Geodesic
Universal Karoubi's characteristic classes of nuclear $C^*$-algebras
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 8, pp. 133-169.

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The main result of this paper is the evaluation of kernels for the Chern character and the universal Karoubi classes of nuclear $C^*$-algebras. It is shown that the kernel of the Chern character coincides with the subgroup of infinitely small elements of the $K_0$-group and the kernel of the universal Karoubi classes coincides with the subgroup of approximately scalar elements of the $K_0$-group. 
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I. M. Nikonov. Universal Karoubi's characteristic classes of nuclear $C^*$-algebras. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 8, pp. 133-169. http://geodesic.mathdoc.fr/item/FPM_2007_13_8_a9/

[1] Vinberg E. B., Lektsii po algebre, MTsNMO, M., 1995

[2] Zhuraev Yu. I., Kharakteristicheskie klassy modulei nad nekommutativnymi algebrami, Dis. $\dots$ kand. fiz.-mat. nauk, M., 1987

[3] Zhuraev Yu. I., Mischenko A. S., Solovëv Yu. P., “O kharakteristicheskikh klassakh v algebraicheskoi $K$-teorii”, Tiraspolskii simpozium po obschei topologii i eë prilozheniyam, Shtiintsa, Kishinëv, 1985, 91–92

[4] Zhuraev Yu. I., Mischenko A. S., Solovëv Yu. P., “O kharakteristicheskikh klassakh v algebraicheskoi $K$-teorii”, Vestn. Mosk. un-ta. Ser. 1, Matematika, mekhanika, 1986, no. 1, 75–76 | MR | Zbl

[5] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, T. 1, Nauka, M., 1981

[6] Korneeva E. V., Kharakteristicheskie klassy v nekommutativnoi differentsialnoi geometrii, Dis. $\dots$ kand. fiz.-mat. nauk, M., 2003 | Zbl

[7] Lyumis L., Vvedenie v abstraktnyi garmonicheskii analiz, Izd. inostr. lit., M., 1956

[8] Mërfi Dzh., $C^*$-algebry i teoriya operatorov, Faktorial, M., 1997

[9] Milnor Dzh., Algebraicheskaya $K$-teoriya, Mir, M., 1974 | Zbl

[10] Nikonov I. M., “Primer netrivialnogo kharakteristicheskogo klassa gruppovoi algebry $\mathbb C[\mathbb Z_3]$”, Vestn. Mosk. un-ta. Ser. 1, Matematika, mekhanika, 2002, no. 4, 58–60 | MR | Zbl

[11] Nikonov I. M., Yadro kharaktera Karubi dlya poluprostykh algebr, Dep. v VINITI 31.10.2003, No 1896-V2003

[12] Fuks D. B., Kogomologii beskonechnomernykh algebr Li, Nauka, M., 1984 | MR | Zbl

[13] Khelemskii A. Ya., Gomologiya v banakhovykh i topologicheskikh algebrakh, Izd-vo Mosk. un-ta, M., 1986 | MR

[14] Blackadar B., $K$-Theory for Operator Algebras, 2nd edition, Cambridge Univ. Press, 1998 | MR | Zbl

[15] Blackadar B., Rørdam M., “Extending states on preordered semigroups and the existence of quasitraces on $C^*$-algebras”, J. Algebra, 152 (1992), 240–247 | DOI | MR | Zbl

[16] Bratteli O., “Inductive limits of finite dimensional $C^*$-algebras”, Trans. Amer. Math. Soc., 171 (1972), 195–234 | DOI | MR | Zbl

[17] Connes A., Noncommutative Geometry, Academic Press, 1994 | MR | Zbl

[18] Connes A., “On the cohomology of operator algebras”, J. Funct. Anal., 28:2 (1978), 248–253 | DOI | MR | Zbl

[19] Davidson K. R., $C^*$-Algebras by Example, Fields Institute Monographs Ser., 6, Amer. Math. Soc., 1996 | MR | Zbl

[20] Dixmier J., Von Neumann Algebras, North-Holland, Amsterdam, 1981 | MR | Zbl

[21] Dubois-Violette M., Lectures on graded differential algebras and noncommutative geometry, arXiv:math/9912017[math.QA] | MR

[22] Effros E. G., Handelman D. E., Shen C.-L., “Dimension groups and their affine representations”, Amer. J. Math., 102:2 (1980), 385–407 | DOI | MR | Zbl

[23] Effros E. G., Shen C.-L., “Approximately finite $C^*$-algebras and continued fractions”, Indiana Univ. Math. J., 29:2 (1980), 191–204 | DOI | MR | Zbl

[24] Haagerup U., “All nuclear $C^*$-algebras are amenable”, Invent. Math., 74:2 (1983), 305–319 | DOI | MR | Zbl

[25] Haagerup U., Quasitraces on exact $C^*$-algebras are traces, Preprint, 1991

[26] Haagerup U., Thorbjørnsen S., “Random matrices and $K$-theory for exact $C^*$-algebras”, Doc. Math., J. DMV, 4 (1999), 341–450 | MR | Zbl

[27] Karoubi M., Homologie cyclique et $K$-théorie, Astérisque, 149, Soc. Math. France, Paris, 1987 | MR | Zbl

[28] Landi G., An introduction to noncommutative spaces and their geometry, arXiv:hep-th/9701078

[29] Loday J.-L., Cyclic Homology, Grundlehren Math. Wiss., 301, Springer, Berlin, 1992 | MR | Zbl

[30] Popelensky F. Yu., Solovyov Yu. P., “Lie–Cartan pairs and characteristic classes in noncommutative geometry”, Contemporary Geometry and Related Topics, World Scientific, 2004, 351–373 | MR | Zbl