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Comparison Theory and Smooth Minimal C*-Dynamics

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Abstract

We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar’s Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimension functions. These results continue to hold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one. Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C*-algebra. This is also used to construct uncountably many non-Morita-equivalent simple separable amenable C*-algebras with the same K-theory and tracial state space, providing a C*-algebraic analogue of McDuff’s uncountable family of II1 factors. We prove in passing that the range of the radius of comparison is exhausted by simple C*-algebras.

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References

  1. Blackadar, B.: Comparison theory for simple C*-algebras, Operator Algebras and Applications, Vol. 1, 21–54, London Math. Soc. Lecture Note Ser., 135, Cambridge Univ. Press, Cambridge, 1988, pp. 21–54

  2. Blackadar B., Handelman D.: Dimension Functions and Traces on C*-algebras. J. Funct. Anal. 45, 297–340 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brown, N.P., Perera, F., Toms, A.S.: The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras, J. Reine Angew. Math., to appear

  4. Brown, N.P., Toms, A.S.: Three applications of the Cuntz semigroup. Int. Math. Res. Not., Article ID rnm068, (2007), 14 pages

  5. Ciuperca, A., Elliott, G.A.:A remark on invariants for C*-algebras of stable rank one. Int. Math. Res. Not., to appear

  6. Connes A.: Classification of injective factors. Cases II 1, II ∞, III λ, λ  ≠  1. Ann. of Math. (2) 104, 73–115 (1976)

    Article  MathSciNet  Google Scholar 

  7. Coward, K., Elliott, G.A., Ivanescu, C.: The Cuntz semigroup as an invariant for C*-algebras. J. Reine Angew. Math., to appear

  8. Ciuperca, A., Santiago, L., and Robert, L.: The Cuntz semigroup of ideals and quotients, and a generalized Kasparov Stabilization Theorem, J. Op. Th., to appear

  9. Cuntz J.: Dimension Functions on Simple C*-algebras. Math. Ann. 233, 145–153 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  10. 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  ADS  MathSciNet  Google Scholar 

  11. Elliott G.A., Toms A.S.: Regularity properties in the classification program for separable amenable C*-algebras. Bull. Amer. Math. Soc. 45, 229–245 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Giol, J., Kerr, D.: Subshifts and perforation. Preprint, 2007

  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.: Connes’ bicentralizer problem and uniqueness of the injective factor of type III 1. Acta. Math. 158, 95–148 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jacob, B.: A remark on the distance between unitary orbits in ASH algebras with unique trace. Preprint, 2008

  16. Kirchberg E., Rørdam M.: Non-simple purely infinite C*-algebras. Amer. J. Math. 122, 637–666 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lin, Q., Phillips, N.C.: The structure of C*-algebras of minimal diffeomorphisms. In preparation

  18. McDuff D.: Uncountably many II1 factors. Ann. of Math. (2) 90, 372–377 (1969)

    Article  MathSciNet  Google Scholar 

  19. Ng P.W., Winter W.: A note on Subhomogeneous C*-algebras. C. R. Math. Acad. Sci. Soc. R. Can. 28, 91–96 (2006)

    MATH  MathSciNet  Google Scholar 

  20. Popa S.: On a class of II1 factors with Betti numbers invariants. Ann. of Math. (2) 163, 809–899 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Perera F., Toms A.S.: Recasting the Elliott conjecture. Math. Ann. 338, 669–702 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Phillips N.C.: Recursive subhomogeneous algebras. Trans. Amer. Math. Soc. 359, 4595–4623 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Phillips N.C.: Cancellation and stable rank for direct limits of recursive subhomogeneous algebras. Trans. Amer. Math. Soc. 359, 4625–4652 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rieffel M.A.: Dimension and stable rank in the K-theory of C*-algebras. Proc. London Math. Soc. (3) 46, 301–333 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rørdam M (2002) Classification of Nuclear C*-Algebras, Encyclopaedia of Mathematical Sciences 126, Berlin-Heidelberg: Springer-Verlag, 2002

  26. Rørdam M.: The stable and the real rank of \({\mathcal{Z}}\) -absorbing C*-algebras. Int. J. Math. 15, 1065–1084 (2004)

    Article  Google Scholar 

  27. Toms A.S.: On the classification problem for nuclear C*-algebras. Ann. of Math. (2) 167, 1059–1074 (2008)

    Article  MathSciNet  Google Scholar 

  28. Toms A.S.: Flat dimension growth for C*-algebras. J. Funct. Anal. 238, 678–708 (2006)

    MATH  MathSciNet  Google Scholar 

  29. Toms, A.S.: An infinite family of non-isomorphic C*-algebras with identical K-theory. Trans. Amer. Math. Soc., to appear

  30. Toms A.S.: Stability in the Cuntz semigroup of a commutative C*-algebra. Proc. London Math. Soc. (3) 96, 1–25 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Villadsen J.: Simple C*-algebras with perforation. J. Funct. Anal. 154, 110–116 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele St., Toronto, Ontario, Canada, M3J 1P3

    Andrew S. Toms

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  1. Andrew S. Toms
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Correspondence to Andrew S. Toms.

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Communicated by Y. Kawahigashi

This research was supported in part by an NSERC Discovery Grant.

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Toms, A.S. Comparison Theory and Smooth Minimal C*-Dynamics. Commun. Math. Phys. 289, 401–433 (2009). https://doi.org/10.1007/s00220-008-0665-4

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  • DOI: https://doi.org/10.1007/s00220-008-0665-4

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