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Asymptotic unitary equivalence and classification of simple amenable C ∗-algebras

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Abstract

Let C and A be two unital separable amenable simple C ∗-algebras with tracial rank at most one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ϕ 1,ϕ 2:C→A are two unital monomorphisms. We show that there is a continuous path of unitaries {u t :t∈[0,∞)} of A such that

$$\lim_{t\to\infty}u_t^*\varphi_1(c)u_t=\varphi_2(c)\quad\mbox{for all }c\in C$$

if and only if [ϕ 1]=[ϕ 2] in \(KK(C,A),\varphi_{1}^{\ddag}=\varphi_{2}^{\ddag},(\varphi_{1})_{T}=(\varphi _{2})_{T}\) and a rotation related map \(\overline{R}_{\varphi_{1},\varphi_{2}}\) associated with ϕ 1 and ϕ 2 is zero.

Applying this result together with a result of W. Winter, we give a classification theorem for a class \({\mathcal{A}}\) of unital separable simple amenable C ∗-algebras which is strictly larger than the class of separable C ∗-algebras with tracial rank zero or one. Tensor products of two C ∗-algebras in \({\mathcal{A}}\) are again in \({\mathcal{A}}\). Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras for which the state space of K 0 is the same as the tracial state space and also some unital simple ASH-algebras whose K 0-group is ℤ and whose tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are \({\mathcal{Z}}\)-stable are isomorphic to ones with no dimension growth.

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References

  1. Blackadar, B., Rørdam, M.: Extending states on preordered semigroups and the existence of quasitraces on C ∗-algebras. J. Algebra 152, 240–247 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Dadarlat, M., Loring, T.: A universal multicoefficient theorem for the Kasparov groups. Duke Math. J. 84, 355–377 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. de la Harpe, P., Skandalis, G.: Déterminant associé a une trace sur une algébre de Banach. Ann. Inst. Fourier 34, 169–202 (1984)

    MATH  Google Scholar 

  4. Dixmier, J.: On some C ∗-algebras considered by Glimm. J. Funct. Anal. 1, 182–203 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  5. Elliott, G.A.: On the classification of C ∗-algebras of real rank zero. J. Reine Angew. Math. 443, 179–219 (1993)

    MATH  MathSciNet  Google Scholar 

  6. Elliott, G.A., Gong, G.: On the classification of C ∗-algebras of real rank zero, II. Ann. Math. 144, 497–610 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Elliott, G.A., Gong, G., Li, L.: Approximate divisibility of simple inductive limit C ∗-algebras. In: Operator Algebras and Operator Theory, Shanghai, 1997. Contemp. Math., vol. 228, pp. 87–97. Am. Math. Soc., Providence (1998)

    Google Scholar 

  8. Elliott, G.A., Gong, G., Li, L.: Injectivity of the connecting maps in AH inductive limit systems. Can. Math. Bull. 48, 50–68 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Elliott, G.A., Gong, G., Li, L.: On the classification of simple inductive limit C ∗-algebras. II. The isomorphism theorem. Invent. Math. 168, 249–320 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Elliott, G.A., Loring, T.: AF embeddings of C(T 2) with a prescribed K-theory. J. Funct. Anal. 103, 1–25 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Elliott, G.A., Rørdam, M.: Classification of certain infinite simple C ∗-algebras, II. Comment. Math. Helv. 70, 615–638 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  12. Exel, R.: The soft torus and applications to almost commuting metrics. Pac. J. Math. 160, 207–217 (1993)

    MATH  MathSciNet  Google Scholar 

  13. Gong, G.: On the classification of simple inductive limit C ∗-algebras, I. The reduction theorem. Doc. Math. 7, 255–461 (2002)

    MATH  MathSciNet  Google Scholar 

  14. Haagerup, U.: Quasitraces in exact C ∗-algebras are traces. Manuscript distributed at the Operator Algebra Conference in Istanbul (1991)

  15. Jiang, X., Su, H.: On a simple unital projectionless C ∗-algebra. Am. J. Math. 121, 359–413 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kishimoto, A., Kumjian, A.: The Ext class of an approximately inner automorphism, II. J. Oper. Theory 46, 99–122 (2001)

    MathSciNet  Google Scholar 

  17. Li, L.: C ∗-algebra homomorphisms and KK-theory. K-Theory 18, 161–172 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lin, H.: Tracial topological ranks of C ∗-algebras. Proc. Lond. Math. Soc. 83, 199–234 (2001)

    Article  MATH  Google Scholar 

  19. Lin, H.: An Introduction to the Classification of Amenable C ∗-Algebras. World Scientific, River Edge (2001)

    Book  Google Scholar 

  20. Lin, H.: Embedding an AH-algebra into a simple C ∗-algebra with prescribed KK-data. K-Theory 24, 135–156 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lin, H.: Classification of simple C ∗-algebras and higher dimensional noncommutative tori. Ann. Math. 157, 521–544 (2003)

    Article  MATH  Google Scholar 

  22. Lin, H.: Simple AH-algebras of real rank zero. Proc. Am. Math. Soc. 131, 3813–3819 (2003)

    Article  MATH  Google Scholar 

  23. Lin, H.: Classification of simple C ∗-algebras with tracial topological rank zero. Duke Math. J. 125, 91–119 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lin, H.: Unitary equivalences for essential extensions of C ∗-algebras. Proc. Am. Math. Soc. 137, 3407–3420 (2009). arXiv:math/0403236

    Article  MATH  Google Scholar 

  25. Lin, H.: Approximate homotopy of homomorpisms from C(X) into a simple C ∗-algebra. Mem. Am. Math. Soc. 205(963), vi+131 pp. (2010). math.OA/0612125

  26. Lin, H.: Simple nuclear C ∗-algebras of tracial topological rank one. J. Funct. Anal. 251, 601–679 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lin, H.: AF-embedding of crossed products of AH-algebras by ℤ and asymptotic AF-embedding. Indiana Univ. Math. J. 57, 891–944 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lin, H.: Asymptotically unitary equivalence and asymptotically inner automorphisms. Am. J. Math. 131, 1589–1677 (2009)

    Article  MATH  Google Scholar 

  29. Lin, H.: AF-embedding of the crossed products of AH-algebras by finitely generated Abelian groups. Int. Math. Res. Pap. 3, rpn007 (2008)

    Google Scholar 

  30. Lin, H.: Localizing the Elliott conjecture at strongly self-absorbing C ∗-algebras—an appendix. Preprint. arXiv:0709.1654 v2

  31. Lin, H.: Approximate unitary equivalence in simple C ∗-algebras of tracial rank one. Preprint. arXiv:0801.2929

  32. Lin, H.: The Range of approximate unitary equivalence classes of homomorphisms from AH-algebras. Math. Z. 263, 903–922 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  33. Lin, H.: Homotopy of unitaries in simple C ∗-algebras with tracial rank one. J. Funct. Anal. 258, 1822–1882 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. Lin, H., Niu, Z.: Lifting KK-elements, asymptotical unitary equivalence and classification of simple C ∗-algebras. Adv. Math. 219, 1729–1769 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Lin, H., Niu, Z.: The range of classifiable simple amenable C ∗-algebras. Preprint. arXiv:0808.3424

  36. Loring, T.: K-theory and asymptotically commuting matrices. Can. J. Math. 40, 197–216 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  37. Mygind, J.: Classification of certain simple C ∗-algebras with torsion in K 1. Can. J. Math. 53, 1223–1308 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  38. Nielsen, K.E., Thomsen, K.: Limits of circle algebras. Expo. Math. 14, 17–56 (1996)

    MATH  MathSciNet  Google Scholar 

  39. Thomsen, K.: Traces, unitary characters and crossed products by Z. Publ. Res. Inst. Math. Sci. 31, 1011–1029 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  40. Toms, A., Winter, W.: Strongly self-absorbing C ∗-algebras. Trans. Am. Math. Soc. 359, 3999–4029 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  41. Toms, A., Winter, W.: The Elliott conjecture for Villadsen algebras of the first type. J. Funct. Anal. 256, 1311–1340 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  42. Winter, W.: Localizing the Elliott conjecture at strongly self-absorbing C ∗-algebras. Preprint. arXiv:0708.0283v2

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  1. Department of Mathematics, East China Normal University, Shanghai, 200062, China

    Huaxin Lin

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Lin, H. Asymptotic unitary equivalence and classification of simple amenable C ∗-algebras. Invent. math. 183, 385–450 (2011). https://doi.org/10.1007/s00222-010-0280-9

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