The Universal CoeffIcient Theorem 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 $K$-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.