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


Author: Nikolay A. Ivanov
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
Published electronically: May 19, 2010
MathSciNet review: 2661506
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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$.


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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

DOI: https://doi.org/10.1090/S0002-9947-2010-05162-1
Received by editor(s): November 9, 2007
Received by editor(s) in revised form: March 3, 2009
Published electronically: May 19, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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