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 $mathrmCu(A)$ is an object in the category $mathrmCu$ 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 $mathrmCu$-semigroups. The authors establish the existence of tensor products in the category $mathrmCu$ and study the basic properties of this construction. They show that $mathrmCu$ is a symmetric, monoidal category and relate $mathrmCu(Aotimes B)$ with $mathrmCu(A)otimesmathrmCumathrmCu(B)$ for certain classes of $C*$-algebras. As a main tool for their approach the authors introduce the category $mathrmW$ of pre-completed Cuntz semigroups. They show that $mathrmCu$ is a full, reflective subcategory of $mathrmW$. One can then easily deduce properties of $mathrmCu$ from respective properties of $mathrmW$, for example the existence of tensor products and inductive limits. The advantage is that constructions in $mathrmW$ are much easier since the objects are purely algebraic.

Tensor Products and Regularity Properties of Cuntz Semigroups (Memoirs of the American Mathematical Society) PDF

The list of books you might like