A C^{∗}-algebra that isn’t a von Neumann algebra.

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K(H), where H is a separable infinite dimensional Hilbert space is a C^{∗}- algebra but not a von Neumann algebra. In fact if (e_{n})_{n∈N} is a orthonormal basis for H and P_{n} = ∑_{i=1}^{n} e_{i}\overline{⊗}e_{i}, then P_{n} is a finite-rank projection converging strongly to the identity operator I (since for each x ∈ H, I(x) = x = ∑_{i=1}^{∞} < x, e_{i} > e_{i} = lim_{n} P_{n}(x)). If K(H) were a von-Neumann algebra, it should be I ∈ K(H), a contradiction.