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. 
@article{FPM_2007_13_8_a9,
     author = {I. M. Nikonov},
     title = {Universal {Karoubi's} characteristic classes of nuclear $C^*$-algebras},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {133--169},
     publisher = {mathdoc},
     volume = {13},
     number = {8},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2007_13_8_a9/}
}
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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/