The answers to the question prime ideals in C([0,1]) explain why the answer is no, assuming that Yemon Choi was mistaken in his answer in believing that you are only interesting in continuous homomorphims.
In particular it is explained that there are non-maximal prime ideals in the ring $C([0,1])$. Given such a prime $P$, $C([0,1])/P$ will provide a non-trivial $W$.