Classification of simple C*-algebras and higher dimensional noncommutative tori

By Huaxin Lin. 2003

68 citations recorded by the source corpus.

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( k ) ×θ are in fact inductive limits of circle algebras.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/144c8d9863e6bd7f459c2dbc91a7025e31f49eb7

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In Algebra and Number Theory

1. Z. Niu
2. G. Elliott
3. Huaxin Lin
4. N. Christopher Phillips
5. W. Winter
6. Andrew S. Toms
7. Christopher Phillips
8. G. Gong

In Mathematical Physics

1. Z. Niu
2. Huaxin Lin
3. W. Winter
4. G. Elliott
5. N. Christopher Phillips
6. Andrew S. Toms
7. David Kerr
8. Yasuhiko Sato

In Statistical and Nonlinear Physics

1. N. Christopher Phillips
2. Z. Niu
3. G. Elliott
4. Huaxin Lin
5. G. Gong
6. W. Winter
7. Christopher Schafhauser

Citations in the corpus

• Every simple higher dimensional noncommutative torus is an AT algebra (2006, background): “14 of [22] shows that the crossed product of (S(1)) by a minimal rotation is an AT algebra; in this case, most of the entries of the relevant skew symmetric matrix are zero.”
• Approximate Homotopy of Homomorphisms from C(x) Into a Simple C*-Algebra (2006, background): “We refer the reader to [24], [28], [25], [27], [3], [45] and [46] for more information.”
• On tracial approximation (2008, background): “In this way, the higher-dimensional simple noncommutative tori were shown to be AT-algebras in [22] and [29], and the crossed-product C*-algebras arising from certain minimal homeomorphisms were shown to be AH-algebras in [24].”
• A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one. (2018, unknown): “5 of [65].”
• Classification of homomorphisms and dynamical systems (2004, unknown): “with tracial rank zero which satisfy the UCT are classified by their K-theory (see [40] and [ 39 ]).”