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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].

References [Enhancements On Off] (What's this?)

  • [1] C. A. Akemann and G. K. Pedersen, Central sequences and inner derivations of separable $ {C^ * }$-algebras, Amer. J. Math. 101 (1979), 1047-1061. MR 546302 (80m:46051)
  • [2] J. Dixmier, Les $ {C^ * }$-algebres et leurs représentations, Gauthier-Villars, Paris, 1969. MR 0246136 (39:7442)
  • [3] B. E. Johnson, Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972), 685-698. MR 0317050 (47:5598)
  • [4] A. J. Lazar, S.-K. Tsui, and S. Wright, A cohomological characterization of finite-dimensional $ {C^ * }$-algebras, J. Operator Theory 14 (1985), 239-247. MR 808290 (87c:46066)
  • [5] J. Phillips and I. Raeburn, Central cohomology of $ {C^ * }$-algebras, J. London Math. Soc. 28 (1983), 363-375. MR 713391 (85d:46095)
  • [6] -, Perturbations of $ {C^ * }$-algebras II, Proc. London Math. Soc. 43 (1981), 46-72. MR 623718 (83a:46066)
  • [7] D. Voiculescu, A noncommutative Weyl-von Neumann theorem, Rev. Roum. Math. Pures et Appl. 21 (1976), 97-113. MR 0415338 (54:3427)
  • [8] Z. A. Lykova, The structure of Banach algebras with zero central bidimension, preprint, 1989. (Russian) MR 563024 (81m:46098)
  • [9] -, The connection between the cohomological characterization of $ {C^ * }$-algebras and their commutative $ {C^ * }$-subalgebras, Ph.D. thesis, Moscow State University, 1985. (Russian)

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Article copyright: © Copyright 1991 American Mathematical Society

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