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 bi-ideal of B(H), we have F(H) Í I (cf. [Mur, Th. 2.4.7]). Hence K(H) Í I. If I Ë eq 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 Î 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^* Î 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.