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On an approximate automorphism on a $C^{*}$-algebra

Author: Chun-Gil Park
Journal: Proc. Amer. Math. Soc. 132 (2004), 1739-1745
MSC (2000): Primary 47B48, 46L40, 39B52
Published electronically: October 9, 2003
MathSciNet review: 2051135
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Abstract: It is shown that for an approximate algebra homomorphism $f : \mathcal{B} \rightarrow \mathcal{B}$ on a Banach $*$-algebra $\mathcal{B}$, there exists a unique algebra $*$-homomorphism $H : \mathcal{B} \rightarrow \mathcal{B}$ near the approximate algebra homomorphism. This is applied to show that for an approximate automorphism $f : \mathcal{A} \rightarrow \mathcal{A}$on a unital $C^{*}$-algebra $\mathcal{A}$, there exists a unique automorphism $\alpha : \mathcal{A} \rightarrow \mathcal{A}$ near the approximate automorphism. In fact, we show that the approximate automorphism $f : \mathcal{A} \rightarrow \mathcal{A}$ is an automorphism.

References [Enhancements On Off] (What's this?)

  • 1. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, New York, Heidelberg and Berlin, 1973. MR 54:11013
  • 2. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436. MR 95e:47089
  • 3. B. E. Johnson, Approximately multiplicative maps between Banach algebras, J. London Math. Soc. (2) 37 (1988), 294-316. MR 89h:46072
  • 4. R. V. Kadison and G. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249-266. MR 87g:47078
  • 5. R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary Theory, Pure and Applied Mathematics, Vol. 100, Academic Press, New York, 1983. MR 85j:46099
  • 6. C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), 711-720. MR 2003i:46045
  • 7. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. MR 80d:47094

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Additional Information

Chun-Gil Park
Affiliation: Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea

Keywords: Approximate algebra homomorphism, approximate automorphism, $C^{*}$-algebra, stability, functional equation
Received by editor(s): December 2, 2002
Received by editor(s) in revised form: February 3, 2003
Published electronically: October 9, 2003
Additional Notes: This work was supported by Korea Research Foundation Grant KRF-2002-041-C00014. The author would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society

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