The Stable Exotic Cuntz Algebras are Higher-Rank Graph Algebras

For each odd integer $n \geq 3$, we construct a rank-3 graph $\Lambda_n$ with involution $\gamma_n$ whose real C*-algebra $C^*_\mathbb{R}(\Lambda_n, \gamma_n)$ is stably isomorphic to the exotic Cuntz algebra $\mathcal E_n^\mathbb{R}$. This construction is optimal, as we prove that a rank-2 graph with involution $(\Lambda,\gamma)$ can never satisfy $C^*_\mathbb{R}(\Lambda, \gamma)\sim_{ME} \mathcal E_n^\mathbb{R}$, and the first author reached the same conclusion in previous work. Our construction relies on a rank-1 graph with involution $(\Lambda, \gamma)$ whose real C*-algebra $C^*_\mathbb{R}(\Lambda, \gamma)$ is stably isomorphic to the suspension $ S \mathbb{R}$. In the Appendix, we show that the i-fold suspension $S^i \mathbb{R}$ is stably isomorphic to a graph algebra iff $-2 \leq i \leq 1$.