Advertisement

Positivity

, Volume 23, Issue 1, pp 111–123 | Cite as

Generalized norm preserving maps between subsets of continuous functions

  • Bagher Jafarzadeh
  • Fereshteh SadyEmail author
Article
  • 37 Downloads

Abstract

Let X and Y be locally compact Hausdorff spaces. In this paper we study surjections \(T: A \longrightarrow B\) between certain subsets A and B of \(C_0(X)\) and \(C_0(Y)\), respectively, satisfying the norm condition \(\Vert \varphi (Tf, Tg)\Vert _Y=\Vert \varphi (f,g)\Vert _X\), \(f,g \in A\), for some continuous function \(\varphi : {\mathbb {C}}\times {\mathbb {C}}\longrightarrow {\mathbb {R}}^+\). Here \(\Vert \cdot \Vert _X\) and \(\Vert \cdot \Vert _Y\) denote the supremum norms on \(C_0(X)\) and \(C_0(Y)\), respectively. We show that if A and B are (positive parts of) subspaces or multiplicative subsets, then T is a composition operator (in modulus) inducing a homeomorphism between strong boundary points of A and B. Our results generalize the recent results concerning multiplicatively norm preserving maps, as well as, norm additive in modulus maps between function algebras to more general cases.

Keywords

Function algebras positive cone Choquet boundaries Weighted composition operators Norm preserving 

Mathematics Subject Classification

Primary 47B38 46J10 Secondary 47B33 

Notes

Acknowledgements

The authors would like to thank the referee for his/her helpful comments.

References

  1. 1.
    Ghodrat, R.S., Sady, F., Jamshidi, A.: Norm conditions on maps between certain subspaces of continuous functions. Tokyo J. Math. 40, 421–437 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Hatori, O., Lambert, S., Luttman, A., Miura, T., Tonev, T., Yates, R.: Spectral preservers in commutative Banach algebras. Contemp. Math. 547, 103–123 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hatori, O., Miura, T., Takagi, H.: Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving properties. Proc. Am. Math. Soc. 134, 2923–2930 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hatori, O., Hino, K., Miura, T., Oka, H.: Peripherally monomial-preserving maps between uniform algebras. Mediterr. J. Math. 6, 47–59 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hatori, O., Miura, T., Shindo, R., Takagi, H.: Generalizations of spectrally multiplicative surjections between uniform algebras. Rend. Circ. Mat. Palermo 59, 161–183 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hosseini, M., Sady, F.: Multiplicatively range-preserving maps between Banach function algebras. J. Math. Anal. Appl. 357, 314–322 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jamshidi, A., Sady, F.: Extremely strong boundary points and real-linear isometries. Tokyo J. Math. 38, 477–490 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lambert, S., Luttman, A., Tonev, T.: Weakly peripherally-multiplicative mappings between uniform algebras. Contemp. Math. 435, 265–281 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Leibowitz, G.M.: Lectures on Complex Function Algebras. Scott, Foresman and Co., Glenview, Ill (1970)zbMATHGoogle Scholar
  10. 10.
    Miura, T., Tonev, T.: Mappings onto multiplicative subsets of function algebras and spectral properties of their products. Ark. Mat. 53, 329–358 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Molnár, L.: Some characterizations of the automorphisms of \(B(H)\) and \(C(X)\). Proc. Am. Math. Soc. 130, 111–120 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Molnár, L., Szokol, P.: Transformations preserving norms of means of positive operators and nonnegative functions. Integr. Equ. Oper. Theory 83, 271–290 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rao, N.V., Roy, A.K.: Multiplicatively spectrum-preserving maps of function algebras. Proc. Am. Math. Soc. 133, 1135–1142 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rao, N.V., Roy, A.K.: Multiplicatively spectrum-preserving maps of function algebras II. Proc. Edinb. Math. Soc. 48, 219–229 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis, 2nd edn. Wiley, New York (1980)zbMATHGoogle Scholar
  16. 16.
    Tonev, T., Yates, R.: Norm-linear and norm-additive operators between uniform algebras. J. Math. Anal. Appl. 57, 45–53 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Faculty of Mathematical SciencesTarbiat Modares UniversityTehranIran

Personalised recommendations