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 <>