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1. Let X be a compact space and A a unital C∗ -algebra. Show that there exists a unique C -norm on the algebraic tensor product C(X) A, and C(X) ⊗ A can be identified with the algebra C(X, A) of continuous functions f : X → A with norm kf k = maxx∈X kf (x)k. Hint: if π is an irreducible representation of C(X) A then π(C(X)) = C1; conclude that any pure state on C(X) A is a tensor product of pure states. 2. Let A be a unital C∗ -algebra, B ∼ = Matn (C) a subalgebra of A containing the unit of A. Let {eij }i,j be matrix units of B. Show that A ∼ = B ⊗ e11 Ae11 . ∗ 3. Find the Bratteli diagram of the even CAR-algebra and conclude that the algebra is isomorphic to the CAR-algebra. 4. Let A = ∪n An be an AF-algebra, p ∈ A1 a minimal projection in A1 . Find necessary and sufficient conditions on the Bratteli diagram for p to be abelian in A. 1