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

Osamu Hatori; Go Hirasawa; Takeshi Miura

Open Mathematics (2010)

  • Volume: 8, Issue: 3, page 597-601
  • ISSN: 2391-5455

Abstract

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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 T a ^ y = T e ^ y a ^ φ y y K T e ^ y a ^ φ y ¯ y M K for all a ∈ A, where e is unit element of A. If, in addition, T e ^ = 1 and T i e ^ = i on M B, then T is an algebra isomorphism.

How to cite

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Osamu Hatori, Go Hirasawa, and Takeshi Miura. "Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras." Open Mathematics 8.3 (2010): 597-601. <http://eudml.org/doc/269004>.

@article{OsamuHatori2010,
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 \[ \widehat\{T\left( a \right)\}\left( y \right) = \left\lbrace \{\begin\{array\}\{c\}\widehat\{T\left( e \right)\}\left( y \right)\hat\{a\}\left( \{\phi \left( y \right)\} \right) y \in K \hfill \\ \widehat\{T\left( e \right)\}\left( y \right)\overline\{\hat\{a\}\left( \{\phi \left( y \right)\} \right)\} y \in M\_\mathcal \{B\} \backslash K \hfill \\ \end\{array\}\} \right. \] 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.},
author = {Osamu Hatori, Go Hirasawa, Takeshi Miura},
journal = {Open Mathematics},
keywords = {Uniform algebra; Commutative Banach algebra; Maximal ideal space; Shilov boundary; Algebra isomorphism; Norm-additive operator; Norm-linear operator; uniform algebra; semi-simple commutative Banach algebra; algebra isomorphism; norm-additive map},
language = {eng},
number = {3},
pages = {597-601},
title = {Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras},
url = {http://eudml.org/doc/269004},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Osamu Hatori
AU - Go Hirasawa
AU - Takeshi Miura
TI - Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras
JO - Open Mathematics
PY - 2010
VL - 8
IS - 3
SP - 597
EP - 601
AB - 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 \[ \widehat{T\left( a \right)}\left( y \right) = \left\lbrace {\begin{array}{c}\widehat{T\left( e \right)}\left( y \right)\hat{a}\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline{\hat{a}\left( {\phi \left( y \right)} \right)} y \in M_\mathcal {B} \backslash K \hfill \\ \end{array}} \right. \] 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.
LA - eng
KW - Uniform algebra; Commutative Banach algebra; Maximal ideal space; Shilov boundary; Algebra isomorphism; Norm-additive operator; Norm-linear operator; uniform algebra; semi-simple commutative Banach algebra; algebra isomorphism; norm-additive map
UR - http://eudml.org/doc/269004
ER -

References

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  1. [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 Zbl0713.46016
  2. [2] Gleason A.M., A characterization of maximal ideals, J. Analyse Math., 1967, 19, 171–172 http://dx.doi.org/10.1007/BF02788714 Zbl0148.37502
  3. [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 Zbl1102.46032
  4. [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 Zbl1113.46047
  5. [5] Jarosz K., Perturbations of Banach algebras, Lecture Notes in Mathematics 1120, Springer, 1985 Zbl0557.46029
  6. [6] Kahane J.P., Zelazko W, A characterization of maximal ideals in commutative Banach algebras, Studia Math., 1968, 29, 339–343 Zbl0155.45803
  7. [7] Kowalski S., Słodkowski Z., A characterization of maximal ideals in commutative Banach algebras, Studia Math., 1980, 67, 215–223 Zbl0456.46041
  8. [8] Lambert S., Luttman A., Tonev T., Weakly peripherally-multiplicative mappings between uniform algebras, Contemp. Math., 2007, 435, 265–281 Zbl1148.46030
  9. [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 Zbl1151.46036
  10. [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 Zbl1134.46030
  11. [11] Mazur S., Ulam S., Sur les transformationes isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris, 1932, 194, 946–948 Zbl58.0423.01
  12. [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 Zbl1197.47051
  13. [13] Miura T., Honma D., Shindo R., Divisibly norm-preserving maps between commutative Banach algebras, Rocky Mountain J. Math., to appear Zbl1232.46048
  14. [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 Zbl0983.47024
  15. [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 Zbl1068.46028
  16. [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 Zbl1074.46033
  17. [17] Rao N.V., Tonev T.V., Toneva E.T., Uniform algebra isomorphisms and peripheral spectra, Contemp. Math., 2007, 427, 401–416 Zbl1123.46035
  18. [18] Tonev T., The Banach-Stone theorem for Banach algebras, preprint 
  19. [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 Zbl1179.47035
  20. [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 Zbl1171.47032
  21. [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. [22] Zelazko W., A characterization of multiplicative linear functionals in complex Banach algebras, Studia Math., 1968, 30, 83–85 Zbl0162.18504

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