Lifting positive elements in C*-algebras

1991 ◽  
Vol 14 (2) ◽  
pp. 183-191 ◽  
Author(s):  
Kenneth R. Davidson
Keyword(s):  
C Algebras ◽  
Positive Elements

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.

Powers of positive elements in C *-algebras

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

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 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.

REMARKS ON -STABLE PROJECTIONLESS C*-ALGEBRAS

2015 ◽  
Vol 58 (2) ◽  
pp. 273-277 ◽  
Author(s):  
LEONEL ROBERT
Keyword(s):  
Equivalence Classes ◽  
Unitary Equivalence ◽  
C Algebras ◽  
Positive Elements

AbstractIt is shown that ${\mathcal{Z}}$-stable projectionless C*-algebras have the property that every element is a limit of products of two nilpotents. This is then used to classify the approximate unitary equivalence classes of positive elements in such C*-algebras using traces.

The Structure of Positive Elements for C*-Algebras with Real Rank Zero

1997 ◽  
Vol 08 (03) ◽  
pp. 383-405 ◽  
Author(s):  
Francesc Perera
Keyword(s):  
Grothendieck Group ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Positive Elements ◽  
Riesz Decomposition ◽  
Riesz Decomposition Properties ◽  
Rank One

In this paper we give a representation theorem for the Cuntz monoid S(A) of a σ-unital C*-algebra A with real rank zero and stable rank one, which allows to prove several Riesz decomposition properties on the monoid. As a consequence, it is proved that the comparability conditions (FCQ), stable (FCQ) and (FCQ+) are equivalent for simple C*-algebras with real rank zero. It is also shown that the Grothendieck group [Formula: see text] of S(A) is a Riesz group, and lattice-ordered under some additional assumptions on A.

THE STABLE AND THE REAL RANK OF ${\mathcal Z}$-ABSORBING C*-ALGEBRAS

2004 ◽  
Vol 15 (10) ◽  
pp. 1065-1084 ◽  
Author(s):  
MIKAEL RØRDAM
Keyword(s):  
Complex Numbers ◽  
Real Rank ◽  
Real Rank Zero ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Matrix Algebras ◽  
C Algebras ◽  
Positive Elements ◽  
Stably Finite ◽  
Unital Simple

Suppose that A is a C*-algebra for which [Formula: see text], where [Formula: see text] is the Jiang–Su algebra: a unital, simple, stably finite, separable, nuclear, infinite-dimensional C*-algebra with the same Elliott invariant as the complex numbers. We show that: (i) The Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is almost unperforated. (ii) If A is exact, then A is purely infinite if and only if A is traceless. (iii) If A is separable and nuclear, then [Formula: see text] if and only if A is traceless. (iv) If A is simple and unital, then the stable rank of A is one if and only if A is finite. We also characterize when A is of real rank zero.

An Introduction to K-Theory for C*-Algebras

2000 ◽  
Author(s):  
M. Rørdam ◽  
F. Larsen ◽  
N. Laustsen
Keyword(s):  
C Algebras ◽  
K Theory
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