A simple C^{∗}-algebra.

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K(H) is a simple C^{∗}-algebra. For if I is a nonzero closed bi-ideal of K(H), then it is a closed bi-ideal of B(H), so by [Mur, Th. 2.4.7] F (H) ⊆ I, hence K(H) = \overline{F (H)} ⊆ Ī = I. Therefore I = K(H).

Ref.

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