tailieunhanh - Đề tài " Classification of simple C*algebras and higher dimensional noncommutative tori "

We show that unital simple C ∗ -algebras with tracial topological rank zero which are locally approximated by subhomogeneous C ∗ -algebras can be classified by their ordered K-theory. We apply this classification result to show that certain simple crossed products are isomorphic if they have the same ordered K-theory. In particular, irrational higher dimensional noncommutative tori of the form C(Tk ) ×θ Z are in fact inductive limits of circle algebras. Introduction In recent years there has been rapid progress in classification of nuclear simple C . | Annals of Mathematics Classification of simple C -algebras and higher dimensional noncommutative tori By Huaxin Lin Annals of Mathematics 157 2003 521 544 Classification of simple C -algebras and higher dimensional noncommutative tori By Huaxin Lin Abstract We show that unital simple C -algebras with tracial topological rank zero which are locally approximated by subhomogeneous C -algebras can be classified by their ordered K-theory. We apply this classification result to show that certain simple crossed products are isomorphic if they have the same ordered K-theory. In particular irrational higher dimensional noncommutative tori of the form C Tk x are in fact inductive limits of circle algebras. Introduction In recent years there has been rapid progress in classification of nuclear simple C -algebras. In the case that C -algebras are of real rank zero and finite Elliott and Gong EG have proved that simple inductive limits of finite direct sums of homogeneous C -algebras AH for brevity of slow dimension growth with real rank zero can be completely classified up to isomorphism by their scaled ordered K-theory with the reduction of dimension growth proved by G2 and D . In their remarkable paper EG they also showed that the class of AH-algebras that they classified exhausts all possible invariants. So any general classification theorem for simple C -algebras of real rank zero and stable rank one with weakly unperforated K0 will not expand their class. However many interesting simple C -algebras which are important in applications do not arise as inductive limits of finite direct sums of homogeneous C -algebras. Therefore it is extremely important to have a classification theorem which covers C -algebras that are not assumed to be AH-algebras. The main purpose of this paper is to establish such a theorem. Our general classification result covers at least some of the well-known interesting simple C -algebras that are not known to be AH-algebras. Research partially .