Voiculescu's double commutant theorem and the cohomology of -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 -algebras [5], we outlined a proof of the following result: a separable, unital -algebra has continuous trace if and only if all of its central cohomology groups for 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 -subalgebras of the Calkin algebra [7].

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DOI:
https://doi.org/10.1090/S0002-9939-1991-1039262-3

Article copyright:
© Copyright 1991
American Mathematical Society