The stable exotic Cuntz algebras are higher-rank graph algebras

For each odd integer n ≥<!-- ≥ --> 3 n ≥ 3 , we construct a rank-3 graph Λ<!-- Λ --> n Λ _n with involution γ<!-- γ --> n γ _n whose real C ∗<!-- ∗ --> C^* -algebra C R ∗<!-- ∗ --> ( Λ<!-- Λ --> n , γ<!-- γ --> n ) C^*R(Λ _n, γ _n) is stably isomorphic to the exotic Cuntz algebra E n E_n . This construction is optimal, as we prove that a rank-2 graph with involution ( Λ<!-- Λ --> , γ<!-- γ --> ) (Λ ,γ ) can never satisfy C R ∗<!-- ∗ --> ( Λ<!-- Λ --> , γ<!-- γ --> ) ∼<!-- ∼ --> M E E n C^*R(Λ , γ )~ ME 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<!-- Λ --> , γ<!-- γ --> ) (Λ , γ ) whose real C ∗<!-- ∗ --> C^* -algebra C R ∗<!-- ∗ --> ( Λ<!-- Λ --> , γ<!-- γ --> ) C^*R(Λ , γ ) is stably isomorphic to the suspension S R S R . In the Appendix, we show that the i i -fold suspension S i R S^i R is stably isomorphic to a graph algebra iff −<!-- − --> 2 ≤<!-- ≤ --> i ≤<!-- ≤ --> 1 -2 ≤ i ≤ 1 .