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2014 Disjointness preserving linear operators between Banach algebras of vector-valued functions
Taher Ghasemi Honary, Azadeh Nikou, Amir Hossein Sanatpour
Banach J. Math. Anal. 8(2): 93-106 (2014). DOI: 10.15352/bjma/1396640054


We present vector-valued versions of two theorems due to A. Jimenez-Vargas, by showing that, if $B(X,E)$ and $B(Y,F)$ are certain vector-valued Banach algebras of continuous functions and $T:B(X,E)\to B(Y,F)$ is a separating linear operator, then $\widehat{T}:\widehat{B(X,E)}\to \widehat{B(Y,F)}$, defined by $\widehat{T}\hat{f}=\widehat{Tf}$, is a weighted composition operator, where $\widehat{Tf}$ is the Gelfand transform of $Tf$. Furthermore, it is shown that, under some conditions, every bijective separating map $T:B(X,E)\to B(Y,F)$ is biseparating and induces a homeomorphism between the character spaces $M(B(X,E))$ and $M(B(Y,F))$. In particular, a complete description of all biseparating, or disjointness preserving linear operators between certain vector-valued Lipschitz algebras is provided. In fact, under certain conditions, if the bijections $T:Lip^{\alpha}(X,E)\to Lip^{\alpha}(Y,F)$ and $T^{-1}$ are both disjointness preserving, then $T$ is a weighted composition operator in the form $Tf(y)=h(y)(f(\phi(y))),$ where $\phi$ is a homeomorphism from $Y$ onto $X$ and $h$ is a map from $Y$ into the set of all linear bijections from $E$ onto $F$. Moreover, if $T$ is multiplicative then $M(E)$ and $M(F)$ are homeomorphic.


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Taher Ghasemi Honary. Azadeh Nikou. Amir Hossein Sanatpour. "Disjointness preserving linear operators between Banach algebras of vector-valued functions." Banach J. Math. Anal. 8 (2) 93 - 106, 2014.


Published: 2014
First available in Project Euclid: 4 April 2014

zbMATH: 1308.47047
MathSciNet: MR3189541
Digital Object Identifier: 10.15352/bjma/1396640054

Primary: 47B38
Secondary: 46J10 , 47B33 , 47B48

Keywords: biseparating , disjointness preserving , maximal ideal space , separating , Vector-valued Banach algebra , vector-valued Lipschitz algebra

Rights: Copyright © 2014 Tusi Mathematical Research Group


Vol.8 • No. 2 • 2014
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