On Voiculescu's double commutant theorem

For a separable infinite-dimensional Hilbert space H, we consider the full algebra of bounded linear transformations B(H) and the unique nontrivial norm-closed two-sided ideal of compact operators K. We also consider the quotient C*-algebra C = B(H)/K. with quotient map π: B(H)→C. For A any C*-subalgebra of C, the relative commutant is given by A′ = {C ε C: CA = AC for all A in .A}. It was shown by D. Voiculescu that, for A. any separable unital C*-subalgebra of C, (VDCT) A″ = A. In this note, we exhibit a non-separable unital C*-subalgebra A₀ of C for which (VDCT) fails.