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

TL;DR

This paper develops a spectral sequence approach to compute the real K-theory of higher-rank graph C*-algebras, linking algebraic invariants to combinatorial graph data, and provides explicit calculations for specific examples.

Contribution

It introduces a spectral sequence framework for calculating the real K-theory of higher-rank graph C*-algebras with involution, connecting algebraic and combinatorial structures.

Findings

Spectral sequence computes real K-theory from graph data.

Explicit K-theory calculations for certain higher-rank graphs.

Framework generalizes previous methods to real C*-algebras with involution.

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 $E^2$ 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 $K^{CR}(C^*_{\mathbb R}(\Lambda, \gamma))$ for several examples of higher-rank graphs $\Lambda$ with involution.