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Classification of simple $$C^*$$-algebras of tracial topological rank zero. (English) Zbl 1068.46032
As part of Elliott’s classification program for simple separable nuclear $$C^*$$-algebras, specifically the very low rank case (real rank zero and stable rank one), the present paper improves results of M. Dardalat (1995), G. A. Elliott and G. Gong (1996), G. Gong (1997), the author himself (2001, 2003) and M. Dardalat and S. Eilers (2002) in that it is no longer needed to assume that the $$C^*$$-algebra $$A$$ and $$B$$ are direct limits of some (sub)homogeneous algebras or have unique tracial states.
The main classification result is the following: {If $$A$$ and $$B$$ are two simple separable unital nuclear $$C^*$$-algebras with tracial topological rank zero which satisfy the universal coefficient theorem, and if they have isomorphic Elliott invariants, that is, if $$(K_0(A),K_0(A)_+,[1_A],K_1(A))\cong(K_0(B),K_0(B)_+,[1_B],K_1(B))$$, then $$A\cong B$$.}
Its importance lies in giving an abstract condition for a simple separable nuclear $$C^*$$-algebra to be classified by Elliott invariants that has been used by S. Walters to prove that certain crossed products of irrational rotation algebras by finite groups are $$AF$$-algebras, by N. C. Phillips to prove that simple higher-dimensional noncommutative tori are $$AT$$-algebras, and by A. Kishimoto for the study of one-parameter automorphism groups on $$AF$$-algebras.

MSC:
 46L05 General theory of $$C^*$$-algebras 46L35 Classifications of $$C^*$$-algebras 46L80 $$K$$-theory and operator algebras (including cyclic theory)
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References:
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