Classification of extensions of certain C * -algebras by their six term exact sequences in K -theory

We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C*-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism. These results are employed to obtain classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin, and to reprove the classification of purely infinite simple nuclear C*-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to the classification of nuclear C*-algebras.2000 Mathematical Subject Classification: primary 46L35; secondary 19K14, 19K35, 46L80.

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MORPHISMS OF SIMPLE TRACIALLY AF ALGEBRAS

International Journal of Mathematics ◽

10.1142/s0129167x04002636 ◽

2004 ◽

Vol 15 (09) ◽

pp. 919-957 ◽

Cited By ~ 10

Author(s):

MARIUS DADARLAT

Keyword(s):

Existence And Uniqueness ◽

Residually Finite ◽

C Algebras ◽

Finite Dimensional ◽

K Theory ◽

Inner Automorphisms ◽

Af Algebras ◽

Universal Coefficient Theorem

Let A, B be separable simple unital tracially AF C*-algebras. Assuming that A is exact and satisfies the Universal Coefficient Theorem (UCT) in KK-theory, we prove the existence, and uniqueness modulo approximately inner automorphisms, of nuclear *-homomorphisms from A to B with prescribed K-theory data. This implies the AF-embeddability of separable exact residually finite-dimensional C*-algebras satisfying the UCT and reproves Huaxin Lin's theorem on the classification of nuclear tracially AF C*-algebras.

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On the Classification of Simple Stably Projectionless C*-Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-2002-006-7 ◽

2002 ◽

Vol 54 (1) ◽

pp. 138-224 ◽

Cited By ~ 14

Author(s):

Shaloub Razak

Keyword(s):

Building Blocks ◽

Isomorphism Theorem ◽

Inductive Limits ◽

C Algebras ◽

Distinguished Subset ◽

K Theory

AbstractIt is shown that simple stably projectionless C*-algebras which are inductive limits of certain specified building blocks with trivial K-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras.

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

International Journal of Mathematics ◽

10.1142/s0129167x12500784 ◽

2012 ◽

Vol 23 (08) ◽

pp. 1250078 ◽

Cited By ~ 5

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.

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On Rørdam's classification of certain $C^*$-algebras with one non-trivial ideal, II

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15045 ◽

2007 ◽

Vol 101 (2) ◽

pp. 280 ◽

Cited By ~ 9

Author(s):

Gunnar Restorff ◽

Efren Ruiz

Keyword(s):

Classification Theorem ◽

C Algebras ◽

Essential Extensions ◽

Exact Sequences

In this paper we extend the classification results obtained by Rørdam in the paper [16]. We prove a strong classification theorem for the unital essential extensions of Kirchberg algebras, a classification theorem for the non-stable, non-unital essential extensions of Kirchberg algebras, and we characterize the range in both cases. The invariants are cyclic six term exact sequences together with the class of some unit.

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