### The UCT, the Milnor sequence, and a canonical decomposition of the Kasparov groups, by Claude L. Schochet

Suppose that A is a C^*-algebra in the bootstrap category with KK-filtration
{A_i} and B is a C^*-algebra with a countable approximate unit. Then the
graded Kasparov group KK_*(A,B) is described both by the Universal
Coefficient Theorem and by the Milnor Lim^1 sequence. It is demonstrated that
these two descriptions are closely related and that KK_*(A,B) decomposes
unnaturally as the direct sum of the term

Hom_Z (K_*(A), K_*(B) )

which stores index information, the term

Lim Ext_Z (K_*(A_i), K_*(B) )

which is the Z-adic completion of Ext_Z (K_*(A), K_*(B) ), and the term

Lim^1 Hom_Z (K_*(A_i), K_*(B) )

which houses the fine structure of KK_*(A,B) . (These groups depend only
upon K_*(A) and K_*(B).) Further, the Milnor sequence itself splits
unnaturally. Applications are given related to work of Dadarlat-Loring,
Rordam, and Salinas.

This paper has appeared in K-Theory, volume 10, pages 49-72.

Claude L. Schochet <claude@math.wayne.edu>