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On the spectral sequence associated with the Baum-Connes Conjecture for $\mathbb Z^n$

We examine a spectral sequence that is naturally associated with the Baum-Connes Conjecture with coefficients for $\mathbb Z^n$ and also constitutes an instance of Kasparov's construction in his work on equivariant $KK$-theory. For $k\leq n$, we give a partial description of the $k$-th page differential of this spectral sequence, which takes into account the natural $\mathbb Z^k$-subactions. In the special case that the action is trivial in $K$-theory, the associated second page differential is given by a formula involving the second page differentials of the canonical $\mathbb Z^2$-subactions. For $n=2$, we give a concrete realisation of the second page differential in terms of Bott elements. We prove the existence of $\mathbb Z^2$-actions, whose associated second page differentials are non-trivial. One class of examples is given by certain outer $\mathbb Z^2$-actions on Kirchberg algebras, which act trivially on $KK$-theory. This relies on a classification result by Izumi and Matui. A second class of examples consists of certain pointwise inner $\mathbb Z^2$-actions. One instance is given as a natural action on the group $\mathrm{C}^*$-algebra of the discrete Heisenberg group $H_3$. We also compute the $K$-theory of the corresponding crossed product. Moreover, a general and concrete construction yields various examples of pointwise inner $\mathbb Z^2$-actions on amalgamated free product $\mathrm{C}^*$-algebras with non-trivial second page differentials. Among these, there are actions which are universal, in a suitable sense, for pointwise inner $\mathbb Z^2$-actions with non-trivial second page differentials. We also compute the $K$-theory of the crossed products associated with these universal $\mathrm{C}^*$-dynamical systems.