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… 
Perforation conditions and almost algebraic order in Cuntz semigroups
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
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.
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
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
TLDR
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
A bivariant theory for the Cuntz semigroup and its role for the classification programme of C*-algebras
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
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
. 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
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
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
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
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
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
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
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
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
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
...
...