On the K-theory of higher rank graph C*-algebras

TL;DR

This paper investigates the K-theory of higher rank graph C*-algebras, providing explicit formulas for the case k=2 and exploring calculations and properties for higher ranks, including the case k=3.

Contribution

It offers explicit K-theory formulas for 2-graph C*-algebras and extends analysis to higher ranks, establishing relationships between K-groups and the unit class.

Findings

Explicit K-theory formulas for 2-graph C*-algebras

Conditions for calculating K-groups for k>2, especially k=3

Equality of torsion-free ranks of K_0 and K_1 for unital algebras

Abstract

Given a row-finite $k$-graph $\Lambda$ with no sources we investigate the $K$-theory of the higher rank graph $C^*$-algebra, $C^*(\Lambda)$. When $k=2$ we are able to give explicit formulae to calculate the $K$-groups of $C^*(\Lambda)$. The $K$-groups of $C^*(\Lambda)$ for $k>2$ can be calculated under certain circumstances and we consider the case $k=3$. We prove that for arbitrary $k$, the torsion-free rank of $K_0(C^*(\Lambda))$ and $K_1(C^*\Lambda))$ are equal when $C^*(\Lambda)$ is unital, and for $k=2$ we determine the position of the class of the unit of $C^*(\Lambda)$ in $K_0(C^*(\Lambda))$.