Boundary $C^{\ast }$-algebras of triangle geometries

Abstract

Let $\Delta$ be a building of type ${\tilde{A}}_2$ and order $q$, with maximal boundary $\Omega$. Let $\Gamma$ be a group of type preserving automorphisms of $\Delta$ which acts regularly on the chambers of $\Delta$. Then the crossed product $C^{\ast }$-algebra $C(\Omega )⋊\Gamma$ is isomorphic to $M_{3(q+1)} \otimes\cl O_{q^2}\otimes\cl O_{q^2}$, where $\cl O_n$ denotes the Cuntz algebra generated by $n$ isometries whose range projections sum to the identity operator.