A primitive C^{∗}-algebra with a unique nontrivial closed bi-ideal (and so that it is not simple).

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Let H be a separable infinite dimensional Hilbert space and A = B(H). Then K(H) is a nontrivial closed bi-ideal of B(H), and if I is a nontrivial closed biideal of B(H), we have F (H) ⊆ I (cf. [Mur, Th. 2.4.7]). Hence K(H) ⊆ I. If I ⊈ K(H), then I has an infinite-rank projection p (cf. [Mur, Cor. 4.1.14]). For each infinite-rank projection q, there exist u ∈ B(H) such that p = u^{∗}u and q = uu^{∗} (if (e_{n}) and (f_{n}) are orthonormal basis for p(H) and q(H) resp., define u(e_{n}) = f_{n} and u = 0 on p(H)^{⊥}) so q = upu^{∗} ∈ I. Hence I = B(H), a contradiction. Since B(H)′ = C1 (For (C1)′ = B(H) and this is because of (C1)” = C1), the identity representation B(H) ⟶ B(H) is a faithful irreducible representation. Hence B(H) is primitive.

Ref.

[Mur] G.J. Murphy, C^{∗}-algebras and operator theory, Academic Press, 1990.