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 Künneth exact
sequence
0→K∗(A) ⊗ K∗(B)
→K∗(A ⊗ B)
→Tor(K∗(A),K∗(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 ⊗𝒦 such that the induced map K∗(F) → K∗(B) is surjective and K∗(F) is
free abelian.
