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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).

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Gelfand-Mazur...and "subspace" should be "subalgebra". –  Michael Feb 19 '14 at 2:47

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