On The Classification of Nuclear C ^* ‐Algebras

Abstract

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.

Classification of simple C*-algebras of tracial topological rank zero

“…With the results in [Ln6] and the introduction of the notion of tracial topological rank (or TAF), the desired uniqueness theorem (5.1) was established in [Ln8]. However, the version of the (half) existence theorem (4.1) in [Ln8] (see also [DE2]) does not provide a true existence theorem for the purpose of classification of simple nuclear C * -algebras with tracial topological rank zero. Extra conditions on the order structure of K 0 (A) were needed in [Ln8], [DE2] and [Ln11].…”

“…However, the version of the (half) existence theorem (4.1) in [Ln8] (see also [DE2]) does not provide a true existence theorem for the purpose of classification of simple nuclear C * -algebras with tracial topological rank zero. Extra conditions on the order structure of K 0 (A) were needed in [Ln8], [DE2] and [Ln11]. The main technical result in this paper is to recover the order information lost in 4.1.…”

“…Many other mathematicians have contributed to this program. (See for example, [BEEK], [BBEK], [D1], [D2], [DE2], [DL1], [DL2], [DG], [Ell1], [Ell2], [Ell3], [Ell4], [EG1], [EE], [EG2], [EGLP], [EGS], [ER], [ES], [G1], [G2], [JS], [BK1], [BK2], [Li1], [Li2], [Ln1]- [Ln9], [Ln11]- [Ln12], [LP1], [LP2], [LP3], [LS], [LqP], [NT], [P1], [P2], [Ro1], [Ro2], [Ro3], [ER], [Su1], [Su2], [Th1], [Th2] -an incomplete list). The application of the classification program began to spread to different fields, in particular, to the study of dynamical systems.…”

Commutative ${C}^*$-subalgebras of simple stably finite ${C}^*$-algebras with real rank zero

“…Finally, the results of this paper are connected to but distinct from the existence and uniqueness theorems found in [6], [7] and [26]. The theorems in these papers require that the domain algebra be a simple TAF C * -algebra.…”

Morphisms of Simple Tracially Af Algebras

“…The algebras in the latter class satisfy a local approximation property similar to the one in the definition of the tracially AF algebras, but with the role of matrix algebras M n (C) played by the Cuntz algebra O 2 . Elaborating on this analogy, and using relative quasidiagonality as a unifying concept, in our joint work with Eilers [10] we reproved the classification of Kirchberg's algebras satisfying the UCT by the same very method that we used in the proof of the classification of separable simple unital nuclear tracially AF algebras satisfying the UCT with K 0 group isomorphic to Q.…”