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.
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