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Proceedings of the American Mathematical Society

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On Voiculescu's double commutant theorem

Authors: C. A. Berger and L. A. Coburn
Journal: Proc. Amer. Math. Soc. 124 (1996), 3453-3457
MSC (1991): Primary 47L05
MathSciNet review: 1346963
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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

\begin{displaymath}\pi \colon B(H)\to \mathcal C.\end{displaymath}

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.

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Additional Information

C. A. Berger
Affiliation: Department of Mathematics and Computer Science, Herbert H. Lehman College, City University of New York, Bronx, New York 10468

L. A. Coburn
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14214

Received by editor(s): May 30, 1995
Additional Notes: This research was partially supported by NSF grant 9500716
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society