Exact Sequences for the Kasparov Groups of Graded Algebras

Source abstract

In [ 11 ] G. G. Kasparov defined the “operator K -functor” KK(A, B) associated with the graded C *-algebras A and B . If the algebras A and B are trivially graded and A is nuclear he proves six term exact sequence theorems. He asks whether this extends to the graded case. Here we prove such “six-term exact sequence” results in the graded case. Our proof does not use nuclearity of the algebra A . This condition is replaced by a completely positive lifting condition (Theorem 1.1). Using our result we may extend the results by M. Pimsner and D. Voiculescu on the K groups of crossed products by free groups to KK groups [ 15 ]. We give however a different way of computing these groups using the equivariant KK -theory developed by G. G. Kasparov in [ 12 ]. This method also allows us to compute the KK groups of crossed products by PSL 2 ( Z ).