Furstenberg Transformations and Approximate Conjugacy

Lin, Huaxin

Canadian journal of mathematics, Tome 60 (2008) no. 1, pp. 189-207

Résumé

Let $\alpha $ and $\beta $ be two Furstenberg transformations on 2-torus associated with irrational numbers ${{\theta }_{1}},\,{{\theta }_{2}}$ , integers ${{d}_{1}},\,{{d}_{2}}$ and Lipschitz functions ${{f}_{1}}\,\text{and}\,{{f}_{2}}$ . It is shown that $\alpha $ and $\beta $ are approximately conjugate in ameasure theoretical sense if (and only if) $\overline{{{\theta }_{1}}\,\pm \,{{\theta }_{2}}}\,=\,0\,\text{in}\,\mathbb{R}\text{/}\mathbb{Z}$ . Closely related to the classification of simple amenable ${{C}^{*}}$ -algebras, it is shown that $\alpha $ and $\beta $ are approximately $K$ -conjugate if (and only if) $\overline{{{\theta }_{1}}\,\pm \,{{\theta }_{2}}}\,=\,0\,\text{in}\,\mathbb{R}\text{/}\mathbb{Z}$ and $|{{d}_{1}}|\,=\,|{{d}_{2}}|$ . This is also shown to be equivalent to the condition that the associated crossed product ${{C}^{*}}$ -algebras are isomorphic.

Détail

DOI : 10.4153/CJM-2008-008-2

Mots-clés : 37A55, secondary, 46L35, Furstenberg transformations, approximate conjugacy

Bibliographie

[1] [1] Dadarlat, M. and Eillers, S., On the classification of nuclear C*-algebras, Proc. LondonMath. Soc. 85(2002), no. 1, 168–210. Google Scholar

[2] [2] Elliott, G. A. and Gong, G., On the classification of C*-algebras of real rank zero, II. Ann. of Math. 144(1996), no. 3, 497–610. Google Scholar

[3] [3] Elliott, G. A., Gong, G. and Li, L., On the classification of simple inductive limit C*-algebras. II. The isomorphism theorem. Invent.Math. 168(2007), no. 2, 249–320. Google Scholar

[4] [4] Furstenberg, H., Strict ergodicity and transformation of the torus. Amer. J. Math. 83(1961), 573–601. Google Scholar

[5] [5] Giordano, T., Putnam, I. and Skau, C., Topological orbit equivalence and C*-crossed products. J. Reine Angew.Math. 469(1995), 51–111. Google Scholar

[6] [6] Ji, R., On the Crossed Product C*-algebras Associated with Furstenberg Transformation on tori. Ph. D. Thesis, State University of New York at Stoney Brook, 1986. Google Scholar

[7] [7] Kodaka, K., Tracial states on crossed products associated with Furstenberg transformations on the 2-torus. Studia Math. 115(1995), no. 2, 183–187. Erratum to: “Tracial states on crossed products associated with Furstenberg transformations on the 2-torus”. Studia Math. (1996), no. 1, 97–98. Google Scholar

[8] [8] Kodaka, K., Anzai and Furstenberg transformations on the 2-torus and topologically quasi-discrete spectrum. Canad. Math. Bull. 38(1995), no. 1, 87–92. Google Scholar

[9] [9] Lin, H., Classification of simple tracially AF C*-algebras. Canad. J. Math. 53(2001), no. 1, 161–194. Google Scholar

[10] [10] Lin, H., The tracial topological ranks of C*-algebras, Proc. LondonMath. Soc. 83(2001), no. 1, 199–234. Google Scholar

[11] [11] Lin, H., An Introduction to the Classification of Amenable C*-algebras, World Scientific, River Edge, NJ, 2001. Google Scholar

[12] [12] Lin, H., Classification of simple C*-algebras and higher dimensional noncommutative tori. Ann. of Math. 157(2003), no. 2, 521–544. Google Scholar

[13] [13] Lin, H., Classification of simple C*-algebras of tracial topological rank zero. Duke Math. J. 125(2004), no. 1, 91–119. Google Scholar

[14] [14] Lin, H., Classification of homomorphisms and dynamical systems, Trans. Amer.Math. Soc. 359(2007), no. 2, 859–895. Google Scholar

[15] [15] Lin, H. and Matui, H., Minimal dynamical systems and approximate conjugacy. Math. Ann. 332(2005), no. 4, 795–822. Google Scholar

[16] [16] Lin, H. and Matui, H., Minimal dynamical systems on the product of the Cantor set and the circle, Comm. Math. Phys. 257(2005), no. 2, 425–471. Google Scholar

[17] [17] Lin, H. and Matui, H., Minimal dynamical systems on the product of the Cantor set and the circle, II. Selecta Math. 12(2006), no. 2, 199–239. Google Scholar

[18] [18] Lin, H. and Phillips, N. C., Crossed products by minimal homeomorphisms , preprint, math.OA/0408291. Google Scholar

[19] [19] Lin, Q. and Phillips, N. C., Ordered K-theory for C*-algebras of minimal homeomorphisms. In: Operator Algebras and Operator Theory. ContemporaryMathematics 228, American Mathematical Society, Providence, RI, 1998 pp. 289–314. Google Scholar

[20] [20] Matui, H., Approximate conjugacy and full groups of Cantor minimal systems, Publ. Res. Inst.Math. Sci. 41(2005), no. 3, 695–722. Google Scholar

[21] [21] Osaka, H. and Phillips, N. C., Furstenberg transformation on irrational rotation algebras. Ergodic Theory Dynam. Systems 26(2006), no. 5, 1623–1651. Google Scholar

[22] [22] Pedersen, G. K., C*-algebras and their Automorphism Groups, Academic Press, London, 1979. Google Scholar

[23] [23] Phillips, N. C., Cancellation and stable rank for direct limits of recursive subhomogeneous algebras, Trans. Amer.Math. Soc., to appear, (math.OA/0101157). Google Scholar

[24] [24] Rouhani, H., A Furstenberg transformation of the 2-torus without quasi-discrete spectrum. Canad. Math. Bull. 33(1990), no. 3, 316–322. Google Scholar

[25] [25] Tomiyama, J., Topological full groups and structure of normalizers in transformation group C*-algebras. Pacific J. Math. 173(1996), no. 2, 571–583. Google Scholar