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

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.

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

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is014009013jkt281 ◽

2014 ◽

Vol 14 (3) ◽

pp. 570-613 ◽

Cited By ~ 3

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.

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On the classification of C*-algebras of real rank zero, III: The infinite case

Operator Algebras and Their Applications II ◽

10.1090/fic/020/02 ◽

1998 ◽

pp. 11-72 ◽

Cited By ~ 1

Author(s):

Ola Bratteli ◽

George Elliott ◽

David Evans ◽

Akitaka Kishimoto

Keyword(s):

Real Rank ◽

Real Rank Zero ◽

Rank Zero ◽

C Algebras ◽

Infinite Case

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On the classification of C*-algebras of real rank zero, IV: Reduction to local spectrum of dimension two

Operator Algebras and Their Applications II ◽

10.1090/fic/020/03 ◽

1998 ◽

pp. 73-95

Author(s):

George Elliott ◽

Guihua Gong ◽

Hongbing Su

Keyword(s):

Real Rank ◽

Local Spectrum ◽

Real Rank Zero ◽

Rank Zero ◽

C Algebras

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The classification of certain ASH C*-algebras of real rank zero

Journal of Topology and Analysis ◽

10.1142/s1793525320500466 ◽

2020 ◽

pp. 1-20

Author(s):

Qingnan An ◽

George A. Elliott ◽

Zhiqiang Li ◽

Zhichao Liu

Keyword(s):

Real Rank ◽

Inductive Limits ◽

Real Rank Zero ◽

Direct Sums ◽

Rank Zero ◽

The Real ◽

C Algebras ◽

Generalized Dimension ◽

Theoretic Classification

In this paper, using ordered total K-theory, we give a K-theoretic classification for the real rank zero inductive limits of direct sums of generalized dimension drop interval algebras.

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Homomorphisms From C(X) Into C*-Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1997-050-9 ◽

1997 ◽

Vol 49 (5) ◽

pp. 963-1009 ◽

Cited By ~ 8

Author(s):

Huaxin Lin

Keyword(s):

Real Rank ◽

Real Rank Zero ◽

Rank Zero ◽

C Algebra ◽

C Algebras ◽

Finite Dimensional ◽

Countable Rank ◽

Rank One ◽

Dimensional Range

AbstractLet A be a simple C*-algebra with real rank zero, stable rank one and weakly unperforated K0(A) of countable rank. We show that a monomorphism Φ: C(S2) → A can be approximated pointwise by homomorphisms from C(S2) into A with finite dimensional range if and only if certain index vanishes. In particular,we show that every homomorphism ϕ from C(S2) into a UHF-algebra can be approximated pointwise by homomorphisms from C(S2) into the UHF-algebra with finite dimensional range.As an application, we show that if A is a simple C*-algebra of real rank zero and is an inductive limit of matrices over C(S2) then A is an AF-algebra. Similar results for tori are also obtained. Classification of Hom (C(X), A) for lower dimensional spaces is also studied.

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Inductive Limits of K-theoretic Complexes with Torsion Coefficients

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is007011012jkt002 ◽

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.

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