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

@article{Robertson2013AffineBT,
  title={Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras},
  author={Guyan Robertson and Tim Steger},
  journal={arXiv: Operator Algebras},
  year={2013}
}
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… 
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References

SHOWING 1-10 OF 29 REFERENCES

FREE-PRODUCT GROUPS, CUNTZ-KRIEGER ALGEBRAS, AND COVARIANT MAPS

A construction is given relating a finitely generated free-product of cyclic groups with a certain Cuntz-Krieger algebra, generalizing the relation between the Choi algebra and 02. It is shown that a…

Groups acting simply transitively on the vertices of a building of typeÃ2, I

If a group Γ acts simply transitively on the vertices of an affine building Δ with connected diagram, then Δ must be of typeÃn−1 for somen⩾2, and Γ must have a presentation of a simple type. The…

A class ofC*-algebras and topological Markov chains

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…

Exact C*-Algebras, Tensor Products, and the Classification of Purely Infinite Algebras

Our survey (and the reference list) does not reflect the history of tensor products of operator algebras. Here we make use of tensor product functors on the category of C-algebras as a unifying…

Actions of Cartan subgroups

Using affine Bruhat-Tits buildings, we associate certain subshifts of finite type to systems arising from the action of a Cartan subgroup of ap-adic semisimple Chevalley group on compact quotient…

Ergodic Theory and Semisimple Groups

1. Introduction.- 2. Moore's Ergodicity Theorem.- 3. Algebraic Groups and Measure Theory.- 4. Amenability.- 5. Rigidity.- 6. Margulis' Arithmeticity Theorems.- 7. Kazhdan's Property (T).- 8. Normal…

Lectures on Buildings

K-Theory for Operator Algebras

I. Introduction To K-Theory.- 1. Survey of topological K-theory.- 2. Overview of operator K-theory.- II. Preliminaries.- 3. Local Banach algebras and inductive limits.- 4. Idempotents and…

Fundamentals of the Theory of Operator Algebras

Buildings and Classical Groups

monograph on Bruhat-TIts buildings attached to classical groups. Both spherical and affine buildings.