A non-unital C^{∗}-algebra with compact primitive ideal space.

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If H is an infinite dimensional Hilbert space, then the non-unital C^{∗}-algebra K(H) is simple. By (CW17), (K(H))′ = C1, so the identity representation

K(H) ⟶ B(H)
T ⟼ T

is a faitful irreducible representation, hence {0} is a primitive ideal of K(H). By (CW19), K(H) is simple, so primitive ideal space of K(H) is {{0}}, a compact space.