Voiculescu’s double commutant theorem and the cohomology of 𝐶*-algebras

Phillips, John; Raeburn, Iain

doi:10.1090/S0002-9939-1991-1039262-3

Voiculescu’s double commutant theorem and the cohomology of $C^ *$-algebras
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by John Phillips and Iain Raeburn PDF

Proc. Amer. Math. Soc. 112 (1991), 139-142 Request permission

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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The connection between the cohomological characterization of

-algebras and their commutative

-subalgebras