1. Let A be a C∗ -algebra, a ∈ A a normal element. Show that
(i) if f ∈ C(SpA∼ (a)) then f (a) ∈ A if and only if f (0) = 0;
(ii) if A is unital then SpA∼ (a) = SpA (a) ∪ {0}, and if f ∈ C(SpA∼ (a)) and g ∈ C(SpA (a))
are such that f (0) = 0 and g = f |SpA (a) , then f (a) = g(a).
2. Show that the category of abelian C∗ -algebras with nondegenerate ∗-homomorphisms
as morphisms is dual to the category of locally compact spaces with proper continuous maps
as morphisms. Hint: C0 (X)∼ = C(X ∼ ), where X ∼ is the one-point compactification of X.
3. Show that for any two elements x and y in a unital algebra we have Sp(xy) ∪ {0} =
Sp(yx) ∪ {0}.
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