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 Filters in C*-Algebras

2013 ◽  
Vol 65 (3) ◽  
pp. 485-509 ◽  
Author(s):  
Tristan Matthew Bice
Keyword(s):  
Unit Ball ◽  
Pure State ◽  
Positive Answer ◽  
The State ◽  
Natural Numbers ◽  
Cardinal Invariant ◽  
Calkin Algebra ◽  
C Algebras ◽  
Positive Elements ◽  
Singer Conjecture

AbstractIn this paper we analyze states on C*-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to investigate the Kadison–Singer conjecture, proving its equivalence to an apparently quite weak paving conjecture and the existence of unique maximal centred extensions of projections coming from ultrafilters on the natural numbers. We then prove that Reid's positive answer to this for q-points in fact also holds for rapid p-points, and that maximal centred filters are obtained in this case. We then show that consistently, such maximal centred filters do not exist at all meaning that, for every pure state on the Calkin algebra, there exists a pair of projections on which the state is 1, even though the state is bounded strictly below 1 for projections below this pair. Next, we investigate towers, using cardinal invariant equalities to construct towers on the natural numbers that do and do not remain towers when canonically embedded into the Calkin algebra. Finally, we show that consistently, all towers on the natural numbers remain towers under this embedding.

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.

Powers of positive elements in C *-algebras

2011 ◽  
Vol 57 (5) ◽  
pp. 481-484
Author(s):  
Hiroki Takamura
Keyword(s):  
C Algebras ◽  
Positive Elements

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.

Equivalent Definitions of Infinite Positive Elements in Simple C*-algebras

2010 ◽  
Vol 53 (2) ◽  
pp. 256-262
Author(s):  
Xiaochun Fang ◽  
Lin Wang
Keyword(s):  
C Algebras ◽  
Positive Elements

AbstractWe prove the equivalence of three definitions given by different comparison relations for infiniteness of positive elements in simple C*-algebras.

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
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