Abstract
Suppose that A and B are C*-algebras, A is separable, and B is stable. The elements of the group E1(A, B) in Connes—Higson E-theory are represented by *-homomorphisms from the suspension of A to the asymptotic algebra 21B. In the paper, an endofunctor M in the category of C* -algebras is constructed and a set of special homotopy classes of *-homomorphisms from A to MUB is defined so that this set endowed with the natural structure of an Abelian group coincides with E1(A,B).
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References
J. Cuntz, “A new look at K K-Theory,” K-Theory 1 (1), 31–51 (1987).
A. Connes and N. Higson, “Déformations, morphismes asymptotiques et K-théorie bivariante,” C. R. Acad. Sci. Paris Sér. I Math. 311 (2), 101–106 (s).
V. M. Manuilov, “Symmetrization of Cuntz’ picture for the Kasparov KK-bifunctor,” Funktsional. Anal. Prilozhen. 52 (3), 32–41 (2018) [Functional Anal. Appl. 52 (3), 186-193 (2018)].
E. Guentner, N. Higson, and J. Trout, Equivariant E-Theory for C*-Algebras, in Mem. Amer. Math. Soc. (Providence, RI, Amer. Math. Soc, 2000), Vol. 148, No. 703.
B. Blackadar, K-Theory for Operator Algebras, in Math. Sci. Res. Inst. Publ. (Cambridge Univ. Press, Cambridge, 1998), Vol. 5.
M. Dădărlat and T. A. Foring, “K-homology, asymptotic representations, and unsuspended E-theory,” J. Fund Anal. 126 (2), 367–383 (1994).
E. C. Fance, Hilbert C* -Modules. A Toolkit for Operator Algebraists, in London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 1995), Vol. 210.
K. K. Jensen and K. Thomsen, Elements of KK-Theory (Birkhäuser Boston, Boston, MA, 1991).
N. Higson and J. Roe, “On the coarse Baum—Connes conjecture,” London Math. Soc. Lecture Note Ser., Vol. 227: Novikov Conjectures, Index Theorems and Rigidity (Cambridge Univ. Press, Cambridge, 1995), Vol. 2, pp. 227–254.
N. Higson and J. Roe, Analytic K-Homology (Oxford Univ. Press, Oxford, 2000).
R. G. Bartle and F. M. Graves, “Mappings between function spaces,” Trans. Amer. Math. Soc. 72 (3), 400–413 (1952).
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Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 4, pp. 561–574.
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Makeev, G.S. Yet Another Description of the Connes—Higson Functor. Math Notes 107, 628–638 (2020). https://doi.org/10.1134/S000143462003030X
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DOI: https://doi.org/10.1134/S000143462003030X