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Equivalence of the C*-Algebras qℂ and C 0(ℝ2) in the Asymptotic Category

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The results of Kasparov, Connes, Higson, and Loring imply the coincidence of the functors [[qℂ ⊗ K, B ⊗ K]] = [[C 0(ℝ2) ⊗ K, B ⊗ K]] for any C*-algebra B; here[[A, B]] denotes the set of homotopy classes of asymptotic homomorphisms from A to B. Inthe paper, this assertion is strengthened; namely, it is shown that the algebras qℂ ⊗ K and C 0(ℝ2) ⊗ K are equivalent in the category whose objects are C*-algebras and morphisms are classes of homotopic asymptotic homomorphisms. Some geometric properties of the obtained equivalence are studied. Namely, the algebras qℂ ⊗ K and C 0(ℝ2) ⊗ K are represented as fields of C*-algebras; it is proved that the equivalence is not fiber-preserving, i.e., is does not take fibers to fibers. It is also proved that the algebras under consideration are not homotopy equivalent.

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Authors and Affiliations

  1. V. A. Steklov Mathematics Institute, Russian Academy of Sciences, Moscow, Russia

    T. V. Shul’man

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  1. T. V. Shul’man
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Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 788–796.

Original Russian Text Copyright ©2005 by T. V. Shul’man.

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Shul’man, T.V. Equivalence of the C*-Algebras qℂ and C 0(ℝ2) in the Asymptotic Category. Math Notes 77, 726–734 (2005). https://doi.org/10.1007/s11006-005-0073-4

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  • DOI: https://doi.org/10.1007/s11006-005-0073-4

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