Algebras Of Higher-rank Graphs Research Articles

The stable exotic Cuntz algebras are higher-rank graph algebras

For each odd integer n ≥ 3 n \geq 3 , we construct a rank-3 graph Λ n \Lambda _n with involution γ n \gamma _n whose real C ∗ C^* -algebra C R ∗ ( Λ n , γ n ) C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n) is stably isomorphic to the exotic Cuntz algebra E n \mathcal E_n . This construction is optimal, as we prove that a rank-2 graph with involution ( Λ , γ ) (\Lambda ,\gamma ) can never satisfy C R ∗ ( Λ , γ ) ∼ M E E n C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n , and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math. 10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution ( Λ , γ ) (\Lambda , \gamma ) whose real C ∗ C^* -algebra C R ∗ ( Λ , γ ) C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma ) is stably isomorphic to the suspension S R S \mathbb {R} . In the Appendix, we show that the i i -fold suspension S i R S^i \mathbb {R} is stably isomorphic to a graph algebra iff − 2 ≤ i ≤ 1 -2 \leq i \leq 1 .

$C^{\star}$-algebras of higher-rank graphs from groups acting on buildings, and explicit computation of their K-theory

$C^{\star}$-algebras of higher-rank graphs from groups acting on buildings, and explicit computation of their K-theory

Simplicity of twisted $C^*$-algebras of higher-rank graphs and crossed products by quasifree actions

We characterise simplicity of twisted C*-algebras of row-finite k-graphs with no sources. We show that each 2-cocycle on a cofinal k-graph determines a canonical second-cohomology class for the periodicity group of the graph. The groupoid of the k-graph then acts on the cartesian product of the infinite-path space of the graph with the dual group of the centre of any bicharacter representing this second-cohomology class. The twisted k-graph algebra is simple if and only if this action is minimal. We apply this result to characterise simplicity for many twisted crossed products of k-graph algebras by quasifree actions of free abelian groups.

A Cuntz--Krieger uniqueness theorem for semigraph $C^*$-algebras

Higher rank semigraph algebras are introduced by mixing concepts of ultragraph algebras and higher rank graph algebras. This yields a kind of higher rank generalisation of ultragraph algebras. We prove Cuntz--Krieger uniqueness theorems for cancelling semigraph algebras and aperiodic saturated semigraph algebras.

E-Theoretic Duality for Higher Rank Graph Algebras

E-Theoretic Duality for Higher Rank Graph Algebras