Tokyo Journal of Mathematics

Norm Conditions on Maps between Certain Subspaces of Continuous Functions

Razieh Sadat GHODRAT, Arya JAMSHIDI, and Fereshteh SADY

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For a locally compact Hausdorff space $X$, let $C_0(X)$ be the Banach space of continuous complex-valued functions on $X$ vanishing at infinity endowed with the supremum norm $\|\cdot\|_X$. We show that for locally compact Hausdorff spaces $X$ and $Y$ and certain (not necessarily closed) subspaces $A$ and $B$ of $C_0(X)$ and $C_0(Y)$, respectively, if $T:A \longrightarrow B$ is a surjective map satisfying one of the norm conditions i) $\|(Tf)^s (Tg)^t\|_Y=\|f^s g^t\|_X$, or ii) $\|\, |Tf|^s+|Tg|^t\,\|_Y=\|\, |f|^s+|g|^t\, \|_X$, \noindent for some $s,t\in \mathbb{N}$ and all $f,g\in A$, then there exists a homeomorphism $\varphi: \mathrm{ch}(B) \longrightarrow \mathrm{ch}(A)$ between the Choquet boundaries of $A$ and $B$ such that $|Tf(y)|=|f(\varphi(y))|$ for all $f\in A$ and $y\in \mathrm{ch}(B)$. We also give a result for the case where $A$ is closed (or, in general, satisfies a special property called Bishop's property) and $T:A \longrightarrow B$ is a surjective map satisfying the inclusion $R_\pi((Tf)^s (Tg)^t) \subseteq R_\pi(f^s g^t)$ of peripheral ranges. As an application, we characterize such maps between subspaces of the form $A_1f_1+A_2f_2+\cdots+A_nf_n$, where for each $1\le i \le n$, $A_i$ is a uniform algebra on a compact Hausdorff space $X$ and $f_i$ is a strictly positive continuous function on $X$. Our results in case (ii) improve similar results in~[30], for subspaces rather than uniform algebras, without the additional assumption that $T$ is $\mathbb{R}^+$-homogeneous.

Article information

Tokyo J. Math., Volume 40, Number 2 (2017), 421-437.

First available in Project Euclid: 9 January 2018

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B38: Operators on function spaces (general) 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
Secondary: 47B33: Composition operators


GHODRAT, Razieh Sadat; SADY, Fereshteh; JAMSHIDI, Arya. Norm Conditions on Maps between Certain Subspaces of Continuous Functions. Tokyo J. Math. 40 (2017), no. 2, 421--437.

Export citation


  • \BibAuthorsD. Blecher and K. Jarosz, Isomorphisms of function modules, and generalized approximation in modulus, Trans. Amer. Math. Soc. 354 (2002), 3663–3701.
  • \BibAuthorsA. Browder, Introduction to Function Algebras, W.A. Benjamin, 1969.
  • \BibAuthorsM. Burgos, A. Jimnez-Vargas and M. Villegas-Vallecillos, Nonlinear conditions for weighted composition operators between Lipschitz algebras, J. Math. Anal. Appl. 359 (2009), 1–14.
  • \BibAuthorsB. Cengiz, Extremely regular function spaces, Pacific J. Math. 49 (1973), 335–338.
  • \BibAuthorsT.W. Gamelin, Uniform Algebras, Prentice-Hall Inc., 1969.
  • \BibAuthorsO. Hatori, S. Lambert, A. Luttman, T. Miura, T. Tonev and R. Yates, Spectral preservers in commutative Banach algebras, Contemp. Math. 547 (2011), 103–123.
  • \BibAuthorsO. Hatori, T. Miura and H. Takagi, Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving properties, Proc. Amer. Math. Soc. 134 (2006), 2923–2930.
  • \BibAuthorsO. Hatori, K. Hino, T. Miura and H. Oka, Peripherally monomial-preserving maps between uniform algebras, Mediterr. J. Math. 6 (2009), 47–59.
  • \BibAuthorsO. Hatori, T. Miura, R. Shindo and H. Takagi, Generalizations of spectrally multiplicative surjections between uniform algebras, Rend. Circ. Mat. Palermo 59 (2010), 161–183.
  • \BibAuthorsM. Hosseini and F. Sady, Multiplicatively and non-symmetric multiplicatively norm-preserving maps, Cent. Eur. J. Math. 8 (2010), 878–889.
  • \BibAuthorsM. Hosseini and F. Sady, Multiplicatively range-preserving maps between Banach function algebra, J. Math. Anal. Appl. 357 (2009), 314–322.
  • \BibAuthorsA. Jamshidi and F. Sady, Extremely strong boundary points and real-linear isometries, Tokyo J. Math. 38 (2015), 477–490.
  • \BibAuthorsK. Jarosz and T.S.S.R.K. Rao, Weak*-extreme points of injective tensor product spaces, Contemp. Math. 328 (2003), 231–237.
  • \BibAuthorsJ. Johnson and T. Tonev, Spectral conditions for composition operators on algebras of functions, Commun. Math. Appl. 3 (2012), 51–59.
  • \BibAuthorsA. Jiménez-Vargas, A. Luttman and M. Villegas-Vallecillos, Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, Rocky Mountain J. Math. 40 (2010), 1903–1922.
  • \BibAuthorsH. Koshimizu, T. Miura, H. Takagi and S.-E. Takahasi, Real-linear isometries between subspaces of continuous functions, J. Math. Anal. Appl. 413 (2014), 229–241.
  • \BibAuthorsS. Kowalski and Z. Słodkowski, A characterization of multiplicative linear functionals in Banach algebras, Studia Math. 67 (1980), 215–223.
  • \BibAuthorsS. Lambert, A. Luttman and T. Tonev, Weakly peripherally-multiplicative mappings between uniform algebras, Contemp. Math. 435 (2007), 265–281.
  • \BibAuthorsK. de Leeuw, W. Rudin and J. Wermer, The isometries of some function spaces, Proc. Amer. Math. Soc. 11 (1960), 694–698.
  • \BibAuthorsG.M. Leibowitz, Lectures on Complex Function Algebras, Scott, Foresman and Co., Glenview, Ill., 1970.
  • \BibAuthorsA. Luttman and S. Lambert, Norm conditions for uniform algebra isomorphisms, Cent. Eur. J. Math. 6 (2008), 272–280.
  • \BibAuthorsA. Luttman and T. Tonev, Uniform algebra isomorphisms and peripheral multiplicativity, Proc. Amer. Math. Soc. 135 (2007), 3589–3598.
  • \BibAuthorsT. Miura and T. Tonev, Mappings onto multiplicative subsets of function algebras and spectral properties of their products, Ark. Mat. 53 (2015), 329–358.
  • \BibAuthorsL. Molnár, Some characterizations of the automorphisms of $B(H)$ and $C(X)$, Proc. Amer. Math. Soc. 133 (2001), 1135–1142.
  • \BibAuthorsM. Nagasawa, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kōdai Math. Semin. Rep. 11 (1959), 182–188.
  • \BibAuthorsN.V. Rao and A.K. Roy, Multiplicatively spectrum-preserving maps of function algebras, Proc. Amer. Math. Soc. 133 (2005), 1135–1142.
  • \BibAuthorsN.V. Rao and A.K. Roy, Multiplicatively spectrum-preserving maps of function algebras II, Proc. Edinb. Math. Soc. 48 (2005), 219–229.
  • \BibAuthorsN.V. Rao, T.V. Tonev and E.T. Toneva, Uniform algebra isomorphisms and peripheral spectra, Contemp. Math. 427 (2007), 401–416.
  • \BibAuthorsA.E. Taylor and D.C. Lay, Introduction to Functional Analysis, 2nd Ed., Wiley, New York, 1980.
  • \BibAuthorsT. Tonev and R. Yates, Norm-linear and norm-additive operators between uniform algebras, J. Math. Anal. Appl. 57 (2009), 45–53.