1. Let {xn }∞
n=1 be a sequence of nonnegative real numbers such that xn+m ≤ xn xm for all
1/n
n, m ∈ N. Show that the sequence {xn }n has finite limit.
2. Let ϕ be a positive linear functional on a (not necessarily unital) C∗ -algebra A. Show
that ϕ is bounded.
3. Let X be a compact space, A = C(X).
(i) Assume I is a closed ideal in A. Put K = {t ∈ X | f (t) = 0 for all f ∈ I}. Show that
I = {f ∈ C(X) | f |K ≡ 0}.
Prove that I is maximal if and only if K consists of one point.
(ii) Let MA be the character space of A. Define a map X → MA , t 7→ χt , by letting
χt (f ) = f (t) for f ∈ C(X). Show that this map is a homeomorphism of X onto MA .
(iii) Let Y be a compact space, B = C(Y ), and π : A → B a unital homomorphism. Show
that there exists a continuous map T : Y → X such that π(f ) = f ◦ T for f ∈ C(X).
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