Boundary C^*-algebras of triangle geometries
classification
🧮 math.OA
math.GRmath.KT
keywords
deltaalgebraboundarygammaomegaotimestypeacts
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Let $\Delta$ be a building of type $\widetilde A_2$ 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 $C^*$-algebra $C(\Omega) \rtimes \Gamma$ 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.
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