## Publicacions Matemàtiques

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

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

Maliheh Hosseini and Fereshteh Sady

#### 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