Universal Karoubi's characteristic classes of nuclear $C^*$-algebras

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.