# Tensor products and regularity properties of Cuntz semigroups

@article{Antoine2014TensorPA, title={Tensor products and regularity properties of Cuntz semigroups}, author={Ramon Antoine and Francesc Perera and Hannes Thiel}, journal={arXiv: Operator Algebras}, year={2014} }

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…

## 57 Citations

Perforation conditions and almost algebraic order in Cuntz semigroups

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2018

For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗ ) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of…

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…

Unitary Cuntz semigroups of ideals and quotients.

- Mathematics
- 2020

We define a notion of ideals in the category of ordered monoids satisfying the Cuntz axioms introduced in [2] and termed Cu$^\sim$. We show that the set of ideals of a Cu$^\sim$-semigroup $S$ has a…

Abstract Bivariant Cuntz Semigroups

- MathematicsInternational Mathematics Research Notices
- 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…

The Cuntz semigroup and domain theory

- MathematicsSoft Comput.
- 2017

Those notions of domain theory that seem to be relevant for the theory of Cuntz semigroups and have sometimes been developed independently in both communities are presented.

A bivariant theory for the Cuntz semigroup

- MathematicsJournal of Functional Analysis
- 2019

A bivariant theory for the Cuntz semigroup and its role for the classification programme of C*-algebras

- Mathematics
- 2016

A bivariant theory for the Cuntz semigroup is introduced and analysed. This is used to define a Cuntz-analogue of K-homology, which turns out to provide a complete invariant for compact Hausdorff…

The equivariant Cuntz semigroup

- Mathematics
- 2015

We introduce an equivariant version of the Cuntz semigroup that takes an action of a compact group into account. The equivariant Cuntz semigroup is naturally a semimodule over the representation…

A total Cuntz semigroup for $C^*$-algebras of stable rank one

- Mathematics
- 2022

. In this paper, we show that for unital, separable C ∗ -algebras of stable rank one and real rank zero, the unitary Cuntz semigroup functor and the functor K ∗ are naturallly equivalent. Then we…

## References

SHOWING 1-10 OF 155 REFERENCES

The cone of functionals on the Cuntz semigroup

- Mathematics
- 2011

The functionals on an ordered semigroup S in the category Cu--a category to which the Cuntz semigroup of a C*-algebra naturally belongs--are investigated. After appending a new axiom to the category…

The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras

- Mathematics
- 2006

Abstract We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C*-algebras. In particular, our results apply to the largest class of simple…

Nuclear dimension and -stability of pure C ∗ -algebras

- Mathematics
- 2010

In this article I study a number of topological and algebraic dimension type properties of simple C*-algebras and their interplay. In particular, a simple C*-algebra is defined to be (tracially)…

The Corona Factorization property, Stability, and the Cuntz semigroup of a C*-algebra

- Mathematics
- 2009

The Corona Factorization Property, originally invented to study extensions of C*-algebras, conveys essential information about the intrinsic structure of the C*-algebras. We show that the Corona…

THE CUNTZ SEMIGROUP OF CONTINUOUS FUNCTIONS INTO CERTAIN SIMPLE C*-ALGEBRAS

- Mathematics
- 2011

This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C0(X, A), where A is a unital, simple, -stable ASH algebra. The computations describe the Cuntz semigroup…

Nuclear dimension and $\mathcal{Z}$-stability of pure C∗-algebras

- Mathematics
- 2012

In this article I study a number of topological and algebraic dimension type properties of simple C∗-algebras and their interplay. In particular, a simple C∗-algebra is defined to be (tracially)…

Recovering the Elliott invariant from the Cuntz semigroup

- Mathematics
- 2011

Let A be a simple, separable C � -algebra of stable rank one. We prove that the Cuntz semigroup of C(T,A) is determined by its Murray-von Neumann semigroup of projections and a certain semigroup of…

K-Theory for operator algebras. Classification of C$^*$-algebras

- Mathematics
- 2009

In this article we survey some of the recent goings-on in the classification programme of C$^*$-algebras, following the interesting link found between the Cuntz semigroup and the classical Elliott…

The equivariant Cuntz semigroup

- Mathematics
- 2015

We introduce an equivariant version of the Cuntz semigroup that takes an action of a compact group into account. The equivariant Cuntz semigroup is naturally a semimodule over the representation…