Ebook: 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 $mathrm{Cu}(Aotimes B)$ 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.