scholarly journals Reduction of filtered K-theory and a characterization of Cuntz-Krieger algebras

2014 ◽  
Vol 14 (3) ◽  
pp. 570-613 ◽  
Author(s):  
Sara E. Arklint ◽  
Rasmus Bentmann ◽  
Takeshi Katsura
Keyword(s):  
Primitive Ideal ◽  
Ideal Space ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
K Theory

AbstractWe show that filtered K-theory is equivalent to a substantially smaller invariant for all real-rank-zero C*-algebras with certain primitive ideal spaces—including the infinitely many so-called accordion spaces for which filtered K-theory is known to be a complete invariant. As a consequence, we give a characterization of purely infinite Cuntz–Krieger algebras whose primitive ideal space is an accordion space.

scholarly journals FILTRATED K-THEORY FOR REAL RANK ZERO C*-ALGEBRAS

2012 ◽  
Vol 23 (08) ◽  
pp. 1250078 ◽  
Author(s):  
SARA ARKLINT ◽  
GUNNAR RESTORFF ◽  
EFREN RUIZ
Keyword(s):  
Primitive Ideal ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebras ◽  
K Theory

The smallest primitive ideal spaces for which there exist counterexamples to the classification of non-simple, purely infinite, nuclear, separable C*-algebras using filtrated K-theory, are four-point spaces. In this article, we therefore restrict to real rank zero C*-algebras with four-point primitive ideal spaces. Up to homeomorphism, there are ten different connected T0-spaces with exactly four points. We show that filtrated K-theory classifies real rank zero, tight, stable, purely infinite, nuclear, separable C*-algebras that satisfy that all simple subquotients are in the bootstrap class for eight out of ten of these spaces.

scholarly journals Inductive Limits of K-theoretic Complexes with Torsion Coefficients

2007 ◽  
Vol 1 (1) ◽  
pp. 145-168
Author(s):  
Søren Eilers ◽  
Andrew S. Toms
Keyword(s):  
Real Rank ◽  
Inductive Limits ◽  
Real Rank Zero ◽  
Rank Zero ◽  
K Theory

AbstractWe present the first range result for the total K-theory of C*-algebras. This invariant has been used successfully to classify certain separable, nuclear C*-algebras of real rank zero. Our results complete the classification of the so-called AD algebras of real rank zero.

On Extensions of Stably FiniteC*-Algebras (II)

2016 ◽  
Vol 59 (2) ◽  
pp. 435-439
Author(s):  
Hongliang Yao
Keyword(s):  
Necessary Condition ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
Approximate Unit ◽  
K Theory ◽  
Stably Finite ◽  
Sufficient And Necessary Condition ◽  
Smallest Ideal

AbstractFor any C*-algebra A with an approximate unit of projections, there is a smallest ideal I of A such that the quotient A/I is stably finite. In this paper a sufficient and necessary condition for an ideal of a C*-algebra with real rank zero to be this smallest ideal is obtained by using K-theory

scholarly journals The Ordered K-theory of a Full Extension

2014 ◽  
Vol 66 (3) ◽  
pp. 596-624 ◽  
Author(s):  
Søren Eilers ◽  
Gunnar Restorff ◽  
Efren Ruiz
Keyword(s):  
Theoretical Description ◽  
Essential Extension ◽  
Real Rank ◽  
Factorization Property ◽  
Real Rank Zero ◽  
Classification Result ◽  
Full Extension ◽  
Cancellation Property ◽  
Rank Zero ◽  
K Theory

AbstractLet be a C*-algebra with real rank zero that has the stable weak cancellation property. Let be an ideal of such that is stable and satisfies the corona factorization property. We prove thatis a full extension if and only if the extension is stenotic and K-lexicographic. As an immediate application, we extend the classification result for graph C*-algebras obtained by Tomforde and the first named author to the general non-unital case. In combination with recent results by Katsura, Tomforde, West, and the first named author, our result may also be used to give a purely K-theoretical description of when an essential extension of two simple and stable graph C*-algebras is again a graph C*- algebra.

scholarly journals CONDITION (K) FOR BOOLEAN DYNAMICAL SYSTEMS

2021 ◽  
pp. 1-25
Author(s):  
TOKE MEIER CARLSEN ◽  
EUN JI KANG
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Directed Graphs ◽  
Ideal Space ◽  
Real Rank ◽  
Topological Dimension ◽  
Real Rank Zero ◽  
Locally Finite ◽  
Rank Zero ◽  
Ideal Property

Abstract We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta )$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta )$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ .

Linear Maps on C*-Algebras Preserving the Set of Operators that are Invertible in

2011 ◽  
Vol 54 (1) ◽  
pp. 141-146
Author(s):  
Sang Og Kim ◽  
Choonkil Park
Keyword(s):  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
Calkin Algebra ◽  
C Algebras ◽  
Linear Maps

AbstractFor C*-algebras of real rank zero, we describe linear maps ϕ on that are surjective up to ideals , and π(A) is invertible in if and only if π(ϕ(A)) is invertible in , where A ∈ and π : → is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra.

scholarly journals Type I $C^*$-algebras of real rank zero

1997 ◽  
Vol 125 (9) ◽  
pp. 2671-2676
Author(s):  
Huaxin Lin
Keyword(s):  
Type I ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebras
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