Boundary C⁎C⁎-algebras of triangle geometries

Let Δ be a building of type A˜2 and order q, with maximal boundary Ω. Let Γ   be a group of type preserving automorphisms of Δ which acts regularly on the chambers of Δ. Then the crossed product C⁎C⁎-algebra C(Ω)⋊ΓC(Ω)⋊Γ is isomorphic to M3(q+1)⊗Oq2⊗Oq2M3(q+1)⊗Oq2⊗Oq2, where OnOn denotes the Cuntz algebra generated by n isometries whose range projections sum to the identity operator.