Topological methods for<i>C</i>^∗-algebras. II. Geometry resolutions and the Künneth formula

Abstract

Let A and B be C*-algebras with A in the smallest subcategory of the category of separable nuclear C*-algebras which contains the separable Type I algebras and is closed under the operations of taking ideals, quotients, extensions, inductive limits, stable isomorphism, and crossed products by Z and by R. Then there is a natural Z/2-graded Knneth exact sequence 0 > K*{A) K*{B) > K*(A <g> B) > 0 Our proof uses the technique of geometric realization. The key fact is that given a unital C*-algebra B, there is a commutative C*-algebra F and an inclusion F- B(g) 3Z~ such that the induced map K*(F) -> K*{B) is sur jective and K*(F) is free abelian. l Introduction* Let A and B be C*-algebras. There is a Z/2graded pairing (defined in 2)