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

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 $T\left(f\right)=T\left(1\right)\mathrm{Re}f+T\left(i\right)\mathrm{Im}f$.