Ten years ago, J. Villadsen constructed the first examples of simple C*-algebras with stable rank other than one or infinity. Villadsen's examples all had a unique tracial state. It is natural to ask whether examples can be found of simple C*-algebras with higher stable rank and more than one tracial state; by building on Villadsen's construction, we describe such examples that admit arbitrary tracial state spaces.

Ten years ago, J. Villadsen constructed the first examples of simple C*-algebras with stable rank other than one or infinity. Villadsen's examples all had a unique tracial state. It is natural to ask whether examples can be found of simple C*-algebras with higher stable rank and more than one tracial state; by building on Villadsen's construction, we describe such examples that admit arbitrary tracial state spaces.