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.
ISBN: | 9781470427979 |
Publication date: | 30th March 2018 |
Author: | Ramon Antoine, Francesc Perera, Hannes Thiel |
Publisher: | American Mathematical Society |
Format: | Paperback |
Pagination: | 191 pages |
Series: | Memoirs of the American Mathematical Society |
Genres: |
Algebraic geometry Algebra Number theory |