Download 1. Let A ⊂ B(H) be a ∗-algebra containing 1 ∈... closure of A in B(H) is a von Neumann algebra.

1. Let A ⊂ B(H) be a ∗-algebra containing 1 ∈ B(H). Show that the weak operator
closure of A in B(H) is a von Neumann algebra.
2. Let H be an infinite dimensional Hilbert space. Show that
(i) the multiplication map (x, y) 7→ xy is so-continuous on B(H)1 × B(H), but not socontinuous on B(H) × B(H)1 ;
(ii) the multiplication map is not wo-continuous on B(H)1 × B(H)1 ;
(iii) the map x 7→ x∗ is not so-continuous on B(H)1 ;
(iv) the map x 7→ x∗ is wo-continuous on B(H).
3. Show that a representation π : A → B(H) of a C∗ -algebra is nondegenerate if and only
if π(A)K(H) is dense in K(H).
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