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1. Let p and q be projections in a unital C∗ -algebra. Show that p + q ≤ 1 if and only if p and q are orthogonal. 2. Let n ≥ 2 and Gn be the closed linear span of elements Tµ Tν∗ ∈ Tn such that |µ| = |ν|. Find the Bratteli diagram of the AF-algebra Gn . 3. Show that if f is a continuous function on the real line then for any element x in a unital C∗ -algebra we have xf (x∗ x) = f (xx∗ )x. Check next that if kxk ≤ 1 then x (1 − xx∗ )1/2 −(1 − x∗ x)1/2 x∗ is a unitary. 4. Show that if 0→I→A→B→0 is a short exact sequence of C -algebras and C is a C∗ -algebra such that the algebraic tensor product B C has a unique C∗ -norm (e.g. either B or C is abelian), then the sequence ∗ 0→I ⊗C →A⊗C →B⊗C →0 is exact. 1