ON A GENERALIZED TRIF'S MAPPING IN BANACH MODULES OVER A C*-ALGEBRA

Title & Authors
ON A GENERALIZED TRIF'S MAPPING IN BANACH MODULES OVER A C*-ALGEBRA
Park, Chun-Gil; Rassias Themistocles M.;

Abstract
Let X and Y be vector spaces. It is shown that a mapping $\small{f\;:\;X{\rightarrow}Y}$ satisfies the functional equation $\small{mn_{mn-2}C_{k-2}f(\frac {x_1+...+x_{mn}} {mn})}$ $(\ddagger)\;+mn_{mn-2}C_{k-1}\;\sum\limits_{i=1}^n\;f(\frac {x_{mi-m+1}+...+x_{mi}} {m}) =k\;{\sum\limits_{1{\leq}i_1<... is additive, and we prove the Cauchy-Rassias stability of the functional equation $\small{(\ddagger)}$ in Banach modules over a unital $\small{C^*-algebra}$. Let A and B be unital $\small{C^*-algebra}$ or Lie $\small{JC^*-algebra}$. As an application, we show that every almost homomorphism h : $\small{A{\rightarrow}B}$ of A into B is a homomorphism when $\small{h(2^d{\mu}y) = h(2^d{\mu})h(y)\;or\;h(2^d{\mu}\;o\;y)=h(2^d{\mu})\;o\;h(y)}$ for all unitaries $\small{{\mu}{\in}A,\;all\;y{\in}A}$, and d = 0,1,2,..., and that every almost linear almost multiplicative mapping $\small{h:\;A{\rightarrow}B}$ is a homomorphism when h(2x)=2h(x) for all $\small{x{\in}A}$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $\small{C^*-algebras}$ or in Lie $\small{JC^*-algebras}$, and of Lie $\small{JC^*-algebra}$ derivations in Lie $\small{JC^*-algebras}$. Keywords Trif's functional equation;Cauchy-Rassias stability;$\small{C^*-algebra homomorphism}$;Lie $\small{JC^*-algebra}$ homomorphism;Lie $\small{JC^*-algebra}$ derivation; Language English Cited by 1. ON FUNCTIONAL INEQUALITIES ASSOCIATED WITH JORDAN-VON NEUMANN TYPE FUNCTIONAL EQUATIONS,; 대한수학회논문집, 2008. vol.23. 3, pp.371-376 2. d-ISOMETRIC LINEAR MAPPINGS IN LINEAR d-NORMED BANACH MODULES,;; 대한수학회지, 2008. vol.45. 1, pp.249-271 3. STABILITY OF A MIXED QUADRATIC AND ADDITIVE FUNCTIONAL EQUATION IN QUASI-BANACH SPACES,;; Journal of applied mathematics & informatics, 2009. vol.27. 5_6, pp.1177-1194 4. STABILITY OF THE CAUCHY FUNCTIONAL EQUATION IN BANACH ALGEBRAS,;; Korean Journal of Mathematics, 2009. vol.17. 1, pp.91-102 5. A CAUCHY-JENSEN FUNCTIONAL INEQUALITY IN BANACH MODULES OVER A$C^*\$-ALGEBRA,;

Journal of applied mathematics & informatics, 2010. vol.28. 1_2, pp.233-241
1.
STABILITY OF THE JENSEN TYPE FUNCTIONAL EQUATION IN BANACH ALGEBRAS: A FIXED POINT APPROACH, Korean Journal of Mathematics, 2011, 19, 2, 149
2.
On the Cauchy–Rassias stability of a generalized additive functional equation, Journal of Mathematical Analysis and Applications, 2008, 339, 1, 372
References
1.
Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431-434

2.
P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approxi- mately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436

3.
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224

4.
D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153

5.
K. W. Jun and Y. H. Lee, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), no. 1, 305-315

6.
R. V. Kadison and G. K. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), no. 2, 249-266

7.
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Elementary Theory, Academic Press, New York, 1983

8.
C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), no. 2, 711-720

9.
C. -G. Park and J. Hou, Homomorphisms between C*-algebras associated with the Trif functional equation and linear derivations on C*-algebras, J. Korean Math. Soc. 41 (2004), no. 4, 461-477

10.
C. -G. Park and W. -G. Park, On the Jensen's equation in Banach modules, Taiwanese J. Math. 6 (2002), no. 4, 523-531

11.
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300

12.
Th. M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aeq. Math. 39 (1990), 292-293; 309

13.
Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130

14.
Th. M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), no. 2, 352-378

15.
Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), no. 1, 264-284

16.
Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers{Ulam stability, Proc. Amer. Math. Soc. 114 (1992), no. 4, 989-993

17.
Th. M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), no. 2, 325-338

18.
T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions, J. Math. Anal. Appl. 272 (2002), no. 2, 604-616

19.
S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960

20.
H. Upmeier, Jordan algebras in analysis, operator theory, and quantum me- chanics, CBMS Regional Conference Series in Mathematics, 67, Amer. Math. Soc., Providence, 1987