The UCT for $C^*$-algebras with finite complexity

A $C^*$-algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov's $KK$-theory to a commutative $C^*$-algebra. This paper is motivated by the problem of establishing the range of validity of the UCT, and in particular, whether the UCT holds for all nuclear $C^*$-algebras. We introduce the idea of a $C^*$-algebra that "decomposes" over a class $\mathcal{C}$ of $C^*$-algebras. Roughly, this means that locally, there are approximately central elements that approximately cut the $C^*$-algebra into two $C^*$-subalgebras from $\mathcal{C}$ that have well-behaved intersection. We show that if a $C^*$-algebra decomposes over the class of nuclear, UCT $C^*$-algebras, then it satisfies the UCT. The argument is based on controlled $KK$-theory, as introduced by the authors in earlier work. Nuclearity is used via Kasparov's Hilbert module version of Voiculescu's theorem, and Haagerup's theorem that nuclear $C^*$-algebras are amenable We say that a $C^*$-algebra has finite complexity if it is in the smallest class of $C^*$-algebras containing the finite-dimensional $C^*$-algebras, and closed under decomposability; our main result implies that all $C^*$-algebras in this class satisfy the UCT. The class of $C^*$-algebras with finite complexity is large, and comes with an ordinal-number invariant measuring the complexity level. We conjecture that a $C^*$-algebra of finite nuclear dimension and real rank zero has finite complexity; this (and several other related conjectures) would imply the UCT for all separable nuclear $C^*$-algebras. We also give new local formulations of the UCT, and some other necessary and sufficient conditions for the UCT to hold for all nuclear $C^*$-algebras.