The Fine Structure of the Kasparov Groups I Continuity of the KK-Pairing, by C. L. Schochet

In this paper it is demonstrated that the Kasparov KK-pairing is continuous with respect to the Salinas topology, so that a KK-equivalence is a homeomorphism. This result is then applied to strengthen the Universal Coefficient Theorem for KK-theory. We show that KK_*(A,B) decomposes to three canonical groups as topological groups. Each of these groups is determined (as topological groups) by the groups K_*(A) and K_*(B).

Next, the special case of K^*(A) = KK_*(A,\Bbb C) is considered. We compute this topological group in terms of K_*(A). A typical result: if A is in the bootstrap category and if K_*(A) is a torsion group, then K^*(A) is homeomorphic to the Pontryagin dual of K_*(A). Finally, we show that the canonical short exact sequence

0 \to K_*(A)_t \to K_*(A) \to K_*(A)_f \to 0 (*)

formed by taking torsion subgroup and torsionfree quotient may be geometrically realized at the level of C^*-algebras. That is, there is a short exact sequence of C^*-algebras

0 \to A\otimes \Cal K \to A_f \to SA_t \to 0

whose associated K_* long exact sequence realizes (*). This and the UCT imply that if A is in the bootstrap category then there is a short exact sequence of topological groups

0 \to K^*(A_f) \to K^*(A) \to K^*(A_t) \to 0

and similarly for KK_*(A,B) if K_*(B) is torsionfree.


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