1. Show that the direct sum of all representations defined by pure states is faithful.
2. Show that if a net of projections eλ converges strongly to the identity, then for any
compact operator T we have kT − T eλ k → 0.
3. Let H be a Hilbert space and K ⊂ H a closed subspace. Assume u is an operator such
that u|K is an isometry and u|K ⊥ = 0 (such operators are called partial isometries). Show
that u∗ is such that u∗ |uK is the inverse of u : K → uK and u∗ |(uK)⊥ = 0.
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