Banach Journal of Mathematical Analysis

Disjointness preserving linear operators between Banach algebras of vector-valued functions

Taher Ghasemi Honary, Azadeh Nikou, and Amir Hossein Sanatpour

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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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Banach J. Math. Anal., Volume 8, Number 2 (2014), 93-106.

First available in Project Euclid: 4 April 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B38: Operators on function spaces (general)
Secondary: 47B33: Composition operators 47B48: Operators on Banach algebras 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]

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


Ghasemi Honary, Taher; Nikou, Azadeh; Sanatpour, Amir Hossein. Disjointness preserving linear operators between Banach algebras of vector-valued functions. Banach J. Math. Anal. 8 (2014), no. 2, 93--106. doi:10.15352/bjma/1396640054.

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