The Stable Exotic Cuntz Algebras are Higher-Rank Graph Algebras

Abstract

For each odd integer n ≥ 3, we construct a rank-3 graph Λn with involution γn whose real C * -algebra C * R (Λn, γn) is stably isomorphic to the exotic Cuntz algebra En. This construction is optimal, as we prove that a rank-2 graph with involution (Λ, γ) can never satisfy C * R (Λ, γ) ∼ M E En, and the first author reached the same conclusion for rank-1 graphs (directed graphs) in [6, Corollary 4.3]. Our construction relies on a rank-1 graph with involution (Λ, γ) whose real C * -algebra C * R (Λ, γ) is stably isomorphic to the suspension SR. In the Appendix, we show that the i-fold suspension S i R is stably isomorphic to a graph algebra iff −2 ≤ i ≤ 1. 1 This class of C * -algebras includes the real C * -algebras C * R (Λ) of a directed graph, introduced in [5], as C * R (Λ) ∼ = C * R (Λ, γ triv ).