Aperiodicity and cofinality for finitely aligned higher-rank graphs†

  title={Aperiodicity and cofinality for finitely aligned higher-rank graphs†},
  author={Peter Lewin and Aidan Sims},
  journal={Mathematical Proceedings of the Cambridge Philosophical Society},
  pages={333 - 350}
  • P. LewinA. Sims
  • Published 6 May 2009
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
Abstract We introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs Λ, and prove that C*(Λ) is simple if and only if Λ is aperiodic and cofinal. The main advantage of our versions of aperiodicity and cofinality over existing ones is that ours are stated in terms of finite paths. To prove our main result, we first characterise each of aperiodicity and cofinality of Λ in terms of the ideal structure of C*(Λ). In an appendix we show how our new cofinality… 

Aperiodicity Conditions in Topological $k$-Graphs

We give two new conditions on topological $k$-graphs that are equivalent to the Yeend's aperiodicity Condition (A). Each of the new conditions concerns finite paths rather than infinite. We use a

Groupoids and C * -algebras for categories of paths

In this paper we describe a new method of defining C*-algebras from oriented combinatorial data, thereby generalizing the constructions of algebras from directed graphs, higher-rank graphs, and


Abstract We construct a representation of each finitely aligned aperiodic k-graph Λ on the Hilbert space $\mathcal{H}^{\rm ap}$ with basis indexed by aperiodic boundary paths in Λ. We show that the


We characterise simplicity of twisted C � -algebras of row-finite k-graphs with no sources. We show that each 2-cocycle on a cofinal k-graph determines a canonical second-cohomology class for the

Higher dimensional generalizations of the Thompson groups

Simplicity of 2-Graph Algebras Associated to Dynamical Systems

We give a combinatorial description of a family of 2-graphs which sub- sumes those described by Pask, Raeburn and Weaver. Each 2-graphwe consider has an associated C � -algebra, denoted C � (�),

Aperiodicity and primitive ideals of row-finite k-graphs

This work describes the primitive ideal space of the C*-algebra of a row-finite k-graph with no sources when every ideal is gauge invariant and proves some new results on aperiodicity.

Simplicity of algebras associated to étale groupoids

We prove that the full C∗-algebra of a second-countable, Hausdorff, étale, amenable groupoid is simple if and only if the groupoid is both topologically principal and minimal. We also show that if G

Remarks on some fundamental results about higher-rank graphs and their C*-algebras

Abstract Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the



Simplicity of C*‐algebras associated to higher‐rank graphs

We prove that if Λ is a row‐finite k‐graph with no sources, then the associated C*‐algebra is simple if and only if Λ is cofinal and satisfies Kumjian and Pask's aperiodicity condition, known as

Simplicity of C*-algebras associated to row-finite locally convex higher-rank graphs

In a previous work, the authors showed that the C*-algebra C*(Λ) of a row-finite higher-rank graph Λ with no sources is simple if and only if Λ is both cofinal and aperiodic. In this paper, we

Graphs, Groupoids, and Cuntz–Krieger Algebras

We associate to each locally finite directed graphGtwo locally compact groupoidsGandG(★). The unit space ofGis the space of one–sided infinite paths inG, andG(★) is the reduction ofGto the space of


NSKI Abstract. We prove versions of the fundamental theorems about Cuntz-Krieger algebras for the C -algebras of row-finite graphs: directed graphs in which each vertex emits at most finitely many

Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras

To an $r$-dimensional subshift of finite type satisfying certain special properties we associate a $C^*$-algebra $\cA$. This algebra is a higher rank version of a Cuntz-Krieger algebra. In


We associate to each row-nite directed graph E a universal Cuntz-Krieger C-algebra C(E), and study how the distribution of loops in E aects the structure of C(E) .W e prove that C(E) is AF if and

The uniqueness of Cuntz-Krieger type algebras

Abstract We introduce a class of C*-algebras which can be viewed as a generalization of the classical Cuntz-Krieger algebras. Our approach is based on a flexible “generators and relations”-concept.


It is shown that no local periodicity is equivalent to the aperiodicity condition for arbitrary nitely-aligned k-graphs. This allows us to conclude that C () is simple if and only if is conal and has


Abstract We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz–Krieger algebras. We describe a variant of the Cuntz–Krieger relations which applies to

Relative Cuntz-Krieger algebras of finitely aligned higher-rank graphs

We define the relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs. We prove versions of the gauge-invariant unique- ness theorem and the Cuntz-Krieger uniqueness theorem