The following is named the Künneth theorem for tensor products in the book by Blackadar on K-theory for operator algebras: If $A$ and $B$ are C*-algebras and $A$ is in the bootstrap class, then there is a natural short exact sequence of the form $$0 \to K_*(A) \otimes K_*(B) \xrightarrow{\alpha} K_*(A \otimes B) \xrightarrow{\sigma} \mathrm{Tor}_1^{\mathbb{Z}}(K_*(A),K_*(B)) \to 0$$ which splits unnaturally.
Question 1) Is there any reasonable description/formula for the map $\sigma$? Since I fear that the answer here is no, let me ask the following variant (which is actually what I need): Given $x \in K_*(A \otimes B)$, is there any reasonable way to decide whether $\sigma(x) = 0$? (And by "reasonable" I mean not "is in the image of $\alpha$", since this is actually the conclusion that I need.)
Question 2) The theorem states that the sequence splits (unnaturally), i.e. there exists a map $\alpha_{A,B}^{-1}$ splitting $\alpha$ (for any fixed choice of $A$ and $B$; I have put this in the notation to account for the unnaturality). The proof of this fact (as provided by Blackadar) does not give a concrete formula for it, so my question is whether one can describe this map somehow in a reasonable way?
