Rank-two graphs whose C∗-algebras are direct limits of circle algebras ✩

We describe a class of rank-2 graphs whose C∗-algebras are AT algebras. For a subclass which we call rank-2 Bratteli diagrams, we compute the K-theory of the C∗-algebra.We identify rank-2 Bratteli diagrams whose C∗-algebras are simple and have real-rank zero, and characterise the K-invariants achieved by such algebras. We give examples of rank-2 Bratteli diagrams whose C∗-algebras contain as full corners the irrational rotation algebras and the Bunce–Deddens algebras. © 2006 Elsevier Inc. All rights reserved