On Voiculescu’s double commutant theorem

Abstract:

For a separable infinite-dimensional Hilbert space $H$, we consider the full algebra of bounded linear transformations $B(H)$ and the unique non-trivial norm-closed two-sided ideal of compact operators $\mathcal K$. We also consider the quotient $C^*$-algebra $\mathcal C=B(H)/\mathcal K$ with quotient map \[ \pi \colon B(H)\to \mathcal C.\] For $\mathcal A$ any $C^*$-subalgebra of $\mathcal C$, the relative commutant is given by $\mathcal A’=\{C\in \mathcal C\colon CA=AC$ for all $A$ in $\mathcal A\}$. It was shown by D. Voiculescu that, for $\mathcal A$ any separable unital $C^*$-subalgebra of $\mathcal C$, \begin{equation*} \mathcal A''=\mathcal A.\tag {VDCT} \end{equation*}

In this note, we exhibit a non-separable unital $C^*$-subalgebra $\mathcal A_0$ of $\mathcal C$ for which (VDCT) fails.