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 .