$C^*$--algebras arising from group actions on the boundary of a triangle building

TL;DR

This paper studies the $C^*$-algebra generated by a group acting on the boundary of an affine building of type $ ilde{A}_2$, showing it is simple, nuclear, and generated by two Cuntz-Krieger algebras, extending previous results from trees.

Contribution

It extends the analysis of $C^*$-algebras from group actions on trees to actions on affine buildings of type $ ilde{A}_2$, demonstrating simplicity and nuclearity.

Findings

The algebra is generated by two Cuntz-Krieger algebras.

The algebra is simple and nuclear.

The group action is minimal, topologically free, and amenable.

Abstract

A subgroup of an amenable group is amenable. The $C^*$-algebra version of this fact is false. This was first proved by M.-D. Choi who proved that the non-nuclear $C^*$-algebra $C^*_r(\ZZ_2*\ZZ_3)$ is a subalgebra of the nuclear Cuntz algebra ${\cal O}_2$. A. Connes provided another example, based on a crossed product construction. More recently J. Spielberg [23] showed that these examples were essentially the same. In fact he proved that certain of the $C^*$-algebras studied by J. Cuntz and W. Krieger [10] can be constructed naturally as crossed product algebras. For example if the group $\Gamma$ acts simply transitively on a homogeneous tree of finite degree with boundary $\Omega$ then $\cross$ is a Cuntz-Krieger algebra. Such trees may be regarded as affine buildings of type $\widetilde A_1$. The present paper is devoted to the study of the analogous situation where a group $\G$ acts simply transitively on the vertices of an affine building of type $\widetilde A_2$ with boundary $\O$. The corresponding crossed product algebra $\cross$ is then generated by two Cuntz-Krieger algebras. Moreover we show that $\cross$ is simple and nuclear. This is a consequence of the facts that the action of $\G$ on $\O$ is minimal, topologically free, and amenable.