Zero product preserving functionals on \(C(\varOmega )\)-valued spaces of functions

Abstract

Let X be a compact Hausdorff space and \(\varOmega \) be a locally compact \(\sigma \)-compact space. In this paper we study (real-linear) continuous zero product preserving functionals \(\varphi : A \longrightarrow {\mathbb {C}}\) on certain subalgebras A of the Fréchet algebra \(C(X,C(\varOmega ))\). The case that \(\varphi \) is continuous with respect to a specified complete metric on A will also be discussed. In particular, for a compact Hausdorff space K we characterize \(\Vert \cdot \Vert \)-continuous linear zero product preserving functionals on the Banach algebra \(C^1([0,1],C(K))\) equipped with the norm \(\Vert f\Vert =\Vert f\Vert _{[0,1]}+\Vert f'\Vert _{[0,1]}\), where \(\Vert \cdot \Vert _{[0,1]}\) denotes the supremum norm. An application of the results is given for continuous ring homomorphisms on such subalgebras.

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References

  1. 1.

    Alaminos, J., Breŝar, M., Ĉerne, M., Extremera, J., Villena, A.R.: Zero product preserving maps on \(C^1[0,1]\). J. Math. Anal. Appl. 347, 472–481 (2008)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Alaminos, J., Extremera, J., Villena, A.R.: Zero product preserving maps on Banach algebras of Lipschitz functions. J. Math. Anal. Appl. 369, 94–100 (2010)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Araujo, J., Beckenstein, E., Narici, L.: Biseparating maps and ring of continuous functions. Manuscr. Math. 62, 257–275 (1988)

    Article  Google Scholar 

  4. 4.

    Dubarbie, L.: Separating maps between spaces of vector-valued absolutely continuous functions. Can. Math. Bull. 53, 466–474 (2010)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Esmaeili, E., Mahyar, M.: Weighted composition operators between vector-valued Lipschitz function spaces. Banach J. Math. Anal. 7, 59–72 (2013)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Font, J.J.: Automatic continuity of certain isomorphisms between regular Banach function algebras. Glasgow Math. J. 39(3), 333–343 (1997)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Font, J.J., Hernandez, S.: On separating maps between locally compact spaces. Arch. Math. 63, 158–165 (1994)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Gau, H.L., Jeang, J.S., Wong, N.C.: Biseparating linear maps between continuous vector-valued function spaces. J. Aust. Math. Soc. 74, 101–109 (2003)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gillman, L., Jerison, M.: Rings of Continuous Functions. D. Van Nostrand Co., New York (1960)

    Book  Google Scholar 

  10. 10.

    Goldmann, H.: Uniform Fréchet Algebras. North Holland, Amsterdam (1990)

    MATH  Google Scholar 

  11. 11.

    Hausner, A.: Ideals in a certain Banach algebra. Proc. Am. Math. Soc. 8, 246–249 (1957)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hernandez, S., Beckenstein, E., Narici, L.: Banach-Stone theorems and separating maps. Manuscr. Math. 86, 409–416 (1995)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Honary, T.G., Nikou, A., Sanatpour, A.H.: Disjointness preserving linear operators between Banach algebras of vector-valued functions. Banach J. Math. Anal. 8(2), 93–106 (2014)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hosseini, M., Sady, F.: Weighted composition operators on \(C(X)\) and \({{\rm Lip}}_c(X, \alpha )\). Tokyo J. Math. 35(1), 71–84 (2012)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Jeang, J.S., Wong, N.C.: Weighted composition operators of \(C_0(X)\)’s. J. Math. Anal. Appl. 201, 981–993 (1996)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Jiménez-Vargas, A., Wang, Ya-Shu: Linear biseparating maps between vector-valued little Lipschitz function spaces. Acta Math. Sin. (Engl. Ser.) 26(6), 1005–1018 (2010)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kantrowitz, R., Neumann, M.M.: Disjointness preserving and local operators on algebras of differentiable functions. Glasgow Math. J. 43(2), 295–309 (2001)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Kestelman, H.: Automorphisms of the feild of complex numbers. Proc. Lond. Math. Soc. 2(53), 1–12 (1951)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Laursen, K.B., Neumann, M.M.: An introduction to local spectral theory. Clarendson Press, Oxford (2000)

    MATH  Google Scholar 

  20. 20.

    Michael, E.: Locally multiplicatively-convex topological algebras. American Mathematical Society, USA (1952)

    Book  Google Scholar 

  21. 21.

    Miura, T.: A representation of ring homomorphisms on unital regular commutative Banach algebras. Math. J. Okayama Univ. 44, 143–153 (2002)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Pourghobadi, Z., Najafi Tavani, M., Sady, F.: Jointly separating maps between vector-valued function spaces. arXiv:1804.10915v1(preprint)

  23. 23.

    Takahasi, S.E., Hatori, O.: A structure of ring homomorphisms on commutative Banach algebras. Proc. Am. Math. Soc. 127(8), 2283–2288 (1999)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for his/her invaluable comments and suggestions.

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Correspondence to Fereshteh Sady.

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Communicated by Vesko Valov.

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Pourghobadi, Z., Sady, F. & Tavani, M.N. Zero product preserving functionals on \(C(\varOmega )\)-valued spaces of functions. Ann. Funct. Anal. 11, 459–472 (2020). https://doi.org/10.1007/s43034-019-00031-2

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Keywords

  • Zero product preserving functionals
  • Vector-valued spaces of functions
  • Ring homomorphisms

Mathematics Subject Classification

  • 47B38
  • 47B48
  • 46J10