Tensor products and regularity properties of Cuntz semigroups

The Cuntz semigroup of a C^*-algebra is an important invariant in the structure and classification theory of C^*-algebras. It captures more information than K-theory but is often more delicate to handle. The authors systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C^*-algebra A, its (concrete) Cuntz semigroup \mathrm{Cu}(A) is an object in the category \mathrm{Cu} of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, the authors call the latter \mathrm{Cu}-semigroups. The authors establish the existence of tensor products in the category \mathrm{Cu} and study the basic properties of this construction. They show that \mathrm{Cu} is a symmetric, monoidal category and relate \mathrm{Cu}(A\otimes B) with \mathrm{Cu}(A)\otimes_{\mathrm{Cu}}\mathrm{Cu}(B) for certain classes of C^*-algebras. As a main tool for their approach the authors introduce the category \mathrm{W} of pre-completed Cuntz semigroups. They show that \mathrm{Cu} is a full, reflective subcategory of \mathrm{W}. One can then easily deduce properties of \mathrm{Cu} from respective properties of \mathrm{W}, for example the existence of tensor products and inductive limits. The advantage is that constructions in \mathrm{W} are much easier since the objects are purely algebraic.