\(K\)-theory for the crossed products by certain actions of \(\mathbb Z^2\)
It is well known that, for an action \(\alpha\) of \(\mathbb Z\) on a \(C^*\)-algebra \(A\), the \(K\)-theory groups of the crossed product \(A\rtimes_\alpha\mathbb Z\) can be evaluated (up to a group extension problem) in terms of the action \(\alpha_*\) on \(K_*(A)\) using the Pimsner-Voiculescu exact sequence, but the case of an action of a more general group (even of \(\mathbb Z^2\)) is more difficult. The author gives a similar solution for an action \(\alpha\) of \(\mathbb Z^2\). In this case, the information about the action \(\alpha_*\) on \(K_*(A)\) is not enough, so the author introduces an obstruction homomorphism \(d_*(\alpha)\). Let \(\alpha^1\), \(\alpha^2\) be the restrictions of \(\alpha\) onto the first and the second copy of \(\mathbb Z\) in \(\mathbb Z^2\). Let \(\text{id}\) denote the identity homomorphism on \(K_*(A)\). Set \(S_*(\alpha)=\text{Ker}(\alpha^1_*-\text{id})\cap\text{Ker}(\alpha^2_*-\text{id})\), \(T_*(\alpha)= K_*(A)/(\text{Im}(\alpha^1_*-\text{id})+\text{Im}(\alpha^2_*-\text{id}))\). The obstruction homomorphism \(d_*(\alpha):S_*(\alpha)\to T_*(\alpha)\) is defined using the snake lemma. It is shown that \(K_*(A\rtimes_\alpha\mathbb Z^2)\) is determined (up to a group extension problem) by \(K_*(A)\), \(\alpha^1_*\), \(\alpha^2_*\) and \(d_*(\alpha)\). The author provides several examples of \(\mathbb Z^2\)-actions with nontrivial obstruction homomorphisms. In particular, the obstruction homomorphism may be nontrivial even if the action is pointwise inner, i.e., if \(\alpha_g\) is inner for any \(g\in\mathbb Z^2\), hence acts trivially on \(K_*(A)\). As an application, the \(K\)-theory groups are computed for several classes of crossed products by \(\mathbb Z^2\).
scientific article; zbMATH DE number 6685591
scientific article