  C*-algebras are rings, sometimes nonunital, obeying certain axioms that
ensure a very well-behaved representation theory upon Hilbert space. Moreover,
there are some well-known features of the representation theory leading to
subtle questions about norms on tensor products of C*-algebras, and thus to the
subclass of nuclear C*-algebras. The question whether all separable nuclear
C*-algebras satisfy the Universal Coefficient Theorem (UCT) remains one of the
most important open problems in the structure and classification theory of such
algebras. One of the most promising ways to test the UCT conjecture depends on
finding C*-algebras that behave as idempotents under the tensor product, and
satisfy certain additional properties. Briefly put, it has been shown that if
there exists a simple, separable, and nuclear C*-algebra that is an idempotent
under the tensor product, satisfies a certain technical property, and is not
one of the already known such elements $\left\{ O_\infty, O_2, \UHF , J, Z, \C,
\compact \right\}$ then the UCT fails. Although we do not disprove the UCT in
this publication, we do find new idempotents in the class of Villadsen
algebras.
