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].References
- Charles A. Akemann and Gert K. Pedersen, Central sequences and inner derivations of separable $C^{\ast }$-algebras, Amer. J. Math. 101 (1979), no. 5, 1047–1061. MR 546302, DOI 10.2307/2374125
- Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition. MR 0246136
- B. E. Johnson, Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972), 685–698. MR 317050, DOI 10.2307/2373751
- A. J. Lazar, S.-K. Tsui, and S. Wright, A cohomological characterization of finite-dimensional $C^\ast$-algebras, J. Operator Theory 14 (1985), no. 2, 239–247. MR 808290
- John Phillips and Iain Raeburn, Central cohomology of $C^{\ast }$-algebras, J. London Math. Soc. (2) 28 (1983), no. 2, 363–375. MR 713391, DOI 10.1112/jlms/s2-28.2.363
- John Phillips and Iain Raeburn, Perturbations of $C^{\ast }$-algebras. II, Proc. London Math. Soc. (3) 43 (1981), no. 1, 46–72. MR 623718, DOI 10.1112/plms/s3-43.1.46
- Dan Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 97–113. MR 415338
- Z. A. Lykova, On conditions for projectivity of Banach algebras of completely continuous operators, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (1979), 8–13, 99 (Russian, with English summary). MR 563024 —, The connection between the cohomological characterization of ${C^ * }$-algebras and their commutative ${C^ * }$-subalgebras, Ph.D. thesis, Moscow State University, 1985. (Russian)
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 112 (1991), 139-142
- MSC: Primary 46L80; Secondary 19K14, 46M20
- DOI: https://doi.org/10.1090/S0002-9939-1991-1039262-3
- MathSciNet review: 1039262