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Common zeros preserving maps on vector-valued function spaces and Banach modules

Maliheh Hosseini and Fereshteh Sady

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Abstract

Let $X$, $Y$ be Hausdorff topological spaces, and let $E$ and $F$ be Hausdorff topological vector spaces. For certain subspaces $A(X, E)$ and $A(Y,F)$ of $C(X,E)$ and $C(Y,F)$ respectively (including the spaces of Lipschitz functions), we characterize surjections $S,T\colon A(X,E) \rightarrow A(Y,F)$, not assumed to be linear, which jointly preserve common zeros in the sense that $Z(f-f') \cap Z(g-g') \neq \emptyset$ if and only if $Z(Sf-Sf') \cap Z(Tg-Tg') \neq \emptyset$ for all $f,f',g,g'\in A(X,E)$. Here $Z(\cdot)$ denotes the zero set of a function. Using the notion of point multipliers we extend the notion of zero set for the elements of a Banach module and give a representation for surjective linear maps which jointly preserve common zeros in module case.

Article information

Source
Publ. Mat., Volume 60, Number 2 (2016), 565-582.

Dates
Received: 16 March 2015
Revised: 22 October 2015
First available in Project Euclid: 11 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.pm/1468242045

Digital Object Identifier
doi:10.5565/PUBLMAT_60216_10

Mathematical Reviews number (MathSciNet)
MR3521501

Zentralblatt MATH identifier
1358.46048

Subjects
Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25] 47B48: Operators on Banach algebras
Secondary: 46J20: Ideals, maximal ideals, boundaries

Keywords
Non-vanishing functions Banach modules maps preserving common zeros vector-valued continuous function point multipliers zero set

Citation

Hosseini, Maliheh; Sady, Fereshteh. Common zeros preserving maps on vector-valued function spaces and Banach modules. Publ. Mat. 60 (2016), no. 2, 565--582. doi:10.5565/PUBLMAT_60216_10. https://projecteuclid.org/euclid.pm/1468242045


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