The \(K\)-theory of certain \(C^*\)-algebras endowed with gauge actions
The author computes the \(K\)-theory of two classes of generalized Cuntz-Krieger algebras: one is the class of Toeplitz algebras of finitely aligned higher-rank graphs introduced by \textit{I.\,Raeburn, A.\,Sims} and \textit{T.\,Yeend} in [J.~Funct.\ Anal.\ 213, No.\,1, 206--240 (2004; Zbl 1063.46041)], the other is that of higher-rank Exel-Laca algebras introduced by the author in [Acta Sci.\ Math.\ 73, No.\,1--2, 209--235 (2007; Zbl 1136.46038)]. The first main result says the following: given a finitely aligned higher-rank graph \(\Lambda\), let \({\mathcal T} C^*(\Lambda)\) be its associated Toeplitz algebra with generators \(\{t_\lambda\}_{\lambda \in \Lambda}\). Then there is an isomorphism of \(K_0({\mathcal T} C^*(\Lambda))\) with \(\bigoplus_{\Lambda^0}\mathbb Z\) which sends \([t_v]\) to \(1_{\{v\}}\) for \(v\in \Lambda^0\). Further, \(K_1({\mathcal T} C^*(\Lambda))=0\). For a higher-rank Exel-Laca algebra \({\mathcal O}_{{\mathcal A}, {\mathbb F}, {\mathbb I}}\) with generators and relations \(({\mathcal A}, {\mathbb F}, {\mathbb I})\), if a certain condition (II) is satisfied, then \(K_0({\mathcal O}_{{\mathcal A}, {\mathbb F}, {\mathbb I}})\) is isomorphic to the linear span over \(\mathbb Z\) of products \({\mathcal Q}_{a_1}\dots {\mathcal Q}_{a_n}\) of elements in \({\mathcal O}_{{\mathcal A}, {\mathcal F}, {\mathbb I}}\), with \(a_i\) elements of the alphabet \(\mathcal A\). Under this isomorphism, the class \([{\mathcal Q}_{a_1}\dots {\mathcal Q}_{a_n}]\) is mapped to the monomial \({\mathcal Q}_{a_1}\dots {\mathcal Q}_{a_n}\). Moreover, \(K_1({\mathcal O}_{{\mathcal A}, {\mathbb F}, {\mathbb I}})=0\). These theorems are corollaries of a more general result proved in the paper. This is established by employing the argument developed by \textit{I.\,Raeburn} and \textit{W.\,Szymanski} for computing the K-theory of (infinite) graph algebras, see [Trans.\ Am.\ Math.\ Soc.\ 356, No.\,1, 39--59 (2004; Zbl 1030.46067)], and techniques used by the author in [loc.\ cit.]. Roughly, the setup is that of a \(C^*\)-algebra \(\mathcal X\) endowed with a gauge-action of \(\mathbb T^d\) and generated by an alphabet \(\mathcal A\) partitioned into \(d\) disjoint subsets such that certain conditions on the generators are satisfied; by Takai-duality, the stabilization of \(\mathcal X\) is isomorphic to a crossed product \(A\rtimes \mathbb{Z}^d\). It turns out that \(A\) is an AF-algebra, and K-theory computations will proceed by successive applications of the Pimsner-Voiculescu sequence.
scientific article; zbMATH DE number 5541657
scientific article