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.
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