Additive Local Multiplications And Zero-preserving Maps On C(X)

Suppose X is a compact Hausdorff space.In this paper,We mainly study that from what topological properties X satisfies,we can deduce that any additive local multiplication on C(X)is a multiplication,and from what topological properties X satisfies,we can deduce that any additive zero-preserving map on CR(X)is a multiplication.In terms of topological properties of X,we find topological conditions on X that are equivalent to each of the following:1.every additive local multiplication on C(X)is a multiplication,2.every additive local multiplication on CR(X)is a multiplication,and 3.every additive map T on C(X)that is zero-preserving(i.e.,f(x)=0 implies(Tf)(x)=0.)has the form T(f)=T(1)Re f+T(i)Im f.