DOI QR코드

DOI QR Code

Poisson Banach Modules over a Poisson C*-Algebr

  • Received : 2004.11.15
  • Published : 2008.12.31

Abstract

It is shown that every almost linear mapping h : $A{\rightarrow}B$ of a unital PoissonC*-algebra A to a unital Poisson C*-algebra B is a Poisson C*-algebra homomorph when $h(2^nuy)\;=\;h(2^nu)h(y)$ or $h(3^nuy)\;=\;h(3^nu)h(y)$ for all $y\;\in\;A$, all unitary elements $u\;\in\;A$ and n = 0, 1, 2,$\codts$, and that every almost linear almost multiplicative mapping h : $A{\rightarrow}B$ is a Poisson C*-algebra homomorphism when h(2x) = 2h(x) or h(3x) = 3h(x for all $x\;\in\;A$. Here the numbers 2, 3 depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings. We prove the Cauchy-Rassias stability of Poisson C*-algebra homomorphisms in unital Poisson C*-algebras, and of homomorphisms in Poisson Banach modules over a unital Poisson C*-algebra.

Keywords

Poisson C*-algebra homomorphism;Poisson Banach module;Poisson C*-algebra;stability;linear functional equation

References

  1. V. A. Faiziev, Th. M. Rassias and P. K. Sahoo, The space of $({\psi},{\gamma})-additive$ mappings on semigroups, Trans. Amer. Math. Soc., 354(2002), 4455-4472. https://doi.org/10.1090/S0002-9947-02-03036-2
  2. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., bf 184(1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  3. K. R. Goodearl and E. S. Letzter, Quantum n-space as a quotient of classical n-space, Trans. Amer. Math. Soc., 352(2000), 5855-5876. https://doi.org/10.1090/S0002-9947-00-02639-8
  4. K. Jun and Y. Lee, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl., 238(1999), 305-315. https://doi.org/10.1006/jmaa.1999.6546
  5. R. V. Kadison and G. Pedersen, Means and convex combinations of unitary operators, Math. Scand., 57(1985), 249-266.
  6. R. V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Elementary Theory, Academic Press, New York, 1983.
  7. S. Oh, C. Park and Y. Shin, Quantum n-space and Poisson n-space, Comm. Algebra, 30(2002), 4197-4209. https://doi.org/10.1081/AGB-120013313
  8. S. Oh, C. Park and Y. Shin, A Poincare-Birkhoff-Witt theorem for Poisson enveloping algebras, Comm. Algebra, 30(2002), 4867-4887. https://doi.org/10.1081/AGB-120014673
  9. C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl., 275(2002), 711-720. https://doi.org/10.1016/S0022-247X(02)00386-4
  10. C. Park, Modified Trif's functional equations in Banach modules over a $C^{\ast}-algebra$ and approximate algebra homomorphisms, J. Math. Anal. Appl., 278(2003), 93-108. https://doi.org/10.1016/S0022-247X(02)00573-5
  11. C. Park and W. Park, On the Jensen's equation in Banach modules, Taiwanese J. Math., 6(2002), 523-531.
  12. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  13. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62(2000), 23-130. https://doi.org/10.1023/A:1006499223572
  14. Th. M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl., 246(2000), 352-378. https://doi.org/10.1006/jmaa.2000.6788
  15. Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251(2000), 264-284. https://doi.org/10.1006/jmaa.2000.7046
  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), 989-993. https://doi.org/10.1090/S0002-9939-1992-1059634-1
  17. Th. M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl., 173(1993), 325-338. https://doi.org/10.1006/jmaa.1993.1070
  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), 604-616. https://doi.org/10.1016/S0022-247X(02)00181-6
  19. P. Xu, Noncommutative Poisson algebras, Amer. J. Math., 116(1994), 101-125. https://doi.org/10.2307/2374983