Does nuclearity imply the Universal Coefficient Theorem (UCT)?

Determine whether every nuclear C*-algebra satisfies the Universal Coefficient Theorem (UCT), i.e., establish whether nuclearity implies the UCT for all nuclear C*-algebras. This question is central to the Elliott classification program, where the UCT enables lifting K-theoretic information to KK-theory.

Background

The Universal Coefficient Theorem (UCT) plays a key role in modern classification results for simple, separable, unital, nuclear C*-algebras by enabling the passage from K-theory to KK-theory. Many landmark classification theorems assume the UCT as a technical hypothesis.

Within the paper’s overview of classification, the authors explicitly note that while all known nuclear C*-algebras satisfy the UCT, it has not been proven that nuclearity itself implies the UCT universally. Resolving this would remove a major standing hypothesis in the classification of nuclear C*-algebras.