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On Voiculescu's double commutant theorem

Authors:	C A Berger L A Coburn

Institution:	Department of Mathematics and Computer Science, Herbert H. Lehman College, City University of New York, Bronx, New York 10468 L. A. Coburn ; Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14214

Abstract:	For a separable infinite-dimensional Hilbert space , we consider the full algebra of bounded linear transformations and the unique non-trivial norm-closed two-sided ideal of compact operators $\mathcal K$ . We also consider the quotient -algebra $\mathcal C=B(H)/\mathcal K$ with quotient map $\begin{displaymath}\pi \colon B(H)\to \mathcal C.\end{displaymath}$ For $\mathcal A$ any -subalgebra of $\mathcal C$ , the relative commutant is given by $\mathcal A'=\{C\in \mathcal C\colon CA=AC$ for all in $\mathcal A\}$ . It was shown by D. Voiculescu that, for $\mathcal A$ any separable unital -subalgebra of $\mathcal C$ , $\begin{equation}\mathcal A'=\mathcal A.\tag {VDCT} \end{equation}$ In this note, we exhibit a non-separable unital -subalgebra $\mathcal A_0$ of $\mathcal C$ for which (VDCT) fails.

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