# Aperiodicity and cofinality for finitely aligned higher-rank graphs†

@article{Lewin2009AperiodicityAC,
title={Aperiodicity and cofinality for finitely aligned higher-rank graphs†},
author={Peter Lewin and Aidan Sims},
journal={Mathematical Proceedings of the Cambridge Philosophical Society},
year={2009},
volume={149},
pages={333 - 350}
}
• 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…
36 Citations
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
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
• Mathematics
Glasgow Mathematical Journal
• 2013
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
• Mathematics
• 2014
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
• Mathematics
• 2010
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 � (�),
• Mathematics, Computer Science
• 2012
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.
• Mathematics
• 2012
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
• Mathematics
Proceedings of the Edinburgh Mathematical Society
• 2013
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

## References

SHOWING 1-10 OF 39 REFERENCES

• Mathematics
• 2007
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
• Mathematics
• 2007
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
• Mathematics
• 1997
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
• Mathematics
• 2000
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
• Mathematics
• 1999
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
• Mathematics
• 1998
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
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
• Mathematics
Proceedings of the Edinburgh Mathematical Society
• 2003
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
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