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 \longmapsto 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.