On The Classification of Nuclear C * ‐Algebras

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.