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

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