$K$-theory for real $k$-graph $C^*$-algebras

We initiate the study of real $C^*$-algebras associated to higher-rank graphs $Λ$, with a focus on their $K$-theory. Following Kasparov and Evans, we identify a spectral sequence which computes the $\mathcal{CR}$ $K$-theory of $C^*_{\mathbb R} (Λ, γ)$ for any involution $γ$ on $Λ$, and show that the $E^2$ page of this spectral sequence can be straightforwardly computed from the combinatorial data of the $k$-graph $Λ$ and the involution $γ$. We provide a complete description of $K^{CR}(C^*_{\mathbb R}(Λ, γ))$ for several examples of higher-rank graphs $Λ$ with involution.

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