Integral Domain and $C^*$-algebra

Question

If $A$ is a commutative $C^*$-algebra with identity, then can it be an integral domain? I think the answer is no, because one can use the Gelfand Transform to get an isomorphism of $A$ with $C(\Omega (A))$ where $\Omega(A)$ is the space of non-trivial complex algebra homomorphisms of $A$. But I cannot show that $C(\Omega(A))$ cannot be an integral domain if $|\Omega(A)|$ $\geq$ $2$. Any other method is also welcome. Thanks for any help.

Answer 1

Suppose $X$ is a normal topological space with more than two points. Then there exist two nonzero functions $f,g\colon X \rightarrow [0,1]$ with $fg=0.$

Proof: Take two distinct points $x,y\in X$ and separate them by opens $U$ and $V$. Then $x$ and $X\backslash U$ are disjoint closed sets and thus there is a continuous function $f:X\rightarrow [0,1]$ with $f(x)=1$ and $f(z)=0$ for $z\in X\backslash U$ (Urysohn's Lemma). Similarly construct a function $g:X\rightarrow [0,1]$ such that $g(y) = 1 $ and $g(z) = 0 $ for $z\in X\backslash V$. Since each $z\in X$ either lies in $X\backslash U$ or in $X\backslash V$ we have $fg=0$.