The 𝐾-theory of Toeplitz 𝐶*-algebras of right-angled Artin groups

Ivanov, Nikolay

doi:10.1090/S0002-9947-2010-05162-1

The $K$-theory of Toeplitz $C^*$-algebras of right-angled Artin groups
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by Nikolay A. Ivanov PDF

Trans. Amer. Math. Soc. 362 (2010), 6003-6027 Request permission

Abstract:

Toeplitz $C^*$-algebras of right-angled Artin groups were studied by Crisp and Laca. They are a special case of the Toeplitz $C^*$-algebras $\mathcal {T}(G, P)$ associated with quasi-lattice ordered groups $(G, P)$ introduced by Nica. Crisp and Laca proved that the so-called “boundary quotients” $C^*_Q(\Gamma )$ of $C^*(\Gamma )$ are simple and purely infinite. For a certain class of finite graphs $\Gamma$ we show that $C^*_Q(\Gamma )$ can be represented as a full corner of a crossed product of an appropriate $C^*$-subalgebra of $C^*_Q(\Gamma )$ built by using $C^*(\Gamma ’)$, where $\Gamma ’$ is a subgraph of $\Gamma$ with one less vertex, by the group $\mathbb {Z}$. Using induction on the number of the vertices of $\Gamma$ we show that $C^*_Q(\Gamma )$ are nuclear and moreover belong to the small bootstrap class. We also use the Pimsner-Voiculescu exact sequence to find their $K$-theory. Finally we use the Kirchberg-Phillips classification theorem to show that those $C^*$-algebras are isomorphic to tensor products of $\mathcal {O}_n$ with $1 \leq n \leq \infty$.

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