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

top
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 ℬ ∖ 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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.