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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Voiculescu's double commutant theorem and the cohomology of $ C\sp *$-algebras


Authors: John Phillips and Iain Raeburn
Journal: Proc. Amer. Math. Soc. 112 (1991), 139-142
MSC: Primary 46L80; Secondary 19K14, 46M20
MathSciNet review: 1039262
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Abstract: In a previous paper, on the central cohomology of $ {C^ * }$-algebras [5], we outlined a proof of the following result: a separable, unital $ {C^ * }$-algebra has continuous trace if and only if all of its central cohomology groups for $ n \geq 1$ vanish. Unfortunately, as was pointed out to us by Professors A. Ja. Helemskii and B. E. Johnson, the proof we outlined was incorrect. Our appeal to [3, Theorem 3.2] was invalid since the algebras we were interested in were not generally commutative. It is the purpose of this note to give a correct proof of this result as well as other interesting cohomological results. Our main tool will be D. Voiculescu's celebrated double commutant theorem for separable $ {C^ * }$-subalgebras of the Calkin algebra [7].


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1039262-3
PII: S 0002-9939(1991)1039262-3
Article copyright: © Copyright 1991 American Mathematical Society