The stable exotic Cuntz algebras are rank-3 graph algebras

The exotic Cuntz algebras E_n (for n>2 odd) are real C*-algebras whose complexification is the classical Cuntz algebra O_n.  However, until very recently, almost nothing has been known about E_n except its K-theory and the fact of its existence.  In joint work with J.L. Boersema and S. Browne, we have constructed E_n as the real C*-algebra of a rank-3 graph with involution. We also prove that this construction is optimal, as the K-theory of E_n precludes it arising from a rank-2 or rank-1 graph. Time permitting, we will also discuss which suspensions of R can be realized as the C*-algebras of directed graphs with involution; some of these suspensions were key ingredients in our construction of the rank-3 graph realizing E_n.