Tokyo Journal of Mathematics

Weighted Composition Operators on $C(X)$ and $\mathrm{Lip}_c(X,\alpha)$

Abstract

Let $A$ and $B$ be subalgebras of $C(X)$ and $C(Y)$, respectively, for some topological spaces $X$ and $Y$. An arbitrary map $T: A\rightarrow B$ is said to be multiplicatively range-preserving if for every $f,g\in A$, $(fg)(X)=(TfTg)(Y)$, and $T$ is said to be separating if $TfTg=0$ whenever $fg=0$. For a given metric space $X$ and $\alpha\in (0,1]$, let Lip$_c(X,\alpha)$ be the algebra of all complex-valued functions on $X$ satisfying the Lipschitz condition of order $\alpha$ on each compact subset of $X$. In this note we first investigate the general form of multiplicatively range-preserving maps from $C(X)$ onto $C(Y)$ for realcompact spaces $X$ and $Y$ (not necessarily compact or locally compact) and then we consider such preserving maps from Lip$_c(X, \alpha)$ onto Lip$_c(Y,\beta)$ for metric spaces $X$ and $Y$ and $\alpha, \beta\in (0,1]$. We show that in both cases multiplicatively range-preserving maps are weighted composition operators which induce homeomorphisms between $X$ and $Y$. We also give a description of a linear separating map $T: A\rightarrow C(Y)$, where $A$ is either $C(X)$ for a normal space $X$ or Lip$_c(X,\alpha)$ for a metric space $X$ and $0<\alpha\le1$ and $Y$ is an arbitrary Hausdorff space.

Article information

Source
Tokyo J. Math., Volume 35, Number 1 (2012), 71-84.

Dates
First available in Project Euclid: 19 July 2012

https://projecteuclid.org/euclid.tjm/1342701345

Digital Object Identifier
doi:10.3836/tjm/1342701345

Mathematical Reviews number (MathSciNet)
MR2945985

Zentralblatt MATH identifier
1252.47032

Citation

HOSSEINI, Maliheh; SADY, Fereshteh. Weighted Composition Operators on $C(X)$ and $\mathrm{Lip}_c(X,\alpha)$. Tokyo J. Math. 35 (2012), no. 1, 71--84. doi:10.3836/tjm/1342701345. https://projecteuclid.org/euclid.tjm/1342701345

References

• J. Araujo, E. Beckenestein and L. Narici, Biseparating maps and homeomorphic real-compactifications, J. Math. Anal. Appl. 192 (1995), 258–265.
• J. Araujo and L. Dubarbie, Biseparating maps between Lipschitz function spaces, J. Math. Anal. Appl. 357 (2009), 191–200.
• J. Araujo and K. Jarosz, Automatic continuity of biseparating maps, Studia Math. 155 (2003), 231–239.
• M. A. Chebotar, W. F. Ke, P. H. Lee and N. C. Wong, Mappings preserving zero products, Studia Math. 155 (2003), 77–94.
• H.G. Dales, Banach algebras and Automatic Continuity, Clarendon Press, Oxford, 2000.
• J. J. Font, Automatic continuity of certain isomorphisms between regular Banach function algebras, Glasgow Math. J. 39 (1997), 333–343.
• H. Goldmann, Uniform Fréchet algebras, North-Holland, Amsterdam, 1990.
• O. 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.
• O. Hatori, T. Miura and H. Takagi, Unital and multiplicatively spectrum preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative, J. Math. Anal. Appl. 326 (2007), 281–269.
• O. Hatori, T. Miura, H. Oka and H. Takagi, Peripheral multiplicativity of maps on uniformly closed algebras of continuous functions which vanish at infinity, Tokyo J. Math. 32 (2009), 91–104.
• M. Hosseini and F. Sady, Multiplicatively range-preserving maps between Banach function algebras, J. Math. Anal. Appl. 357 (2009), 314–322.
• K. Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull. 33 (1990), 139–144.
• A. Jiménez-Vargas, A. Luttman and M. Villegas-Vallecillos, Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, to appear in Rocky Mountain J. Math.
• A. Jiménez-Vargas and M. Villegas-Vallecillos, Lipschitz algebras and peripherally-multiplicative maps, Acta Math. Sin. (Engl. Ser.) 24 (2008), 1233–1242.
• A. Jiménez-Vargas, M. Villegas-Vallecillos and Y.-S. Wang, Banach-Stone theorems for vector-valued little Lipschitz functions, Publ. Math. Debrecen 74 (2009), 81–100.
• R. Kantrowitz and M. M. Neumann, Disjointness preserving and local operators on algebras of differentiable functions, Glasgow Math. J. 43 (2001), 295–309.
• A. Luttman and T. Tonev, Uniform algebra isomorphisms and peripheral multiplicativity, Proc. Amer. Math. Soc. 135 (2007), 3589–3598.
• L. Molnár, Some characterizations of the automorphisms of $B(H)$ and $C(X)$, Proc. Amer. Math. Soc. 130 (2002), 111–120.
• N. V. Rao and A. K. Roy, Multiplicatively spectrum-preserving maps of function algebras, Proc. Amer. Math. Soc. 133 (2005), 1135–1142.
• N. V. Rao and A. K. Roy, Multiplicatively spectrum-preserving maps of function algebras (II), Proc. Edinb. Math. Soc. 48 (2005), 219–229.
• M. D. Weir, Hewitt-Nachbin Spaces, North-Holland, Amsterdam, 1975.