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Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 5 (2010), 297 -- 310

HIGHER *-DERIVATIONS BETWEEN UNITAL C*-ALGEBRAS

M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun

Abstract. Let A, B be two unital C*-algebras. We prove that every sequence of mappings from A into B, H = {h0, h1, ..., hm, ...}, which satisfy hm(3nuy) =Σi+j=mhi(3nu)hj(y) for each m ∈ N0, for all u∈ U(A), all y∈ A, and all n = 0, 1, 2, ..., is a higher derivation. Also, for a unital C*-algebra A of real rank zero, every sequence of continuous mappings from A into B, H = {h0, h1,..., hm, ...}, is a higher derivation when hm(3nuy) =Σi+j=mhi(3nu)hj(y) holds for all u∈ I1(Asa), all y∈ A, all n = 0, 1, 2, ... and for each m ∈ N0. Furthermore, by using the fixed points methods, we investigate the Hyers-Ulam-Rassias stability of higher *-derivations between unital C*-algebras.

2010 Mathematics Subject Classification: 39B52; 39B82; 46B99; 17A40.
Keywords: Alternative fixed point; Hyers--Ulam--Rassias stability; Higher *-derivation.

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References

  1. R. Badora, On approximate ring homomorphisms, J. Math. Anal. Appl. 276 (2002), 589--597. MR1944777. Zbl 1014.39020.

  2. J. Baker, J. Lawrence and F. Zorzitto, The stability of the equation f(x+y)=f(x)f(y), Proc. Amer. Math. Soc. 74(2) (1979), 242--246. MR0524294. Zbl 0397.39010.

  3. D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385--397. MR0031194. Zbl 0033.37702.

  4. L. Brown and G. Pedersen, Limits and C*-algebras of low rank or dimension, J. Oper. Theory 61(2) (2009), 381--417. MR2501012. Zbl pre05566809.

  5. L. Cădariu and V. Radu, On the stability of the Cauchy functional equation:a fixed point approach, Grazer Mathematische Berichte 346 (2004), 43--52. MR2089527. Zbl 1060.39028.

  6. L. Cădariu, V. Radu, The fixed points method for the stability of some functional equations, Carpathian Journal of Mathematics 23 (2007), 63--72. MR2305836. Zbl 1196.39013.

  7. L. Cădariu, V. Radu, Fixed points and the stability of quadratic functional equations, Analele Universitatii de Vest din Timisoara 41 (2003), 25--48. MR2245911. Zbl 1103.39304.

  8. L. Cădariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4 (2003), Art. ID 4. MR1965984. Zbl 1043.39010.

  9. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002. MR1904790. Zbl 1011.39019.

  10. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222--224. MR0004076. JFM 67.0424.01.

  11. D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 no. 2-3 (1992), 125--153. MR1181264. Zbl 0806.47056.

  12. D.H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998. MR1639801. Zbl 0907.39025.

  13. B. E. Johnson, Approximately multiplicative maps between Banach algebras, J. London Math. Soc. 37(2) (1988), 294–-316. MR0928525. Zbl 0652.46031.

  14. K. Jun, Y. Lee, A generalization of the Hyers–Ulam–Rassias stability of Jensen’s equation, J. Math. Anal. Appl. 238 (1999), 305–-315. MR1711432. Zbl 0933.39053.

  15. S.-M. Jung, Hyers--Ulam--Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001. MR1841182. Zbl 0980.39024.

  16. R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Elementary Theory, Academic Press, New York, 1983. MR0719020. Zbl 0888.46039.

  17. T. Miura, S.-E. Takahasi and G. Hirasawa, Hyers--Ulam--Rassias stability of Jordan homomorphisms on Banach algebras, J. Inequal. Appl. 4 (2005), 435--441. MR2210730. Zbl 1104.39026.

  18. C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, Bull. Sci. Math. 132(2) (2008), 87--96. MR2387819. Zbl 1140.39016.

  19. C. Park, D.-H. Boo and J.-S. An, Homomorphisms between C*-algebras and linear derivations on C*-algebras, J. Math. Anal. Appl. 337(2) (2008), 1415--1424. MR2386388. Zbl 1147.39011.

  20. C. Park, Homomorphisms between Poisson JC*-algebras, Bull. Braz. Math. Soc. 36 (2005) 79--97. MR2132832. Zbl 1091.39007.

  21. C. Park and W. Park, On the Jensen’s equation in Banach modules, Taiwanese J. Math. 6 (2002), 523--531. MR1937477. Zbl 1035.39017.

  22. C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in C*-algebras: a fixed point approach, Abstract and Applied Analysis Volume 2009 (2009), Article ID 360432, 17 pages. MR2485640. Zbl 1167.39020.

  23. V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91--96. MR2031824. Zbl 1051.39031.

  24. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297--300. MR0507327. Zbl 0398.47040.

  25. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62(1) (2000), 23--130. MR1778016. Zbl 0981.39014.

  26. I. A. Rus, Principles and Applications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979. (in Romanian).

  27. S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed. Wiley, New York, 1940. MR0280310.



M. Eshaghi Gordji R. Farokhzad Rostami
Department of Mathematics, Semnan University, Department of Mathematics,
P.O. Box 35195-363, Semnan, Iran. Shahid Beheshti University,
e-mail: madjid.eshaghi@gmail.com Tehran, Iran.

S. A. R. Hosseinioun
Department of Mathematics,
Shahid Beheshti University,
Tehran, Iran.



http://www.utgjiu.ro/math/sma