Disjointness preserving maps between vector-valued group algebras

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Abstract

Let G be a locally compact abelian group and B be a commutative Banach algebra. Let \(L^{1}(G, B)\) be the Banach algebra of B-valued Bochner integrable functions on G. In this paper we provide a complete description of continuous disjointness preserving maps on \(L^{1}(G, B)\)-algebras based on a scarcely used tool: the vector-valued Fourier transform. We also present necessary and sufficient conditions for these operators to be compact.

Keywords

Locally compact abelian group Vector-valued group algebras Disjointness preserving mappings 

Mathematics Subject Classification

Primary 47B38 Secondary 43A20 43A22 43A25 

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References

  1. 1.
    Abramovich, Y.: Multiplicative representation of disjointness preserving operators. Indag. Math. 45, 265–279 (1983)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Abramovich, Y., Veksler, A.I., Koldunov, A.V.: On operators preserving disjointness. Soviet Math. Dokl. 248, 1033–1036 (1983)MATHGoogle Scholar
  3. 3.
    Araujo, J.: Separating maps and linear isometries between some spaces of continuous functions. J. Math. Anal. Appl. 226, 23–39 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Arendt, W.: Spectral properties of Lamperti operators. Indiana Univ. Math. J. 32, 199–215 (1983)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Arendt, W., Hart, D.R.: The spectrum of quasi-invertible disjointness preserving operators. J. Funct. Anal. 68, 149–167 (1986)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chan, J.T.: Operators with disjoint support property. J. Oper. Theory 24(2), 383–391 (1990)MathSciNetMATHGoogle Scholar
  7. 7.
    de Pagter, B.: A note on disjointness preserving operators. Proc. Am. Math. Soc. 90, 543–549 (1984)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Font, J.J., Hernández, S.: Separating maps between locally compact spaces. Arch. Math. (Basel) 63, 158–165 (1994)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Font, J.J., Hernández, S.: Automatic continuity and representation of certain linear isomorphisms between group algebras. Indag. Math. 6(4), 397–409 (1995)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hausner, A.: On a homomorphism between generalized group algebras. Bull. Am. Math. Soc. 67, 138–141 (1961)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis II. Springer, New York (1970)MATHGoogle Scholar
  13. 13.
    Huijsmans, C., de Pagter, B.: Invertible disjointness preserving operators. Proc. Edinb. Math. Soc. 37, 125–132 (1993)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jamison, J.E., Rajagopalan, M.: Weighted composition operator on \(C(X, E)\). J. Oper. Theory 19(2), 307–317 (1988)MATHGoogle Scholar
  16. 16.
    Jarosz, K.: Automatic continuity of separating linear isomorphisms. Canad. Math. Bull. 33(2), 139–144 (1990)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Laursen, K.B., Neumann, M.: Introduction to Local Spectral Theory. Oxford University Press, Oxford (2000)MATHGoogle Scholar
  18. 18.
    Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  19. 19.
    Munkres, J.R.: Topology: a First Course. Prentice-Hall Inc, Englewood Cliffs, N.J. (1975)MATHGoogle Scholar
  20. 20.
    Rudin, W.: Fourier Analysis on Groups. Wiley-Interscience, New York (1962)MATHGoogle Scholar
  21. 21.
    Ruess, W.M., Summers, W.H.: Compactness in spaces of vector-valued continuous functions and asymptotic almost periodicity. Math. Nachr. 135, 7–33 (1988)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Tewari, U.B., Dutta, M., Vaidya, D.P.: Multipliers of group algebras of vector-valued functions. Proc. Am. Math. Soc. 81, 223–229 (1981)MathSciNetCrossRefMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of MathematicsK. N. Toosi University of TechnologyTehranIran
  2. 2.Departamento de Matemáticas (IMAC)Universitat Jaume ICastellónSpain

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