# HIGHER-RANK GRAPHS AND THEIR $C^*$-ALGEBRAS

@article{Raeburn2001HIGHERRANKGA,
title={HIGHER-RANK GRAPHS AND THEIR \$C^*\$-ALGEBRAS},
author={Iain Raeburn and Aidan Sims and Trent Yeend},
journal={Proceedings of the Edinburgh Mathematical Society},
year={2001},
volume={46},
pages={99 - 115}
}
• Published 31 July 2001
• Mathematics
• Proceedings of the Edinburgh Mathematical Society
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 graphs with sources, and describe a local convexity condition which characterizes the higher-rank graphs that admit a non-trivial Cuntz–Krieger family. We then prove versions of the uniqueness theorems and classifications of ideals for the $C^*$-algebras generated by Cuntz–Krieger families. AMS 2000…
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## References

SHOWING 1-10 OF 15 REFERENCES

• 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
• 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
Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C *-algebra, C * (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C
• 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
• 1994
Let r+ be the positive cone in a totally ordered abelian group F. We construct crossed products by actions of r1" as endomorphisms of C- algebras, and give criteria which ensure a given
• 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
• Mathematics
• 2001
We build upon Mac Lane's definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid G. We show that the existing notions of product systems
• Mathematics
• 1980
In this paper we present a class of C*-algebras and point out its close relationship to topological Markov chains, whose theory is part of symbolic dynamics. The C*-algebra construction starts from a
0. Since their invention about twenty years ago Cuntz-Krieger algebras OA corresponding to finite, square 0–1 matrices [3] have attracted immense interest. It was remarked in their original paper by
• Mathematics
Abstract Let $\Gamma$ be a torsion free lattice in $G=\text{PGL}\left( 3,\mathbb{F} \right)$ where $\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits