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On The Classification of Nuclear C * ‐Algebras
Author(s) -
Dadarlat Marius,
Eilers Søren
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/85.1.168
Subject(s) - morphism , mathematics , element (criminal law) , realization (probability) , simple (philosophy) , cover (algebra) , pure mathematics , mathematics subject classification , point (geometry) , zero (linguistics) , algebra over a field , discrete mathematics , geometry , statistics , philosophy , linguistics , epistemology , mechanical engineering , political science , law , engineering
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.