Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지

HOMOMORPHISMS BETWEEN POISSON BANACH ALGEBRAS AND POISSON BRACKETS

  • PARK, CHUN-GIL (Department of Mathematics, Chungnam National University) ;
  • WEE, HEE-JUNG (Department of Mathematics, Chungnam National University)
  • Received : 2003.10.15
  • Accepted : 2004.02.05
  • Published : 2004.03.25
Download PDF
⟨ Previous Next ⟩

Abstract

It is shown that every almost linear mapping $h:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ of a unital Poisson Banach algebra ${\mathcal{A}}$ to a unital Poisson Banach algebra ${\mathcal{B}}$ is a Poisson algebra homomorphism when h(xy) = h(x)h(y) holds for all $x,y{\in}\;{\mathcal{A}}$, and that every almost linear almost multiplicative mapping $h:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ is a Poisson algebra homomorphism when h(qx) = qh(x) for all $x\;{\in}\;{\mathcal{A}}$. Here the number q is in the functional equation given in the almost linear almost multiplicative mapping. We prove that every almost Poisson bracket $B:{\mathcal{A}}\;{\times}\;{\mathcal{A}}\;{\rightarrow}\;{\mathcal{A}}$ on a Banach algebra ${\mathcal{A}}$ is a Poisson bracket when B(qx, z) = B(x, qz) = qB(x, z) for all $x,z{\in}\;{\mathcal{A}}$. Here the number q is in the functional equation given in the almost Poisson bracket.

Keywords

Acknowledgement

Supported by : Korea Science & Engineering Foundation

References

  1. J. Math. Anal. Appl. v.184 A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings Gavruta, P.
  2. Trans. Amer. Math. Soc. v.352 Letzter Quantum n-space as a quotient of classical nspace Goodearl, K.R.;Letzter, E.S.
  3. Math. Scand. v.57 Means and convex combinations of unitary operators Kadison, R.V.;Pedersen, G.
  4. Comm. Algebra v.30 Quantum n-space and Poisson n-space Oh, S.;Park, C.;Shin, Y.
  5. Comm. Algebra v.30 A Poincare-Birkhoff-Witt theorem for Poisson enveloping algebras Oh, S.;Park, C.;Shin, Y.
  6. J. Math. Anal. Appl. v.275 On the stability of the linear mapping in Banach modules Park, C.
  7. J. Math. Anal. Appl. v.278 Modified Trif's functional equations in Banach modules over a C-algebra and approximate algebra homomorphisms Park, C.
  8. Proc. Amer. Math. Soc. v.72 On the stability of the linear mapping in Banach spaces Rassias, Th.M.
  9. J. Math. Anal. Appl. v.272 On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions Trif, T.
  10. Amer. J. Math. v.116 Noncommutative Poisson algebras Xu, P.