advertisement

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). 1