1. Let X be a compact space, A = C(X). For an ideal I put
K(I) = {t ∈ X | f (t) = 0 for all f ∈ I}.
It is a closed subset of X. For a closed subset K of X put
I(K) = {f ∈ A | f |K = 0}.
It is an ideal in A.
(i) Show that K(I(K)) = K and I(K(I)) = I. Thus the map I 7→ K(I) is a bijection
between ideals in A and closed subsets of X, and the inverse map is K 7→ I(K).
In particular, maximal ideals correspond to one-point sets. In other words, any character
of A has the form χt , χt (f ) = f (t), for some point t ∈ X. Therefore the map X → Â,
t 7→ χt , is a homeomorphism.
(ii) Let B = C(Y ) for a compact space Y . Show that any unital homomorphism A → B
has the form f 7→ f ◦ T , where T : Y → X is a continuous map.
1