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

DOI:
https://doi.org/10.1090/S0002-9939-96-03531-9

MathSciNet review:
1346963

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Abstract | References | Similar Articles | Additional Information

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 . We also consider the quotient -algebra with quotient map

For any -subalgebra of , the relative commutant is given by for all in . It was shown by D. Voiculescu that, for any *separable* unital -subalgebra of ,

In this note, we exhibit a *non-separable* unital -subalgebra of for which (VDCT) fails.

**[1]**C. A. Berger and L. A. Coburn,*Toeplitz operators and quantum mechanics*, J. Funct. Analysis**68**(1986) 273--299. MR**88b:46098****[2]**C. A. Berger and L. A. Coburn,*Toeplitz operators on the Segal-Bargmann space*, Trans. AMS**301**(1987) 813--829. MR**88c:47044****[3]**O. Bratteli and D. Robinson,*Operator algebras and quantum statistical mechanics*, II, Springer, 1981. MR**82k:82013****[4]**L. A. Coburn and J. Xia,*Toeplitz algebras and Rieffel deformations*, Comm. Math. Physics,**168**(1995) 23--38.**[5]**B. E. Johnson and S. K. Parrott,*Operators commuting with a von Neumann algebra modulo the set of compact operators*, J. Funct. Analysis**11**(1972) 39--61. MR**49:5869****[6]**S. Popa,*The commutant modulo the set of compact operators of a von Neumann algebra*, J. Funct. Analysis**71**(1987) 393--408. MR**88b:46091****[7]**D. Voiculescu,*A non-commutative Weyl-von Neumann theorem*, Rev. Roumaine Math. Pures Appl.**21**(1976), 97--113. MR**54:3427**

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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

DOI:
https://doi.org/10.1090/S0002-9939-96-03531-9

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