  We introduce the notion of a decomposable $C^*$-algebra. Roughly, this means
that there is an almost central element that approximately cuts the
$C^*$-algebra into two finite-dimensional subalgebras with well-behaved
intersection. There are many examples of decomposable (and simple)
$C^*$-algebras including Cuntz algebras, and crossed products associated to
Cantor minimal systems.
  Kirchberg has shown that to establish the Universal Coefficient Theorem (UCT)
of Rosenberg and Schochet for all separable nuclear $C^*$-algebras, it suffices
to establish it for all Kirchberg algebras with zero $K$-theory. Our main
result shows that if a separable $C^*$-algebra has zero $K$-theory and is
decomposable, then it satisfies the UCT. Combining our result with Kirchberg's,
it follows that the UCT for all separable nuclear $C^*$-algebras is equivalent
to the statement that all Kirchberg algebras with zero $K$-theory are
decomposable.
