Geodesic
Symmetrization of Cuntz' Picture for the Kasparov $KK$-Bifunctor
Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 3, pp. 32-41.

Voir la notice de l'article provenant de la source Math-Net.Ru

Given $C^*$-algebras $A$ and $B$, we generalize the notion of a quasi-homomorphism from $A$ to $B$ in the sense of Cuntz by considering quasi-homomorphisms from some $C^*$-algebra $C$ to $B$ such that $C$ surjects onto $A$ and the two maps forming the quasi-homomorphism agree on the kernel of this surjection. Under an additional assumption, the group of homotopy classes of such generalized quasi-homomorphisms coincides with $KK(A, B)$. This makes the definition of the Kasparov bifunctor slightly more symmetric and provides more flexibility in constructing elements of $KK$-groups. These generalized quasi-homomorphisms can be viewed as pairs of maps directly from $A$ (instead of various $C$'s), but these maps need not be $*$-homomorphisms.
Keywords: $C^*$-algebra, Kasparov's $KK$-bifunctor
Mots-clés : quasi-homomorphism. 
@article{FAA_2018_52_3_a2,
     author = {V. M. Manuilov},
     title = {Symmetrization of {Cuntz'} {Picture} for the {Kasparov} $KK${-Bifunctor}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {32--41},
     publisher = {mathdoc},
     volume = {52},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2018_52_3_a2/}
}
TY  - JOUR
AU  - V. M. Manuilov
TI  - Symmetrization of Cuntz' Picture for the Kasparov $KK$-Bifunctor
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2018
SP  - 32
EP  - 41
VL  - 52
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2018_52_3_a2/
LA  - ru
ID  - FAA_2018_52_3_a2
ER  - 
%0 Journal Article
%A V. M. Manuilov
%T Symmetrization of Cuntz' Picture for the Kasparov $KK$-Bifunctor
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2018
%P 32-41
%V 52
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2018_52_3_a2/
%G ru
%F FAA_2018_52_3_a2
V. M. Manuilov. Symmetrization of Cuntz' Picture for the Kasparov $KK$-Bifunctor. Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 3, pp. 32-41. http://geodesic.mathdoc.fr/item/FAA_2018_52_3_a2/

[1] R. G. Bartle, L. M. Graves, “Mappings between function spaces”, Trans. Amer. Math. Soc., 72 (1952), 400–413 | DOI | MR | Zbl

[2] J. Cuntz, “A new look at $KK$-theory”, $K$-theory, 1:1 (1987), 31–51 | DOI | MR | Zbl

[3] J. Cuntz, “A general construction of bivariant $K$-theories on the category of $C^*$-algebras”, Operator algebras and operator theory, Contemp. Math., 228, Amer. Math. Soc., Providence, RI, 1998, 31–43 | DOI | MR | Zbl

[4] B. Magajna, “Hilbert $C^*$-modules in which all closed submodules are complemented”, Proc. Amer. Math. Soc., 125:3 (1997), 849–852 | DOI | MR | Zbl

[5] V. Manuilov, A $KK$-like picture for $E$-theory of $C^*$-algebras, arXiv: 1703.10781

[6] G. K. Pedersen, “Pullback and pushout constructions in $C^*$-algebra theory”, J. Funct. Anal., 167:2 (1999), 243–344 | DOI | MR | Zbl

[7] K. Thomsen, “Homotopy classes of $*$-homomorphisms between stable $C^*$-algebras and their multiplier algebras”, Duke Math. J., 61:1 (1990), 67–104 | DOI | MR | Zbl