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

Abstract

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