# Abstract Bivariant Cuntz Semigroups

@article{Antoine2018AbstractBC, title={Abstract Bivariant Cuntz Semigroups}, author={Ramon Antoine and Francesc Perera and Hannes Thiel}, journal={International Mathematics Research Notices}, year={2018} }

We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups $S$ and $T$, there is another Cuntz semigroup $[\![ S,T ]\!] $ playing the role of morphisms from $S$ to $T$. Applied to $C^{\ast }$-algebras $A$ and $B$, the semigroup $[\![ \operatorname{Cu}(A),\operatorname{Cu}(B) ]\!] $ should be considered as the target in analogs of the universal coefficient theorem for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups…

## 13 Citations

### MF traces and the Cuntz semigroup

- Mathematics
- 2017

A trace $\tau$ on a separable C*-algebra $A$ is called matricial field (MF) if there is a trace-preserving morphism from $A$ to $Q_\omega$, where $Q_\omega$ denotes the norm ultrapower of the…

### Abstract bivariant Cuntz semigroups II

- MathematicsForum Mathematicum
- 2020

Abstract We previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an…

### Covering dimension of Cuntz semigroups II

- MathematicsInternational Journal of Mathematics
- 2021

We show that the dimension of the Cuntz semigroup of a [Formula: see text]-algebra is determined by the dimensions of the Cuntz semigroups of its separable sub-[Formula: see text]-algebras. This…

### Uniformly Based Cuntz Semigroups and Approximate Intertwinings

- MathematicsInternational Journal of Mathematics
- 2022

. We study topological aspects of the category of abstract Cuntz semigroups, termed Cu. We provide a suitable setting in which we are able to uniformly control how to approach an element of a…

### Cuntz semigroups of ultraproduct C∗ ‐algebras

- MathematicsJournal of the London Mathematical Society
- 2020

We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the…

### A categorical study on the generalized type semigroup

- Mathematics
- 2021

. In this short note, we show that the generalized type semigroup W ( X, Γ) introduced by the author in [7] belongs to the category W introduced in [1]. In particular, we demonstrate that W ( X, Γ)…

### C*-algebras of stable rank one and their Cuntz semigroups

- MathematicsDuke Mathematical Journal
- 2022

The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: We answer affirmatively, for the class of stable rank one C*-algebras, a…

### The Global Glimm Property

- Mathematics
- 2022

. It is known that a C ∗ -algebra with the Global Glimm Property is nowhere scattered (it has no elementary ideal-quotients), and the Global Glimm Problem asks if the converse holds. We provide a new…

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### Abstract bivariant Cuntz semigroups II

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Abstract We previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an…

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