Controlled $KK$-theory II: a Mayer-Vietoris principle and the UCT

Rufus Willett, Guoliang Yu

Published 2021-04-21Version 1

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.