1. Show that if representations π1 : A → B(H1 ) and π2 : A → B(H2 ) are irreducible, then
the space of intertwiners is either zero or consists of scalar multiples of a unitary operator.
Hint: if T : H1 → H2 is an intertwiner, consider T T ∗ and T ∗ T .
2. Assume πi : A → B(Hi ), i = 1, . . . , n, are mutually inequivalent irreducible representations. Consider the representation π = π1 ⊕· · ·⊕πn . Show that π(A)00 = B(H1 )⊕· · ·⊕B(Hn ).
Prove then that for any f.d. Ki ⊂ Hi and Si ∈ B(Hi ), i = 1, . . . , n, there exists a ∈ A such
that πi (a)|Ki = Si |Ki for i = 1, . . . , n.
3. Complete the proof of Theorem 10.1.8 by proving the following result. Assume H =
⊕i∈I Hi and A ⊂ ⊕i∈I K(Hi ). Denote by πi the representation of A on Hi , so πi (a) = a|Hi .
Assume the representations πi are irreducbile and mutually inequivalent. Show that A =
⊕i∈I K(Hi ).
4. Assume A = ⊕i∈I Ai , and π : A → B(H) is an irreducible representation. Prove that
there exists i0 ∈ I such that π|Ai = 0 for all i 6= i0 .
1