Let B(H) be the set of bounded linear operators on a hilbert space H. Let F be a unital commutative subspace of B(H).
Give an example of a homomorphism h from F to the complex numbers such that h is surjective.
Now fix T in F. Show that for all w in the spectrum of T, there is a homomorphism h such that h(T)=w
For the first, I'm stumped. For the second I have reasoned thus far: h(T)=w implies h(Iw-T)=0 implies Iw-T is in the ker(h).