Abstract

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

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