Abstract
AbstractWe initiate the program of extending to higher-rank graphs (k-graphs) the geometric classification of directed graph $C^*$ -algebras, as completed in Eilers et al. (2016, Preprint). To be precise, we identify four “moves,” or modifications, one can perform on a k-graph $\Lambda $ , which leave invariant the Morita equivalence class of its $C^*$ -algebra $C^*(\Lambda )$ . These moves—in-splitting, delay, sink deletion, and reduction—are inspired by the moves for directed graphs described by Sørensen (Ergodic Th. Dyn. Syst. 33(2013), 1199–1220) and Bates and Pask (Ergodic Th. Dyn. Syst. 24(2004), 367–382). Because of this, our perspective on k-graphs focuses on the underlying directed graph. We consequently include two new results, Theorem 2.3 and Lemma 2.9, about the relationship between a k-graph and its underlying directed graph.
Highlights
Recent years have seen a number of breakthroughs in the classification of C∗-algebras by K-theoretic invariants
The proof of the K-theoretic classification of simple Cuntz–Krieger algebras draws heavily on the dynamical characterization of Cuntz–Krieger algebras as arising from one-sided shifts of finite type [CK80]. As this dynamical characterization holds in the nonsimple case as well, Cuntz–Krieger algebras were a natural setting for a first foray into classification of nonsimple C∗-algebras, and many classes of nonsimple
Taking inspiration from [Dri99, BP04, CG06, Sø13], we identify four moves on row-finite, source-free k-graphs Λ which preserve the Morita equivalence class of C∗(Λ)
Summary
Recent years have seen a number of breakthroughs in the classification of C∗-algebras by K-theoretic invariants. Taking inspiration from [Dri, BP04, CG06, Sø13], we identify four moves (sink deletion, in-splitting, reduction, and delay) on row-finite, source-free k-graphs Λ which preserve the Morita equivalence class of C∗(Λ). The fact that this move does not change the Morita equivalence class of the k-graph C∗-algebra is established in Theorem 5.5. We turn to “reduction” in Section 6, where we identify when contraction (reduction) of a “complete edge” (see Definition 6.0.1) in a k-graph produces a kgraph (Theorem 6.3) In this case, the C∗-algebra of the resulting k-graph is always Morita equivalent to the original k-graph C∗-algebra, by Theorem 6.4. Throughout the paper, we include examples showcasing the moves and indicating the necessity of our hypotheses
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