# Additive Local Multiplications and zero-preserving maps on $C(X)$

## TL;DR

This paper characterizes topological conditions on compact Hausdorff spaces that determine when additive local multiplications on continuous function spaces are actual multiplications, and describes zero-preserving maps in terms of real and imaginary parts.

## Contribution

It establishes topological criteria for when additive local multiplications are genuine multiplications and characterizes zero-preserving maps on $C(X)$.

## Key findings

- Topological conditions equivalent to all additive local multiplications being multiplications.
- Characterization of zero-preserving maps as linear combinations of real and imaginary parts.
- Conditions under which additive maps on $C(X)$ are necessarily of a specific form.

## Abstract

Suppose $X$ is a compact Hausdorff space. In terms of topolocical properties of $X$, we find topological conditions on $X$ that are equivalent to each of the following: 1. every additive local multiplication on $C\left( X\right) $ is a multiplication, 2. every additive local multiplication on $C_{R}\left( X\right) $ is a multiplication, and 3. every additive map on $C\left( X\right) $ that is zero-preserving (i.e., $f\left( x\right) =0$ implies $\left( Tf\right) \left( x\right) =0$) has the form $T\left( f\right) =T\left( 1\right) \operatorname{Re}f+T\left( i\right) \operatorname{Im}f$.