TAILIEUCHUNG - Đề tài " On the classification problem for nuclear C -algebras "

We exhibit a counterexample to Elliott’s classification conjecture for simple, separable, and nuclear C∗ -algebras whose construction is elementary, and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves. The consequences for the program to classify nuclear C∗ -algebras are far-reaching: one has, among other things, that existing results on the classification of simple, unital AH algebras via the Elliott invariant of K-theoretic data are the best possible, and that these cannot be improved by the addition of continuous homotopy invariant. | Annals of Mathematics On the classification problem for nuclear CD-algebras By Andrew S. Toms Annals of Mathematics 167 2008 1029-1044 On the classification problem for nuclear G -algebras By Andrew S. Toms Abstract We exhibit a counterexample to Elliott s classification conjecture for simple separable and nuclear C -algebras whose construction is elementary and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves. The consequences for the program to classify nuclear C -algebras are far-reaching one has among other things that existing results on the classification of simple unital AH algebras via the Elliott invariant of K-theoretic data are the best possible and that these cannot be improved by the addition of continuous homotopy invariant functors to the Elliott invariant. 1. Introduction Elliott s program to classify nuclear C -algebras via K-theoretic invariants see E2 for an overview has met with considerable success since his seminal classification of approximately finite-dimensional AF algebras via their scaled ordered K0-groups E1 . Classification results of this nature are existence theorems asserting that isomorphisms at the level of certain invariants for C -algebras in a class B are liftable to isomorphisms at the level of the algebras themselves. Obtaining such theorems usually requires proving a uniqueness theorem for B . a theorem which asserts that two isomorphisms between members A and B of B which agree at the level of the said invariants differ by a locally inner automorphism. Elliott s program began in earnest with his classification of simple circle algebras of real rank zero in 1989 he conjectured shortly thereafter that the topological K-groups the Choquet simplex of tracial states and the natural connections between these objects would form a complete invariant for the class of separable .

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