A \(C^*\)-algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov's \(K\)-theory to a commutative \(C^*\)-algebra. This book 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.

The authors 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 \(C^*\)-sub-algebras from \(C\) that have well-behaved intersection. The authors 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 a Mayer-Vietoris principle in the framework of controlled \(K\)-theory; the latter was introduced by the authors in an 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.

The authors 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; their 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. They 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. The authors 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.