J. Villadsen constructed examples now known as Villadsen algebras, which form an exciting class of C*-algebras: it provides examples of C*-algebras for which the $K_0$-group is not weakly unperforated or simple C*-algebras with stable rank other than one. We use Villadsen construction to build a C*-algebra, which is idempotent in the sense that the algebra is isomorphic to its tensor product with itself. The Villadsen algebras are conjectured classifiable by sufficiently many invariants; hence Villadsen idempotents should play an important role in studying Villadsen algebras.
This is joint work with Dan Kucerovsky, UNB.