Let $\alpha$ be the action of $\mathbf{Z}_2$ on $\mathbf{S}^1$ (the circle seen in the complex plane) which sends $z$ to $\bar{z}$. Consider the crossed product $A = C(\mathbf{S}^1)\rtimes_{\alpha} \mathbf{Z}_2$. Let $B$ be another C*-algebar such that $K_*(B)$ is torsion-free. Does the following hold?: \begin{align} K_0(A\otimes B) \simeq (K_0(A)\otimes K_0(B))\oplus(K_1(A)\otimes K_1(B))\\ K_1(A\otimes B) \simeq (K_0(A)\otimes K_1(B))\oplus(K_1(A)\otimes K_0(B)) \end{align} Said another way: Can I use the Künneth formula to compute $K_*(A\otimes B)$ given the $A$ described above?
2026-03-25 11:00:59.1774436459
$C(\mathbf{S}^1)\rtimes_{\alpha} \mathbf{Z}_2$ in Künneth formula
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