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

Abstract

Given a row-finite $k$-graph $\Lambda$ with no sources we investigate the $K$-theory of the higher rank graph $C^{\ast }$-algebra, $C^{\ast }(\Lambda )$. When $k=2$ we are able to give explicit formulae to calculate the $K$-groups of $C^{\ast }(\Lambda )$. The $K$-groups of $C^{\ast }(\Lambda )$ 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 $K_0(C^{\ast }(\Lambda ))$ and $K_1(C^{\ast }\Lambda ))$ are equal when $C^{\ast }(\Lambda )$ is unital, and for $k=2$ we determine the position of the class of the unit of $C^{\ast }(\Lambda )$ in $K_0(C^{\ast }(\Lambda ))$.

23 pages. To appear in the New York Journal of Mathematics (http://nyjm.albany.edu:8000/). Revisions include: a different numbering system for sections, theorems and related parts; correction of typographical errors; re-organisation of results; and addition of examples (Section 5)

References in corpus (3)

• Higher rank graph C*-algebras
• Simplicity of C*-algebras associated to higher-rank graphs
• Removing sources from higher-rank graphs

Cited by in corpus (20)

• Crossed products of k-graph C*-algebras by Z^l
• Type III von Neumann Algebras associated with ${Ø}_{\theta }$
• Topological higher-rank graphs and the $C^{\ast }$-algebras of topological 1-graphs
• Removing sources from higher-rank graphs
• Rank-two graphs whose C^*-algebras are direct limits of circle algebras
• Iterating the Cuntz-Nica-Pimsner construction for compactly aligned product systems
• Aperiodicity and the primitive ideal space of a row-finite $k$-graph $C^{\ast }$-algebra
• Homotopy of product systems and K-theory of Cuntz-Nica-Pimsner algebras
• Crossed products and twisted $k$-graph algebras
• $K$-theory for real $k$-graph $C^{\ast }$-algebras
• C*-algebras of higher-rank graphs from groups acting on buildings, and explicit computation of their K-theory
• C*-algebras associated to coverings of k-graphs
• On groupoids and $C^{\ast }$-algebras from self-similar actions
• Dirichlet forms and ultrametric Cantor sets associated to higher-rank graphs
• The K-theory of the C*-algebras of 2-rank graphs associated to complete bipartite graphs
• Computing the fundamental group of a higher-rank graph
• AF $C^{\ast }$-algebras from non AF groupoids
• Stably finite extensions of rank-two graph C*-algebras
• Simplicity of 2-graph algebras associated to Dynamical Systems
• C^*-algebras from k group representations