scholarly journals The Cuntz semigroup as an invariant for C*-algebras

2008 ◽  
Vol 2008 (623) ◽  
Author(s):  
Kristofer T. Coward ◽  
George A. Elliott ◽  
Cristian Ivanescu
Keyword(s):  
C Algebras ◽  
Cuntz Semigroup

scholarly journals The Cone of Functionals on the Cuntz Semigroup

2013 ◽  
Vol 113 (2) ◽  
pp. 161 ◽  
Author(s):  
Leonel Robert
Keyword(s):  
Distributive Lattice ◽  
Ordered Semigroup ◽  
Riesz Decomposition Property ◽  
Decomposition Property ◽  
C Algebra ◽  
C Algebras ◽  
Riesz Decomposition ◽  
Cuntz Semigroup

The functionals on an ordered semigroup $S$ in the category $\mathbf{Cu}$ - a category to which the Cuntz semigroup of a C*-algebra naturally belongs - are investigated. After appending a new axiom to the category $\mathbf{Cu}$, it is shown that the "realification" $S_{\mathsf{R}}$ of $S$ has the same functionals as $S$ and, moreover, is recovered functorially from the cone of functionals of $S$. Furthermore, if $S$ has a weak Riesz decomposition property, then $S_{\mathsf{R}}$ has refinement and interpolation properties which imply that the cone of functionals on $S$ is a complete distributive lattice. These results apply to the Cuntz semigroup of a C*-algebra. At the level of C*-algebras, the operation of realification is matched by tensoring with a certain stably projectionless C*-algebra.

scholarly journals An Equivariant Theory for the Bivariant Cuntz Semigroup

2018 ◽  
Vol 61 (2) ◽  
pp. 573-598
Author(s):  
Gabriele N. Tornetta
Keyword(s):  
Compact Group ◽  
Group Actions ◽  
Crossed Products ◽  
C Algebras ◽  
Regularity Properties ◽  
Cuntz Semigroup ◽  
The One ◽  
Equivariant Extension ◽  
Equivariant Theory

AbstractWe provide an equivariant extension of the bivariant Cuntz semigroup introduced in previous work for the case of compact group actions over C*-algebras. Its functoriality properties are explored, and some well-known classification results are retrieved. Connections with crossed products are investigated, and a concrete presentation of equivariant Cuntz homology is provided. The theory that is here developed can be used to define the equivariant Cuntz semigroup. We show that the object thus obtained coincides with the one recently proposed by Gardella [‘Regularity properties and Rokhlin dimension for compact group actions’, Houston J. Math.43(3) (2017), 861–889], and we complement their work by providing an open projection picture of it.

scholarly journals A revised augmented Cuntz semigroup

2021 ◽  
Vol 127 (1) ◽  
pp. 131-160
Author(s):  
Leonel Robert ◽  
Luis Santiago
Keyword(s):  
Stable Rank ◽  
Inductive Limits ◽  
C Algebras ◽  
Exact Functor ◽  
Cw Complexes ◽  
Rank One ◽  
Cuntz Semigroup ◽  
Original Construction

We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of $1$-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: we show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.

scholarly journals Nonseparable Uhf Algebras Ii: Classification

2015 ◽  
Vol 117 (1) ◽  
pp. 105 ◽  
Author(s):  
Ilijas Farah ◽  
Takeshi Katsura
Keyword(s):  
Density Character ◽  
Car Algebra ◽  
C Algebras ◽  
Uncountable Cardinal ◽  
Cuntz Semigroup ◽  
Af Algebras

For every uncountable cardinal $\kappa$ there are $2^\kappa$ nonisomorphic simple AF algebras of density character $\kappa$ and $2^\kappa$ nonisomorphic hyperfinite ${\rm II}_1$ factors of density character $\kappa$. These estimates are maximal possible. All C*-algebras that we construct have the same Elliott invariant and Cuntz semigroup as the CAR algebra.

scholarly journals Edwards' condition for quasitraces on C*-algebras

2020 ◽  
pp. 1-23
Author(s):  
Ramon Antoine ◽  
Francesc Perera ◽  
Leonel Robert ◽  
Hannes Thiel
Keyword(s):  
Additional Structure ◽  
Realization Problem ◽  
C Algebras ◽  
Positive Elements ◽  
Cuntz Semigroup

Abstract We prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with respect to every quasitrace. This condition is a key ingredient in the study of the realization problem of functions on the cone of quasitraces as ranks of positive elements. In the course of our investigation, we identify additional structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of quasitraces.

scholarly journals COMPLETIONS OF MONOIDS WITH APPLICATIONS TO THE CUNTZ SEMIGROUP

2011 ◽  
Vol 22 (06) ◽  
pp. 837-861 ◽  
Author(s):  
RAMON ANTOINE ◽  
JOAN BOSA ◽  
FRANCESC PERERA
Keyword(s):  
C Algebras ◽  
Categorical Framework ◽  
Cuntz Semigroup

We provide an abstract categorical framework that relates the Cuntz semigroups of the C*-algebras A and [Formula: see text]. This is done through a certain completion of ordered monoids by adding suprema of countable ascending sequences. Our construction is rather explicit and we show it is functorial and unique up to isomorphism. This approach is used in some applications to compute the stabilized Cuntz semigroup of certain C*-algebras.

scholarly journals Comparison Properties of the Cuntz Semigroup and Applications to C* -algebras

2018 ◽  
Vol 70 (1) ◽  
pp. 26-52
Author(s):  
Joan Bosa ◽  
Henning Petzka
Keyword(s):  
Factorization Property ◽  
C Algebra ◽  
C Algebras ◽  
Algebraic Setting ◽  
Cuntz Semigroup ◽  
Comparison Property

AbstractWe study comparison properties in the category Cu aiming to lift results to the C* -algebraic setting. We introduce a new comparison property and relate it to both the corona factorization property (CFP) and ω-comparison. We show differences of all properties by providing examples that suggest that the corona factorization for C* -algebras might allow for both finite and infinite projections. In addition, we show that Rørdam's simple, nuclear C* -algebra with a finite and an inifnite projection does not have the CFP.

scholarly journals On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*-algebras

2015 ◽  
Vol 58 (2) ◽  
pp. 402-414 ◽  
Author(s):  
Aaron Peter Tikuisis ◽  
Andrew Toms
Keyword(s):  
Lower Semicontinuous ◽  
Positive Operators ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Simple Algebras ◽  
Cuntz Semigroup ◽  
Unital Simple ◽  
Densely Defined

AbstractWe examine the ranks of operators in semi-finite C*-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple C*-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray–von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first author, this shows that slow dimension growth coincides with Z-stability for approximately subhomogeneous algebras.

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