The $K$-theory of Toeplitz $C^*$-algebras of right-angled Artin groups
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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 $1 \leq n \leq \infty$.References
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Additional Information
- Nikolay A. Ivanov
- Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6
- Address at time of publication: 18 Momina Krepost Str., apt. 3, Veliko Turnovo, 5000 Bulgaria
- Email: nikolay.antonov.ivanov@gmail.com
- Received by editor(s): November 9, 2007
- Received by editor(s) in revised form: March 3, 2009
- Published electronically: May 19, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 6003-6027
- MSC (2010): Primary 19K99, 46L80; Secondary 46L35
- DOI: https://doi.org/10.1090/S0002-9947-2010-05162-1
- MathSciNet review: 2661506