Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras

Osamu Hatori 1 , Go Hirasawa 2 , and Takeshi Miura 3
  • 1 Niigata University
  • 2 Ibaraki University
  • 3 Yamagata University

Abstract

Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M B → M A and a closed and open subset K of M B such that $$ \widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right. $$ for all a ∈ A, where e is unit element of A. If, in addition, $$ \widehat{T\left( e \right)} = 1 $$ and $$ \widehat{T\left( {ie} \right)} = i $$ on M B, then T is an algebra isomorphism.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Ellis A.J., Real characterizations of function algebras amongst function spaces, Bull. London Math. Soc., 1990, 22, 381–385 http://dx.doi.org/10.1112/blms/22.4.381

  • [2] Gleason A.M., A characterization of maximal ideals, J. Analyse Math., 1967, 19, 171–172 http://dx.doi.org/10.1007/BF02788714

  • [3] Hatori O., Miura T., Takagi H., Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving property, Proc. Amer. Math. Soc., 2006, 134, 2923–2930 http://dx.doi.org/10.1090/S0002-9939-06-08500-5

  • [4] Hatori O., Miura T., Takagi T., Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative, J. Math. Anal. Appl., 2007, 326, 281–296 http://dx.doi.org/10.1016/j.jmaa.2006.02.084

  • [5] Jarosz K., Perturbations of Banach algebras, Lecture Notes in Mathematics 1120, Springer, 1985

  • [6] Kahane J.P., Zelazko W, A characterization of maximal ideals in commutative Banach algebras, Studia Math., 1968, 29, 339–343

  • [7] Kowalski S., Słodkowski Z., A characterization of maximal ideals in commutative Banach algebras, Studia Math., 1980, 67, 215–223

  • [8] Lambert S., Luttman A., Tonev T., Weakly peripherally-multiplicative mappings between uniform algebras, Contemp. Math., 2007, 435, 265–281

  • [9] Luttman A., Lambert S., Norm conditions for uniform algebra isomorphisms, Cent. Eur. J. Math., 2008, 6, 272–280 http://dx.doi.org/10.2478/s11533-008-0016-x

  • [10] Luttman A., Tonev T., Uniform algebra isomorphisms and peripheral multiplicativity, Proc. Amer. Math. Soc., 2007, 135, 3589–3598 http://dx.doi.org/10.1090/S0002-9939-07-08881-8

  • [11] Mazur S., Ulam S., Sur les transformationes isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris, 1932, 194, 946–948

  • [12] Miura T., Honma D., A generalization of peripherally-multiplicative surjections between standard operator algebras, Cent. Eur. J. Math., 2009, 7, 479–486 http://dx.doi.org/10.2478/s11533-009-0033-4

  • [13] Miura T., Honma D., Shindo R., Divisibly norm-preserving maps between commutative Banach algebras, Rocky Mountain J. Math., to appear

  • [14] Molnár L., Some characterizations of the automorphisms of B(H) and C(X), Proc. Amer. Math. Soc., 2002, 130, 111–120 http://dx.doi.org/10.1090/S0002-9939-01-06172-X

  • [15] Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras, Proc. Amer. Math. Soc., 2005, 133, 1135–1142 http://dx.doi.org/10.1090/S0002-9939-04-07615-4

  • [16] Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras. II, Proc. Edinburgh Math. Soc., 2005, 48, 219–229 http://dx.doi.org/10.1017/S0013091504000719

  • [17] Rao N.V., Tonev T.V., Toneva E.T., Uniform algebra isomorphisms and peripheral spectra, Contemp. Math., 2007, 427, 401–416

  • [18] Tonev T., The Banach-Stone theorem for Banach algebras, preprint

  • [19] Tonev T., Luttman A., Algebra isomorphisms between standard operator algebras, Studia Math., 2009, 191, 163–170 http://dx.doi.org/10.4064/sm191-2-4

  • [20] Tonev T., Yates R., Norm-linear and norm-additive operators between uniform algebras, J. Math. Anal. Appl., 2009, 357, 45–53 http://dx.doi.org/10.1016/j.jmaa.2009.03.039

  • [21] Väisälä J., A proof of the Mazur-Ulam theorem, Amer. Math. Monthly, 2003, 110–7, 633–635 http://dx.doi.org/10.2307/3647749

  • [22] Zelazko W., A characterization of multiplicative linear functionals in complex Banach algebras, Studia Math., 1968, 30, 83–85

OPEN ACCESS

Journal + Issues

Search