APPROXIMATE BI-HOMOMORPHISMS AND BI-DERIVATIONS IN C*-TERNARY ALGEBRAS

Title & Authors
APPROXIMATE BI-HOMOMORPHISMS AND BI-DERIVATIONS IN C*-TERNARY ALGEBRAS
Bae, Jae-Hyeong; Park, Won-Gil;

Abstract
In this paper, we prove the generalized Hyers-Ulam stability of bi-homomorphisms in $\small{C^*}$-ternary algebras and of bi-derivations on $\small{C^*}$-ternary algebras for the following bi-additive functional equation f(x + y, z - w) + f(x - y, z + w) = 2f(x, z) - 2f(y, w). This is applied to investigate bi-isomorphisms between $\small{C^*}$-ternary algebras.
Keywords
bi-additive mapping;$\small{C^*}$-ternary algebra;
Language
English
Cited by
1.
ON THE STABILITY OF BI-DERIVATIONS IN BANACH ALGEBRAS,;;

대한수학회보, 2011. vol.48. 5, pp.959-967
2.
GENERALIZED ULAM-HYERS STABILITY OF $C^{\star}$-TERNARY ALGEBRA 3-HOMOMORPHISMS FOR A FUNCTIONAL EQUATION,;;

충청수학회지, 2011. vol.24. 2, pp.147-162
3.
APPROXIMATE BI-HOMOMORPHISMS AND BI-DERIVATIONS IN C*-TERNARY ALGEBRAS: REVISITED,;;;;;

Korean Journal of Mathematics, 2013. vol.21. 2, pp.161-170
1.
APPROXIMATE BI-HOMOMORPHISMS AND BI-DERIVATIONS IN C*-TERNARY ALGEBRAS: REVISITED, Korean Journal of Mathematics, 2013, 21, 2, 161
2.
C *-Ternary 3-Homomorphisms on C *-Ternary Algebras, Results in Mathematics, 2014, 66, 1-2, 87
3.
On the Asymptoticity Aspect of Hyers-Ulam Stability of Quadratic Mappings, Journal of Inequalities and Applications, 2010, 2010, 1, 454875
4.
C∗-ternary 3-derivations on C∗-ternary algebras, Journal of Inequalities and Applications, 2013, 2013, 1, 124
5.
Stability of bi-θ-derivations on JB*-triples: Revisited, International Journal of Geometric Methods in Modern Physics, 2014, 11, 03, 1450015
6.
Comment to “Approximate bihomomorphisms and biderivations in 3-Lie algebras” [Int. J. Geom. Methods Mod. Phys. 10 (2013) 1220020], International Journal of Geometric Methods in Modern Physics, 2017, 14, 05, 1750079
7.
A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C∗-ternary algebras, Journal of Mathematical Physics, 2010, 51, 10, 103508
8.
Generalized ulam-hyers stability of C*-Ternary algebra n-Homomorphisms for a functional equation, Journal of Inequalities and Applications, 2011, 2011, 1, 30
References
1.
V. Abramov, R. Kerner, and B. Le Roy, Hypersymmetry: a $Z_3$-graded generalization of supersymmetry, J. Math. Phys. 38 (1997), no. 3, 1650-1669.

2.
M. Amyari and M. S. Moslehian, Approximate homomorphisms of ternary semigroups, Lett. Math. Phys. 77 (2006), no. 1, 1-9.

3.
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.

4.
J.-H. Bae, On the stability of 3-dimensional quadratic functional equation, Bull. Korean Math. Soc. 37 (2000), no. 3, 477–486.

5.
J.-H. Bae and K.-W. Jun, On the generalized Hyers-Ulam-Rassias stability of a quadratic functional equation, Bull. Korean Math. Soc. 38 (2001), no. 2, 325–336.

6.
J.-H. Bae and W.-G. Park, Generalized Jensen's functional equations and approximate algebra homomorphisms, Bull. Korean Math. Soc. 39 (2002), no. 3, 401–410.

7.
J.-H. Bae and W.-G. Park, On the solution of a bi-Jensen functional equation and its stability, Bull. Korean Math. Soc. 43 (2006), no. 3, 499–507.

8.
D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237.

9.
A. Cayley, On the 34 concomitants of the ternary cubic, Amer. J. Math. 4 (1881), no. 1-4, 1–15.

10.
Y. L. Daletskii and L. Takhtajan, Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys. 39 (1997), no. 2, 127–141.

11.
Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431–434.

12.
P. Gavrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431–436.

13.
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222–224.

14.
K.-W. Jun, S.-M. Jung, and Y.-H. Lee, A generalization of the Hyers-Ulam-Rassias stability of a functional equation of Davison, J. Korean Math. Soc. 41 (2004), no. 3, 501–511.

15.
M. Kapranov, I. M. Gelfand, and A. Zelevinskii, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, Berlin, 1994.

16.
R. Kerner, The cubic chessboard: Geometry and physics, Classical Quantum Gravity 14 (1997), A203–A225.

17.
R. Kerner, Ternary algebraic structures and their applications in physics, Proc. BTLP, 23rd International Conference on Group Theoretical Methods in Physics, Dubna, Russia, 2000; http://arxiv.org/abs/math-ph/0011023v1.

18.
E. H. Lee, I.-S. Chang, and Y.-S. Jung, On stability of the functional equations having relation with a multiplicative derivation, Bull. Korean Math. Soc. 44 (2007), no. 1, 185–194.

19.
Y.-H. Lee and K.-W. Jun, A note on the Hyers-Ulam-Rassias stability of Pexider equation, J. Korean Math. Soc. 37 (2000), no. 1, 111–124.

20.
Y. W. Lee, Stability of a generalized quadratic functional equation with Jensen type, Bull. Korean Math. Soc. 42 (2005), no. 1, 57–73.

21.
T. Miura, S.-M. Jung, and S.-E. Takahasi, Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations y' = ${\lambda}$y, J. Korean Math. Soc. 41 (2004), no. 6, 995–1005.

22.
M. S. Moslehian, Almost derivations on $C^{\ast}$-ternary rings, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 135–142.

23.
A. Najati, G. Z. Eskandani, and C. Park, Stability of homomorphisms and derivations in proper $JCQ^{\ast}$-triples associated to the pexiderized Cauchy type mapping, Bull. Korean Math. Soc. 46 (2009), no. 1, 45–60.

24.
C. Park, Isomorphisms between $C^{\ast}$-ternary algebras, J. Math. Phys. 47 (2006), no. 10, 12 pp.

25.
C. Park and J. S. An, Isomorphisms in quasi-Banach algebras, Bull. Korean Math. Soc. 45 (2008), no. 1, 111–118.

26.
C. Park and Th. M. Rassias, d-Isometric linear mappings in linear d-normed Banach modules, J. Korean Math. Soc. 45 (2008), no. 1, 249–271.

27.
C.-G. Park and J. Hou, Homomorphisms between $C^{\ast}$-algebras associated with the Trif functional equation and linear derivations on $C^{\ast}$-algebras, J. Korean Math. Soc. 41 (2004), no. 3, 461–477.

28.
C.-G. Park and W.-G. Park, On the stability of the Jensen's equation in a Hilbert module, Bull. Korean Math. Soc. 40 (2003), no. 1, 53–61.

29.
K.-H. Park and Y.-S. Jung, Stability of a cubic functional equation on groups, Bull. Korean Math. Soc. 41 (2004), no. 2, 347–357.

30.
W.-G. Park and J.-H. Bae, On the stability of involutive A-quadratic mappings, Bull. Korean Math. Soc. 43 (2006), no. 4, 737–745.

31.
W.-G. Park and J.-H. Bae, Quadratic functional equations associated with Borel functions and module actions, Bull. Korean Math. Soc. 46 (2009), no. 3, 499–510.

32.
W.-G. Park and J.-H. Bae, Quadratic functional equations associated with Borel functions and module actions, Bull. Korean Math. Soc. 46 (2009), no. 3, 499–510.

33.
A. Prastaro, Geometry of PDEs and Mechanics, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.

34.
J. M. Rassias and H.-M. Kim, Approximate homomorphisms and derivations between $C^{\ast}$-ternary algebras, J. Math. Phys. 49 (2008), no. 6, 10 pp.

35.
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297–300.

36.
Th. M. Rassias and P. ˇSemrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), no. 4, 989-993.

37.
J. Rho and H. J. Shin, Approximation of Cauchy additive mappings, Bull. Korean Math. Soc. 44 (2007), no. 4, 851–860.

38.
S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.

39.
L. Vainerman and R. Kerner, On special classes of n-algebras, J. Math. Phys. 37 (1996), no. 5, 2553–2565.

40.
H. Zettl, A characterization of ternary rings of operators, Adv. in Math. 48 (1983), no. 2, 117–143.