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

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 $Γ$ acts simply transitively on a homogeneous tree of finite degree with boundary $Ω$ 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 $Ø$. 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 $Ø$ is minimal, topologically free, and amenable.