Superstability of derivations on Banach -algebras

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1 Jang Advances in Difference Equations 2017) 2017:193 DOI /s R E S E A R C H Open Access Superstability of derivations on Banach -algebras Sun Young Jang * * Correspondence: jsym@ulsan.ac.kr Department of Mathematics, University of Ulsan, Ulsan, 44610, Korea Abstract In this paper, we show that approximate derivations on Banach -algebras are exactly derivations and also show that approximate quadratic -derivations on Banach -algebras are exactly quadratic -derivations by the fixed point theorem. MSC: Primary 46S40; 39B52; 47H10; 39B62; 26E50; 47S40 Keywords: derivation; quadratic derivation; superstability; fixed point theorem 1 Introduction Let A be a Banach -algebra. A map δ : A A is called a derivation on A if it satisfies the following property: δλa + b)= λδa)+ δb), 1.1) δab)= δa)b + aδb) 1.2) for all a, b A and λ C. Equation1.2) is called the derivation property. If δ satisfies the additional condition δa )=δa) for all a A, thenδ is called a -derivation on A. SakaishowedthatifA is a C -algebra, then the -derivation δ is bounded. And also he showed that δx)=ad ih x)=ihx xh) for some self-adjoint element h in the enveloping von Neumann algebra A of the C -algebra A. If the self-adjoint element h is in the multiplier algebra MA) ofa, δ is called an inner -derivation on A. Furthermore, if we put U t = exp ith for h in MA)andt R,thenU t can be a unitary operator and generate a oneparameter group of -automorphisms on A. So bounded derivations of C -algebras have deep relations with generators of C -dynamical systems. Besides these, since derivations play an important role in the classifications of operator algebras, the theory of bounded -derivations of C -algebras is very important in the theory of quantum mechanics and operator algebras [1 3]. We can say that a mathematical property is stable if any mathematical object satisfying a certain mathematical property approximately is near to the object exactly satisfying the mathematical property. On the stability of the functional equation, Ulam was the first beginner. He suggested the question Under what condition is there an additive mapping near an approximately additive mapping? in Hyers [4] gaveanaffirmativeanswer for the problem of Ulam on the case of Banach spaces after one year. Since then, there The Authors) This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 JangAdvances in Difference Equations 2017) 2017:193 Page 2 of 10 have been a lot of results obtained related to the stability problems of various functional equations for instances, [5 13]). In particular, some of the important functional equations are the following functional equations: f x + y)=f x)+f y), 1.3) f x + y)+f x y)=2f x)+2f y) 1.4) which are called Cauchy additive functional equation and Cauchy quadratic functional equation, respectively. It is said that a mathematical property is superstable if every mathematical object satisfying approximately the property is an exact object satisfying it. The superstability phenomenon was first investigated by Baker, Lawrence and Zorzitto, etc. [14 17]. They showed the superstability of the exponential functional equation from the vector space to the set of real numbers. In the proof of the superstability of the exponential functional equation, the multiplicative property of the norm was the necessary condition. We say that when E is a normed algebra and xy = x y for all x, y E, the norm is multiplicative. In this paper we define functional equations of a -derivation and a quadratic -derivation. And we show that the stability of -derivations and -quadratic derivations on Banach -algebras without the multiplicative property of the norm are superstable by using the fixed point theorem. In order to use the fixed point theory, we should introduce a fundamental result in the fixed point theory. Let S be a set. A function d : S S [0, ] is called a generalized metric ons if d satisfies 1) dx, y)=0if and only if x = y; 2) dx, y)=dy, x) for all x, y S; 3) dx, z) dx, y)+dy, z) for all x, y, z S. Theorem 1.1 [18, 19]) Let S, d) be a complete generalized metric space, and let J : S S be a strictly contractive mapping with Lipschitz constant L <1.Then, for each given element x S, either d J n x, J n+1 x ) = for all non-negative integers n or there exists a positive integer n 0 such that 1) dj n x, J n+1 x)<, n n 0 ; 2) the sequence {J n x} converges to a point y in S; 3) Jy )=y ; 4) y is the unique fixed point of J in the set T = {y S dj n 0x, y)< }; 5) dy, y ) 1 dy, Jy) for all y T. 1 L 2 Superstability of derivations on Banach -algebras Let A denote a -Banachalgebra with the unit e in Section 1 and Section 2. Theorem 2.1 Let ψ 1 : A A [0, ) and ψ 2 : A [0, ) be functions. Suppose that f : A A is a mapping such that f λx + y) λf x) f y) ψ1 x, y), 2.1)

3 JangAdvances in Difference Equations 2017) 2017:193 Page 3 of 10 f xy) xf y) fx)y ψ1 x, y), 2.2) f x ) f x) ψ 2 x) 2.3) for all λ T and x, y A. If there exist a natural number s N and 0<L <1such that s 1 ψ 1 sx, sy) <Lψ 1 x, y), s 1 ψ 1 sx, y) <Lψ 1 x, y), s 1 ψ 1 x, sy) <Lψ 1 x, y), and s 1 ψ 2 sx)<lψ 2 x) for all x, y A, then f is a -derivation on A. Proof If we put x = y and λ =1in2.1), then we have f 2x) 2f x) ψ1 x, x) 2.4) for all x A.Wecanseethat n 1 f nx) nf x) ψ 1 jx, x) 2.5) j=1 for all x, y A and n 2 by using the induction. We put s 1 x)= ψ 1 jx, x) j=1 for x A.Thenwehave f sx) sf x) x). 2.6) Let S be the set of all functions r : A A. Wedefineafunctiond : S S [0, ] as follows: du, v)=inf { α >0: } ux) vx) α x), x A for u, v S. We can easily show that S, d) is a generalized complete metric space. Define afunctionh : S S by Hu)x)=s 1 usx). If we put du, v)=αu, v S), then we can have Hu)x) Hv)x) = s 1 usx) vsx) α s 1 sx) Lα x). It follows that for u, v S d Hu), Hv) ) Ldu, v). 2.7) Therefore H is a strictly contractive mapping on S with Lipschitz constant L.By2.6), Hf )x) f x) = s 1 f sx) f x) = s 1 f sx) sf x) s 1 x).

4 JangAdvances in Difference Equations 2017) 2017:193 Page 4 of 10 This means that dhf ), f ) 1/ s.bytheorem1.1, H has a unique fixed point h : A A in the set U = { u S : d u, Hf ) ) < }. We see that for each x A hx)= lim m Hm f x) ) = lim m s m f s m x ). 2.8) From 2.6)wecanhave hλx + y) λhx) hy) = lim s n f s n λx + y) ) λf s n x ) f s n y ) s n ψ 1 s n x, s n y ) Ln ψ 1 x, y)=0 for all x, y A and λ T.Next,letλ = λ 1 +iλ 2 C,whereλ 1, λ 2 R.Letμ 1 = λ 1 [λ 1 ]and μ 2 = λ 2 [λ 2 ], where [λ] denotes the integer part of λ.onecanrepresentμ i as μ i = λ i,1+λ i,2 2 such that λ i,j T 1 i, j 2). Since we show that hλx + y)=λhx)+hy) forλ T, we can infer that hλx)=hλ 1 x)+ihλ 2 x) = [λ 1 ]hx)+hμ 1 x) ) +i [λ 2 ]hx)+hμ 2 x) ) = [λ 1 ]hx)+ 1 ) 2 hλ 1,1x + λ 1,2 x) +i [λ 2 ]hx)+ 1 ) 2 hλ 2,1x + λ 2,2 x) = [λ 1 ]hx)+ 1 2 λ 1,1hx)+ 1 ) 2 λ 1,2hx) +i [λ 2 ]hx)+ 1 2 λ 2,1hx)+ 1 ) 2 λ 2,2hx) = λ 1 hx)+iλ 2 hx) = λhx) for all x A and λ C.Soh is a C-linear map on A. For the involution of h,wecanhave h x ) hx) = lim s n f s n x ) f s n x ) s n ψ 2 s n x ) Ln ψ 2 x)=0. Next, we are going to prove the derivation property of h.replacingxby s n x and y by s n y in 2.2) and dividing by s 2n,weget f s n xs n y) x f sn y) f sn x) y s 2n s n s n 1 s ψ 2n 2 s n x, s n y ) L 2n ψ 2 x, y).

5 JangAdvances in Difference Equations 2017) 2017:193 Page 5 of 10 By taking n,wehave hxy)=xhy)+hx)y 2.9) for all x, y A. It follows that h is a -derivation on A.Next,ifwereplacex by s n x in 2.2) and divide by s n,weget f s n xy) xf y) f sn x) y s n s n 1 s ψ n 2 s n x, y ) L n ψ 2 x, y) for all x, y A and all n N.Bytakingn,wehave hxy)=xf y)+hx)y 2.10) for all x, y A.Fixm N. And considering the following equations xf s m y ) = h s m xy ) hx)s m y = s m xf y) 2.11) for all x, y A,wecangetxf y)=x f sm y) s m for all x, y A and each m N.Bytakingm, we have xf y)=xhy). If we put x = e, thenhy)=f y) for all y A. Sof is an exactly - derivation on A. 3 Superstability of quadratic -derivations on Banach -algebras Definition 3.1 A mapping δ : A A is a -quadratic derivation of A if a map δ satisfies the following properties: for all a, b A and λ C, 1) δa + b)+δa b) 2δa) 2δb)=0; 2) δ is quadratic homogeneous, that is, δλa)=λ 2 δa); 3) δab)=δa)b 2 + a 2 δb); 4) δa )=δa). Theorem 3.2 Let ψ 1 : A A [0, ) and ψ 2 : A [0, ) be functions. Suppose that f : A A is a mapping such that f x + y)+f x y) 2f x) 2fy) ψ1 x, y), 3.1) f xy) x 2 f y) fx)y 2 ψ 1 x, y), 3.2) f λx) λ 2 f x) ψ 2 x), 3.3) f x ) f x) ψ2 x) 3.4) for all λ C and x, y A. If there exist a natural number s N and 0<L <1such that 2 2s ψ 1 2 s x,2 s y)<lψ 1 x, y), 2 2s ψ 1 2 s x, y) <Lψ 1 x, y), 2 2s ψ 1 x,2 s y)<lψ 1 x, y), and 2 2s ψ 2 2 s x)<lψ 2 x) for all x, y A, then f is a -quadratic derivation on A. Proof If we put x = y and λ =1in3.1), then we have f 2x) 4f x) ψ1 x, x), x A.

6 JangAdvances in Difference Equations 2017) 2017:193 Page 6 of 10 By induction on n,we cansee that f 2 n x ) 2 2n f x) n 1 2 2n i) ψ 1 2 i x,2 i x ) 3.5) for all x, y A and n 2. For simplicity, if we put i=0 s 1 x)= 2 2s i) ψ 1 2 i x,2 i x ), 3.6) then we have i=0 f 2 s x ) 2 2s f x) x). Let S be the set of all functions u : A A. Wedefineafunctiond : S S [0, ] as follows: du, v)=inf { α >0: ux) vx) α x), x A }. We can easily show that S, d) is a generalized complete metric space. Define a function H : S S by Hu)x)=2 2s u2 s x). If we put du v)=α for u, v S, then we can have Hu)x) Hv)x) =2 2s u 2 s x ) v 2 s x ) α2 2s 2 s x ) Lα x). It follows that for u, v S d Hu), Hv) ) Ldu, v). 3.7) Hence H is a strictly contractive mapping on X with Lipschitz constant L. Wehavethat for x A Hf )x) f x) = 2 2s f 2 s x ) f x) =2 2s f 2 s ) 2 2s f x) 2 2s x). This means that dhf ), f ) 1/2 2s.ByTheorem1.1, H has a unique fixed point h : A A in the set U = { u X : d u, Hf ) ) < }, 3.8) and for each x A hx)= lim m Hm f x) ) = lim 2 2sm f 2 sm x ). 3.9) From 3.9), we can have hx + y)+hx y) 2hx) 2hy) = lim 2 2sn f 2 sn x + y)+f 2 sn x y) ) 2f 2 sn x ) 2f 2 sn y )

7 JangAdvances in Difference Equations 2017) 2017:193 Page 7 of ns ψ 1 2 ns x,2 ns y ) Ln ψ 1 x, y)=0 for all x, y A.Soh is a quadratic map on A.Sincewecanhavethat hλx) λ 2 hx) = lim 2 2ns f 2 ns λx) ) λ 2 f 2 ns x ) 2 2ns ψ 2 2 ns x ) Ln ψ 2 x)=0, it follows that h is quadratic homogeneous. Next, if we replace x by 2 ns x in 3.2) and divide by 2 2sn,thenweget f 2 ns xy) x 2 f y) f 2ns x) y 2 2 2ns 2 2ns 1 2 ψ 2ns 1 2 ns x, y ) L n ψ 1 x, y) 3.10) for all x, y A and all n N.Bytakingn,wehave hxy)=x 2 f y)+hx)y ) for all x, y A.Fixm N. If we consider the following equations x 2 f 2 ms y ) = h 2 ms xy ) h 2 ms x ) y 2 =2 2ms x 2 f y)+h 2 ms x ) y 2 h 2 ms x ) y 2 =2 2ms x 2 f y) for all x, y A, thenwehavex 2 f y)=x 2 f 2 ms y) for all x, y A and for each m N. Taking 2 2ms m,wehavex 2 f y)=x 2 hy). If we put x = e, thenhy) =f y) for all y A. Sof is a -quadratic derivation on A. 4 Derivations on C -ternary algebras A C -ternary algebra is a complex Banach space A equipped with a ternary product x, y, z) [x, y, z]ofa 3 into A satisfying the following properties: 1) [λx + u, y, z]=λ[x, y, z]+[u, y, z] for all λ C; 2) [x, λy + u, z]= λ[x, y, z]+[x, u, z] for all λ C; 3) [x, y, λz + u]=λ[x, y, z]+[x, y, u] for all λ C; 4) [x, y,[z, w, v]] = [x,[w, z, y], v]=[[x, y, z], w, v]; 5) [x, y, z] x y z ; 6) [x, x, x] = x 3 see [20, 21]) for x, y, z, u, v, w A. WesaythataC -ternary algebra A has the unit e if there exists a unique element e A such that x =[x, e, e]=[e, e, x] for all x A. Alsowecandefinean

8 JangAdvances in Difference Equations 2017) 2017:193 Page 8 of 10 involution on the C -ternary algebra A with the unit e such as [e, x, e]=x for each x A. A mapping δ : A A is called a C -ternary derivation if it satisfies δ [x, y, z] ) = [ δx), y, z ] + [ x, δy), z ] + [ x, y, δz) ], δλx + y)= λδx)+δy) for all x, y, z A and λ C. In addition, if δ satisfies that δ[e, x, e]) = [e, δx), e]onthec - ternary algebra A with the unit e, then it is said that δ is an involutive C -ternary derivation on A. Theorem 4.1 Let A be a C -ternary algebra with the unit e. Let ψ 1 : A 2 [0, ) and ψ 2 : A 3 [0, ) be functions. Suppose that f : A A is a mapping such that f λx + y) λf x) fy) ψ1 x, y), 4.1) f [x, y, z] ) [ f x), y, z ] [ x, f y), z ] [ x, y, f z) ] ψ 2 x, y, z), 4.2) ) [ ] f [e, y, e] e, f y), e ψ2 e, y, e) 4.3) for all λ T. Also suppose that there exist a natural number s N and 0<L <1such that s i+j) ψ 1 s i x, s j y)<l i+j ψ 1 x, y), s i+j+k) ψ 2 i sx, s j y, s k z)<l i+j+k ψ 2 x, y, z) for all x, y, z A and i, j, k =0,1.Then f is an involutive C -ternary derivation on A. Proof If we put s 1 x)= ψ 1 jx, x) j=1 for x A,thenwehave f sx) sf x) x) 4.4) similar to Theorem 2.1. LetS be the set of all functions r : A A. Wedefineafunction d : S S [0, ] as follows: du, v)=inf { α >0: ux) vx) α x), x A } for u, v S. We can easily show that S, d) is a generalized complete metric space. Define afunctionh : S S by Hu)x)=s 1 usx). When du, v)=αu, v S), then we can have Hu)x) Hv)x) = s 1 usx) vsx) α s 1 sx) Lα x). It follows that for u, v S d Hu), Hv) ) Ldu, v). 4.5)

9 JangAdvances in Difference Equations 2017) 2017:193 Page 9 of 10 Therefore, H is a strictly contractive mapping on S with Lipschitz constant L.By4.4), Hf )x) f x) = s 1 f sx) f x) = s 1 f sx) sf x) s 1 x). This means that dhf ), f ) 1/ s.bytheorem1.1, H has a unique fixed point h : A A in the set U = { u S : d u, Hf ) ) < }. We see that for each x A hx)= lim m Hm f x) ) = lim m s m f s m x ). 4.6) We can see that h is a C-linear map on A similartotheproofoftheorem2.1. Now we are going to prove the C -ternary derivation property of h. h [x, y, z] ) [ hx), y, z ] [ x, hy), z ] [ x, y, hz) ] = lim s 3n f s 3n [x, y, z] ) s 2n[ f s n x ), y, z ] s 2n[ x, f s n y ), z ] s 2n[ x, y, f s n z )] s 3n ψ 1 s n x, s n y, s n z ) L3n ψ 1 x, y, z)=0. Hence we have h [x, y, z] ) = [ hx), y, z ] + [ x, hy), z ] + [ x, y, hz) ] 4.7) for all x, y, z A. For the involution of h,wecanhave h [e, x, e] ) [ e, hx), e ] = lim s 3n f s 3n [e, x, e] ) s 2n[ e, f s n x ), e ] s 3n ψ 1 s n e, s n x, s n e ) L3n ψ 1 e, x, e)=0. So h is an involutive C -ternary derivation on A. Replacing y by s n y and z by s n z in 4.2), dividing by s 2n,andlettingn go to the infinity, we get lim s 2n f [ x, s n y, s n z ]) [ f x), s n y, s n z ] s n[ x, f s n y ), z ] s n[ x, y, f s n z )]) = lim s 2n f s 2n [x, y, z] ) s 2n[ f x), y, z ] s n[ x, f s n y ), z ] s n[ x, y, f s n z )] ) s 2n ψ 1 x, s n y, s n z ) L2n ψ 1 x, y, z)=0.

10 JangAdvances in Difference Equations 2017) 2017:193 Page 10 of 10 So we have h [x, y, z] ) = [ f x), y, z ] + [ x, hy), z ] + [ x, y, hz) ] 4.8) for all x, y, z A. In4.7) and4.8) weputf x) hx) instead of y and z, thenwecanget hx) fx) =0.Sof is an exactly involutive C -ternary derivation on A. Acknowledgements This paper was partially supported by the Research Fund of University of Ulsan, The author would like to thank the referees for their useful comments. Competing interests The author declares that she has no competing interests. Author s contributions The author 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. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 3 January 2017 Accepted: 15 June 2017 References 1. Bratteli, O: Derivation, Dissipation and Group Actions on C -Algebras. Lecture Notes in Math., vol Springer, Berlin 1986) 2. Bratteli, O, Kishimoto, A, Robinson, DW: Approximately inner derivations. Math. Scand. 103, ) 3. Jang, SY: Approximate -derivations on fuzzy Banach -algebras. Adv. Differ. Equ. doi: / Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, ) 5. Czerwik, S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 62, ) 6. Cădariu, L, Găvruta, L, Găvruta, P: On the stability of an affine functional equation. J. Nonlinear Sci. Appl. 6, ) 7. Hyers, DH, Isac, G, Rassias, TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel 1998) 8. Jung, SM, Popa, D, Rassias, MT: On the stability of the linear functional equation in a single variable on complete metric groups. J. Glob. Optim. 59, ) 9. Kannappan, P: Functional Equations and Inequalities with Applications. Springer, Berlin 2009) 10. Miheţ, D, Radu, V: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, ) 11. Moslehian, MS, Rahbarnia, F, Sahoo, PK: Approximate double centralizers are exact double centralizers. Bol. Soc. Mat. Mexicana 13, ) 12. Skof, F: Proriet locali e approssimazione di operatori. Rend. Semin. Mat. Fis. Milano 53, ) 13. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, ) 14. Baker, JA, Lawrence, J, Zorzitto, F: The stability of the equation f x + y)= f x)f y).proc. Am. Math. Soc. 74, ) 15. Baker, JA: The stability of the cosine equation. Proc. Am. Math. Soc. 80, ) 16. Cholewa, PW: The stability of the sine equation. Proc. Am. Math. Soc. 88, ) 17. Szèkelyhidi, L: The stability of the sine and cosine functional equations. Proc. Am. Math. Soc. 110, ) 18. Cădariu, L, Radu, V: Fixed points and the stability of Jensen s functional equation. J. Inequal. Pure Appl. Math. 41), Article ID ) 19. Diaz, J, Margolis, B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull.Am.Math.Soc.74, ) 20. Amyari, M, Moslehian, MS: Approximately ternary semigroup homomorphisms. Lett. Math. Phys. 77, ) 21. Zettl, H: A characterization of ternary rings of operators. Adv. Math. 48, )