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

By Iain Raeburn, Aidan Sims, Trent Yeend. Proceedings of the Edinburgh Mathematical Society, 2001

153 citations recorded by the source corpus.

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 Mathematics subject classification: Primary 46L05

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/568a8555366a3d0de928c83005d93bf2b9c655ce

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Aidan Sims
2. Trent Yeend
3. M. Tomforde
4. L. O. Clark
5. Iain Raeburn
6. G. Pino
7. Astrid an Huef
8. John Clark

In Mathematical Physics

1. Aidan Sims
2. Trent Yeend
3. M. Tomforde
4. L. O. Clark
5. Iain Raeburn
6. G. Pino
7. Astrid an Huef
8. John Clark

Citations in the corpus

• Groupoid models for the C*-algebras of topological higher-rank graphs (2006, background): “In recent years there has been significant interest in different generalizations of Cuntz-Krieger algebras, including the C-algebras of higher-rank graphs [12, 25, 26, 6] and the C-algebras of topological graphs [10, 11].”
• On the K-theory of higher rank graph C ∗ -algebras (2004, background): “For further examples of k-graphs see, for example, [13, 20, 21, 29].”
• A groupoid generalisation of Leavitt path algebras (2014, background): “Versions of these theorems have been established for many generalisations of Cuntz-Krieger algebras [6, 9, 11, 12, 13, 19, 20].”
• C*-algebras associated to product systems of hilbert bimodules (2007, background): “Xp do not form a product system; rather, they play a rˆole similar to that played by the sets �≤m for a locally convex k-graph in [ 29 ].”
• Kumjian-Pask algebras of higher-rank graphs (2011, background): “We find the finite path formulation of aperiodicity from [32] easier to understand, and it allows us to borrow arguments from [17] which do not require readers to know about the different formulations in [21] and [30].”