A $C^*$-algebra satisfies the UCT if it is K-theoretically the same as a commutative $C^*$-algebra, in some sense. Whether or not all (separable) nuclear $C^*$-algebras satisfy the UCT is an important open problem; in particular, it is the last remaining ingredient needed to prove the ‘best possible’ classification result for simple nuclear $C^*$-algebras in the sense of the Elliott classification program.
We introduce a notion of a ‘decomposition’ of a $C^*$-algebra over a class of $C^*$-algebras. Roughly, this means that there are almost central elements of the $C^*$-algebra that cut it into two pieces from the class, with well-behaved intersection. Our main result shows that the class of nuclear $C^*$-algebras that satisfy the UCT is closed under decomposability.
Decomposability introduces a natural ‘complexity hierarchy’ on the class of $C^*$-algebras: one starts with finite-dimensional $C^*$-algebras, and the ‘complexity rank’ of a $C^*$-algebra is roughly the number of decompositions one needs to get to down to the finite-dimensional level. There are interesting examples: we show that all UCT Kirchberg (i.e. purely infinite, separable, simple, unital, nuclear) $C^*$-algebras have complexity rank one or two, and characterize when each of these cases occur. The UCT for all nuclear $C^*$-algebras thus becomes equivalent to the statement that all Kirchberg algebras have finite complexity rank.
This is based on joint work with Arturo Jaime, and with Guoliang Yu.