Tensor products and regularity properties of Cuntz semigroups

The Cuntz semigroup of a C ∗ C^* -algebra is an important invariant in the structure and classification theory of C ∗ C^* -algebras. It captures more information than K K -theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C ∗ C^* -algebra A A , its (concrete) Cuntz semigroup C u ( A ) Cu(A) is an object in the category C u Cu of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu (2008). To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter C u Cu -semigroups. We establish the existence of tensor products in the category C u Cu and study the basic properties of this construction. We show that C u Cu is a symmetric, monoidal category and relate C u ( A ⊗ B ) Cu(A⊗ B) with C u ( A ) ⊗