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

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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(₂*₃) is a subalgebra of the nuclear Cuntz algebra O₂. 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 is a Cuntz-Krieger algebra. Such trees may be regarded as affine buildings of type A₁. The present paper is devoted to the study of the analogous situation where a group acts simply transitively on the vertices of an affine building of type A₂ with boundary Ø. The corresponding crossed product algebra is then generated by two Cuntz-Krieger algebras. Moreover we show that is simple and nuclear. This is a consequence of the facts that the action of on Ø is minimal, topologically free, and amenable.