Real and complex K-theory for higher rank graph algebras arising from cube complexes

TL;DR

This work computes both real and complex K-theory for infinite families of higher-rank graph C*-algebras arising from cube complexes, using the Evans spectral sequence and its real analogue to obtain KO and KU groups. By analyzing rank-3 and rank-4 graphs with and without a non-trivial involution, the authors express K^{CR} in terms of gcd-based invariants g,h,k and decompose the algebras into products of real Cuntz algebras, thereby resolving open questions and correcting prior results. The combined real/complex approach enables complete classification of the algebras up to isomorphism in many cases, demonstrating that isomorphism types depend only on the gcd data, and highlighting both the power and limitations of the spectral-sequence method when higher ranks induce additional extension and differential phenomena. The results advance the classification program for purely infinite simple C*-algebras and illustrate a concrete, computable bridge between cube-complex constructions, higher-rank graphs, and K-theory invariants, with implications for future computations in higher ranks.”

Abstract

Using the Evans spectral sequence and its counter-part for real $K$-theory, we compute both the real and complex $K$-theory of several infinite families of $C^*$-algebras based on higher-rank graphs of rank $3$ and $4$. The higher-rank graphs we consider arise from double-covers of cube complexes. By considering the real and complex $K$-theory together, we are able to carry these computations much further than might be possible considering complex $K$-theory alone. As these algebras are classified by $K$-theory, we are able to characterize the isomorphism classes of the graph algebras in terms of the combinatorial and number-theoretic properties of the construction ingredients.

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Theorems & Definitions (20)

• Proposition 3.1
• proof
• Remark 3.2
• Remark 3.3
• Lemma 3.4
• proof
• Proposition 4.1
• Remark 4.2
• Remark 4.3
• proof : Proof of Proposition \ref{['rank-3-involution']}