  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.
