Equivariant KK-theory Research Articles

Exact Sequences for the Kasparov Groups of Graded Algebras

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 PSL2(Z).