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

By Guyan Robertson, T. Steger. 2013

160 citations recorded by the source corpus.

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 particular, it is simple, purely infinite and nuclear. We study an example: if $\G$ is a group acting freely on the vertices of an $\wt A_2$ building, with finitely many orbits, and if $\Omega$ is the boundary of that building, then $C(\Om)\rtimes \G$ is the algebra associated to a certain two dimensional subshift.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cd347015bf22b2ea1dcf2ff483f20b69e256778f

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In Algebra and Number Theory

1. Aidan Sims
2. Trent Yeend
3. L. O. Clark
4. M. Tomforde
5. Cynthia Farthing
6. P. Muhly
7. Ben Steinberg
8. N. Fowler

In Geometry and Topology

1. Aidan Sims
2. Trent Yeend
3. L. O. Clark
4. P. Muhly
5. M. Tomforde
6. Ben Steinberg
7. Efren Ruiz
8. Toke Meier Carlsen

In Mathematical Physics

1. Aidan Sims
2. Trent Yeend
3. L. O. Clark
4. M. Tomforde
5. P. Muhly
6. Cynthia Farthing
7. Ben Steinberg
8. N. Fowler

Citations in the corpus

• HIGHER-RANK GRAPHS AND THEIR $C^*$-ALGEBRAS (2001, unknown): “Keywords higher, graphs, their, algebras, c, rank Disciplines Engineering | Science and Technology Studies Publication Details Raeburn, I., Sims, A. & Yeend, T. (2003).”
• A groupoid generalisation of Leavitt path algebras (2014, background): “Shortly afterwards, Kumjian and Pask introduced a sort of higher-dimensional graph [11], now known as a k-graph, and an associated class of C∗algebras, as a flexible visual model for the higher-rank Cuntz-Krieger algebras discovered by Robertson and Steger [22].”
• Product systems over right-angled Artin semigroups (2001, background): “It also holds when the vertex matrices Mi of the directed graphs Ei satisfy Robertson and Steger’s conditions (H0), (H1a), (H1b) and (H1c) from [ 16 ]; that is, if the Mi’s are pairwise commutingf0; 1g-matrices such MiMj is af0; 1g-matrix whenever i<j ,a ndMiMjMl is a f0; 1g-matrix whenever i< j<l .”
• Higher-Rank Graph C *-Algebras: An Inverse Semigroup and Groupoid Approach (2004, background): “In [2], Kumjian and Pask introduced the notion of a higher-rank graph in order to capture the essential features of the C-algebras that Robertson and Steger associated to buildings [14, 15, 16, 17] and to provide links between these and higher order shift dynamical systems.”