Tensor products and regularity properties of Cuntz semigroups
By Ramon Antoine, F. Perera, Hannes Thiel. 2014
109 citations recorded by the source corpus.
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. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups. We establish the existence of tensor products in the category $Cu$ and study the basic properties of this construction. We show that $Cu$ is a symmetric, monoidal category and relate $Cu(A\otimes B)$ with $Cu(A)\otimes_{Cu}Cu(B)$ for certain classes of C*-algebras. As a main tool for our approach we introduce the category $W$ of pre-completed Cuntz semigroups. We show that $Cu$ is a full, reflective subcategory of $W$. One can then easily deduce properties of $Cu$ from respective properties of $W$, e.g. the existence of tensor products and inductive limits. The advantage is that constructions in $W$ are much easier since the objects are purely algebraic. We also develop a theory of $Cu$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C*-algebra has a natural product giving it the structure of a $Cu$-semiring. We give explicit characterizations of $Cu$-semimodules over such $Cu$-semirings. For instance, we show that a $Cu$-semigroup $S$ tensorially absorbs the $Cu$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/28de2762e2fd17c8836bb8b53af92c27d3ecb511
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In Algebra and Number Theory
In Mathematical Physics
Citations in the corpus
- Sequentially split ∗-homomorphisms between C*-algebras (2015, background): “ inductive limit of 1-NCCW complexes. This shows (14). (15): Let S be either one of the following classes of C∗-algebras: • all finite-dimensional C∗-algebras; • all C∗-algebras isomorphic to F1 ⊕F2 ⊗C[0,1], where F1 and F2 are finite-dimensional; • All unital 1-NCCW-complexes (also knownas Elliott-Th”
- C*-algebras of stable rank one and their Cuntz semigroups (2018, unknown): “ Preliminaries In this section we recall basic notions concerning the Cuntz semigroup of a C*-algebra and the category Cu it belongs to. For a fuller account, we refer the reader to [APT11], [ORT11], [APT18], and the references therein. 2.1. The Cuntz semigroup. Let A be a C*-algebra. Denote by A + ”