A 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 (en)n Î N is a orthonormal basis for H and Pn = åi = 1n ei[`(Ä)] ei, then Pn is a finite-rank projection converging strongly to the identity operator I (since for each x Î H, I(x) = x = åi = 1¥ < x,ei > ei = limn Pn(x)). If K(H) were a von-Neumann algebra, it should be I Î K(H), a contradiction.


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On 22 Feb 2001, 00:19.