Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Embeddability of higher-rank graphs in groupoids, and the structure of their C*-algebras

Nathan Brownlowe, Alex Kumjian, David Pask, Aidan Sims

Abstract

We show that the C*-algebra of a row-finite source-free k-graph is Rieffel-Morita equivalent to a crossed product of an AF algebra by the fundamental group of the k-graph. When the k-graph embeds in its fundamental groupoid, this AF algebra is a Fell algebra; and simple-connectedness of a certain sub-1-graph characterises when this Fell algebra is Rieffel--Morita equivalent to a commutative C*-algebra. We provide a substantial suite of results for determining if a given k-graph embeds in its fundamental groupoid, and provide a large class of examples, arising via work of Cartwright, Robertson, Steger et al. from the theory of $\tilde{A_2}$-groups, that do embed.

Embeddability of higher-rank graphs in groupoids, and the structure of their C*-algebras

Abstract

We show that the C*-algebra of a row-finite source-free k-graph is Rieffel-Morita equivalent to a crossed product of an AF algebra by the fundamental group of the k-graph. When the k-graph embeds in its fundamental groupoid, this AF algebra is a Fell algebra; and simple-connectedness of a certain sub-1-graph characterises when this Fell algebra is Rieffel--Morita equivalent to a commutative C*-algebra. We provide a substantial suite of results for determining if a given k-graph embeds in its fundamental groupoid, and provide a large class of examples, arising via work of Cartwright, Robertson, Steger et al. from the theory of -groups, that do embed.
Paper Structure (15 sections, 32 theorems, 58 equations)

This paper contains 15 sections, 32 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Background and preliminary results
  3. Higher rank graphs
  4. Fundamental groupoids, fundamental groups and universal covers
  5. Simply connected k-graphs
  6. The path groupoid and the C*-algebraof a k-graph
  7. Embedding results for higher-rank graphs
  8. Non-embeddings
  9. Embedding singly connected higher-rank graphs
  10. More-general embedding results
  11. C*-algebraic results
  12. A_2-tilde-groups
  13. A_2-tilde-group basics
  14. The 2-graph associated to an A_2-tilde-group
  15. The covering 2-graph

Key Result

Proposition 2.7

Let $(\Lambda,d)$ be a $k$-graph and let $f : \mathbb{N}^\ell \to \mathbb{N}^k$ be an affine map with $f(0)=p \in \mathbb{N}^k$. Set $f^* (\Lambda ) = \{ ( \lambda , n ) : d ( \lambda ) = f(n) \} \subseteq \Lambda \times \mathbb{N}^\ell$. Then $f^*(\Lambda)$ is an $\ell$-graph, with structure maps and $d_{f^*(\Lambda)} (\lambda,n) = n$. We have $f^*(\Lambda)^0 = \{ ( \lambda, 0 ) : \lambda \in

Theorems & Definitions (109)

  • Definition 2.2
  • Example 2.4: Skew-product graphs
  • Example 2.5: Monoidal $2$-graphs
  • Remark 2.6
  • Proposition 2.7: Affine pullbacks
  • proof
  • Example 2.8: Crossed-product graph
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • ...and 99 more