The Stable Exotic Cuntz Algebras are Higher-Rank Graph Algebras

For each odd integer $n\ge 3$, we construct a rank-3 graph ${\mathrm{\Lambda }}_n$ with involution ${\gamma }_n$ whose real C*-algebra $C_{\mathbb{R}}^{\ast }({\mathrm{\Lambda }}_n,{\gamma }_n)$ 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 $(\mathrm{\Lambda },\gamma )$ can never satisfy $C_{\mathbb{R}}^{\ast }(\mathrm{\Lambda },\gamma ){\sim }_{ME}{\mathcal{E}}_n^{\mathbb{R}}$, and the first author reached the same conclusion in previous work. Our construction relies on a rank-1 graph with involution $(\mathrm{\Lambda },\gamma )$ whose real C*-algebra $C_{\mathbb{R}}^{\ast }(\mathrm{\Lambda },\gamma )$ 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 $-2\le i\le 1$.