,

z_,z^*_,z_

–

x,x^*_,x

,f(y,y,y), z,z^*_,z

– x,x^*_,x

, y,y^*_,y

,f(z,z,z)

≤θ 

i=

x_i^p· y_i^q· z_i^r ()

for all λ,μ,ν ∈S^ and all x,x,x,y,y,y,z,z,z ∈A. Then there exists a unique C^*-ternary-derivationδ:A^→A such that

f(x,y,z) –δ(x,y,z)≤ θ

^– ^p+q+rx^p· y^q· z^r ()

for all x,y,z∈A.

Proof By the same method as in the proof of [, Theorem .], we obtain a -C-linear mappingδ:A^→Asatisfying (). The mappingδ(x,y,z) :=lim_j→∞ ^

^jf(^jx, ^jy, ^jz) for all x,y,z∈A.

It follows from () that

δ [x,y,z], [x,y,z], [x,y,z] –

δ(x,x,x), y,y^*_,y

, z,z^*_,z

–

x_,x^*_,x_

,δ(y_,y_,y_),

z_,z^*_,z_

–

x_,x^*_,x_ ,

y_,y^*_,y_

,δ(z_,z_,z_)

= lim

n→∞



^nf ^n[x_,y_,z_], ^n[x_,y_,z_], ^n[x_,y_,z_] –

f ⁿx, ⁿx, ⁿx

,

ⁿy, ⁿy^*_, ⁿy

,

ⁿz, ⁿz^*_, ⁿz

–

ⁿx, ⁿx^*_, ⁿx

,f ⁿy, ⁿy, ⁿy

,

ⁿz, ⁿz^*_, ⁿz

–

ⁿx, ⁿx^*_, ⁿx

,

ⁿy, ⁿy^*_, ⁿy

,f ⁿz, ⁿz, ⁿz

≤ lim

n→∞

θ^n(p+q+r)

^n 

i=

xi^p· yi^q· zi^r= 

for allx,x,x,y,y,y,z,z,z∈A.

Now, letT:A^→Abe another -derivation satisfying (). Then we have δ(x,y,z) –T(x,y,z)= 

ⁿδ ⁿx, ⁿy, ⁿz

–T ⁿx, ⁿy, ⁿz

≤ 

ⁿδ ⁿx, ⁿy, ⁿz

–f ⁿx, ⁿy, ⁿz

+ 

ⁿf ⁿx, ⁿy, ⁿz

–T ⁿx, ⁿy, ⁿz

≤ θ(p+q+r–)n+

^– ^p+q+r x^p· y^q· z^r,

which tends to zero as n→ ∞ for all x,y,z∈A. So, we can conclude thatδ(x,y,z) = T(x,y,z) for allx,y,z∈A. This proves the uniqueness ofδ.

Therefore, the mappingδ:A^→Ais a uniqueC^*-ternary -derivation satisfying ().

Corollary . Let∈(,∞),and let f :A^→A be a mapping satisfying D_λ,μ,νf(x_,x_,y_,y_,z_,z_)≤

and

f [x,y,z], [x,y,z], [x,y,z] –

f(x,x,x), y,y^*_,y

, z,z^*_,z

–

x,x^*_,x

,f(y,y,y), z,z^*_,z

– x,x^*_,x

, y,y^*_,y

,f(z,z,z)≤

for all λ,μ,ν ∈S^ and all x_,x_,x_,y_,y_,y_,z_,z_,z_ ∈A. Then there exists a unique C^*-ternary-derivationδ:A^→A such that

f(x,y,z) –δ(x,y,z)≤

 for all x,y,z∈A.

Competing interests

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

1Department of Mathematics, Sirjan University of Technology, Sirjan, Iran.²Department of Mathematics, University of Yazd, P.O. Box 89195-741, Yazd, Iran.³Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea. ⁴Department of Mathematics, University of Seoul, Seoul, 130-743, Korea.

Acknowledgements

CP was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299) and DYS was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

Received: 31 October 2012 Accepted: 5 March 2013 Published: 25 March 2013

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doi:10.1186/1029-242X-2013-124

Cite this article as:Dehghanian et al.:C^*-ternary3-derivations onC^*-ternary algebras.Journal of Inequalities and Applications20132013:124.