A non-separable von Neumann algebra with a (unique) separable closed ∗-bi-ideal.

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Let H be a separable infinite dimensional Hilbert space and (x_{n}) be a dense sequence in H. Then K(H) which is the closed linear span of rank-one projections, is the closure of the linear span of x_{n}\overline{⊗}x_{n} with rational coefficients,hence it is separable. If (e_{n}) is an orthonormal basis for H and for each subset S of the natural numbers N,

P_{S}(e_{n}) = \{ \begin{matrix}e_{n} & n ∈ S \\ 0 & otherwise \end{matrix} ,

then ∥P_{S} − P_{S′}∥ = 1, for S ≠ S′. Thus {P_{S}}_{S∈2^{N}} cannot be in the closure of any countable sequence of B(H). Thus B(H) isn’t separable. Note that for x and y in H the rank one operator x\overline{⊗}y is defined by

(x\overline{⊗}y)(z) =< z, y > x.