2007/08/22 by Ivanov, Nikolay A.
#19K99 #46L80 #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.0708.2944
To a graph Γ one can associate a C^*-algebra C^*(Γ) generated by isometries. Such C^*-algebras were studied recently by Crisp and Laca. They are a special case of the Toeplitz C^*-algebras T(G, P) associated to quasi-latice ordered groups (G, P) introduced by Nica. Crisp and Laca proved that the so called "boundary quotients" C^*q(Γ) of C^*(Γ) are simple and purely infinite. For a certain class of finite graphs Γ we show that C^*q(Γ) can be represented as a full corner of a crossed product of an appropriate C^*-subalgebra of C^*q(Γ) built by using C^*(Γ'), where Γ' is a subgraph of Γ with one less vertex, by the group ℤ. Using induction on the number of the vertices of Γ we show that C^*q(Γ) are nuclear and belong to the small bootstrap class. This also enables us to 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 On for 1 ≤ n ≤ ∞.