Coincidence of essential commutant and the double commutant relation in the Calkin algebra

Let B be a von Neumann algebra on a separable Hilbert space H. We show that, if the dimension of B as a linear space is infinite, then it has a proper C ∗ -subalgebra A whose essential commutant in B(H) coincides with the essential commutant of B. Moreover, if π is the quotient map from B(H) to the...

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Bibliographic Details
Published inJournal of functional analysis Vol. 197; no. 1; pp. 140 - 150
Main Author Xia, Jingbo
Format Journal Article
LanguageEnglish
Published Elsevier Inc 10.01.2003
Subjects
Calkin algebra
Double commutant relation
Essential commutant
Double commutant relation
Calkin algebra
Essential commutant
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Summary:Let B be a von Neumann algebra on a separable Hilbert space H. We show that, if the dimension of B as a linear space is infinite, then it has a proper C ∗ -subalgebra A whose essential commutant in B(H) coincides with the essential commutant of B. Moreover, if π is the quotient map from B(H) to the Calkin algebra B(H)/ K(H) , then π( A)≠ π( B) and { π( A)}″= π( B).
ISSN:0022-1236
1096-0783
DOI:10.1016/S0022-1236(02)00034-4