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$K$-theory for real $k$-graph $C^*$-algebras (2108.09303v2)

Published 20 Aug 2021 in math.OA and math.KT

Abstract: We initiate the study of real $C*$-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 $\mathcal{CR}$ $K$-theory of $C*_{\mathbb R} (\Lambda, \gamma)$ for any involution $\gamma$ on $\Lambda$, and show that the $E2$ 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 $K{CR}(C*_{\mathbb R}(\Lambda, \gamma))$ for several examples of higher-rank graphs $\Lambda$ with involution.

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