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The structure of crossed products of irrational rotation algebras by finite subgroups of \(\text{SL}_2(\mathbb Z)\). (English) Zbl 1202.46081

The authors consider crossed products \(A_\theta\rtimes_{\alpha}F\), where \(A_\theta\) is an irrational rotation algebra and \(F\) is a finite subgroup of \(\text{SL}\) acting via the canonical action of \(\text{SL}\) on \(A_\theta\). (Such subgroups are all cyclic of order \(2\), \(3\), \(4\) or \(6\).) The main results can be summarized as follows. The crossed product \(A_\theta\rtimes_{\alpha}F\) is always an AF algebra and
\[ \begin{aligned} K_{0}(A_\theta\rtimes_{\alpha}\mathbb{Z}_{2})&\cong \mathbb{Z}^{6},\\ K_{0}(A_\theta\rtimes_{\alpha}\mathbb{Z}_{3})&\cong \mathbb{Z}^{8},\\ K_{0}(A_\theta\rtimes_{\alpha}\mathbb{Z}_{4})&\cong \mathbb{Z}^{9},\\ K_{0}(A_\theta\rtimes_{\alpha}\mathbb{Z}_{6})&\cong \mathbb{Z}^{10}. \end{aligned} \]
Furthermore, for \(k,l=2,3,4,6\), \(A_\theta\rtimes_{\alpha}\mathbb{Z}_{k}\) is isomorphic to \(A_{\rho}\rtimes_{\alpha}\mathbb{Z}_{l}\) if and only if \(k=l\) and \(\rho=\pm\theta\mod {\mathbb{Z}}\).
As a corollary, the authors deduce that the fixed point algebras \(A_\theta^{F}\) for the actions \(\alpha:G\to\text{Aut}A_\theta\) are also AF algebras.
The proof is involved and is accomplished through a number of intermediate results that will be of interest and useful on their own. There isn’t room to list many here, but an example is the following. The authors show that if \(\omega_{0}\) and \(\omega_{1}\) are suitably homotopic Borel \(2\)-cocycles on a locally compact group \(G\) satisfying the Baum-Connes conjecture with coefficients, then \(K_{*}(C^{*}_{r}(G,\omega_{0})) \cong K_{*}(C^{*}_{r}(G,\omega_{1}))\).
The first part of the paper is devoted to computing the \(K\)-theory of the crossed products. The authors then show that the crossed products satisfy the Universal Coefficient Theorem and the Tracial Rokhlin Property. From this, they deduce that the crossed products have tracial rank zero. Then they can apply H.-X.Lin’s classification result [Duke Math.J.125, No.1, 91–119 (2004; Zbl 1068.46032)] to conclude that the crossed products are AF.

MSC:

46L55 Noncommutative dynamical systems
46L80 \(K\)-theory and operator algebras (including cyclic theory)

Citations:

Zbl 1068.46032
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References:

[1] DOI: 10.1093/qmath/21.3.277 · Zbl 0208.03603 · doi:10.1093/qmath/21.3.277
[2] DOI: 10.1093/qmath/21.2.203 · Zbl 0194.33701 · doi:10.1093/qmath/21.2.203
[3] Boca F., Math. 492 pp 179– (1997)
[4] DOI: 10.1142/S0129167X91000090 · Zbl 0759.46063 · doi:10.1142/S0129167X91000090
[5] Bratteli O., J. Oper. Th. 27 pp 53– (1992)
[6] DOI: 10.1007/BF02097244 · Zbl 0772.46040 · doi:10.1007/BF02097244
[7] Brenken B. A., Pacific J. Math. 111 pp 257– (1984)
[8] DOI: 10.1007/s002200050255 · Zbl 0916.46057 · doi:10.1007/s002200050255
[9] DOI: 10.1023/A:1017916521415 · Zbl 1010.19004 · doi:10.1023/A:1017916521415
[10] Chabert J., Doc. Math. 6 pp 127– (2001)
[11] Chabert J., Publ. Math. Inst. Hautes E’t. Sci. 97 pp 239– (2003)
[12] DOI: 10.1007/s00039-004-0467-6 · Zbl 1063.46056 · doi:10.1007/s00039-004-0467-6
[13] Connes A., C. R. Acad. Sci. Paris Sér. A-B 290 pp A599– (1980)
[14] DOI: 10.1006/jfan.1996.3010 · Zbl 0873.22003 · doi:10.1006/jfan.1996.3010
[15] DOI: 10.1023/B:KTHE.0000015339.41953.04 · Zbl 1051.55009 · doi:10.1023/B:KTHE.0000015339.41953.04
[16] Echterho{\currency} S., Mem. Amer. Math. Soc. 123 pp 586– (1996)
[17] DOI: 10.1090/S0002-9947-01-02794-5 · Zbl 0980.46049 · doi:10.1090/S0002-9947-01-02794-5
[18] Echterho{\currency} S., J. Oper. Th. 45 pp 131– (2001)
[19] DOI: 10.1006/jfan.1998.3295 · Zbl 0909.46055 · doi:10.1006/jfan.1998.3295
[20] DOI: 10.1016/0021-8693(76)90242-8 · Zbl 0323.46063 · doi:10.1016/0021-8693(76)90242-8
[21] DOI: 10.2307/2946553 · Zbl 0847.46034 · doi:10.2307/2946553
[22] Farsi C., C. R. Math. Rep. Acad. Sci. Canada 14 pp 75– (1991)
[23] Farsi C., Canad. J. Math. 44 pp 1167– (1992) · Zbl 0789.46053 · doi:10.4153/CJM-1992-070-4
[24] DOI: 10.1007/BF01445133 · Zbl 0791.46048 · doi:10.1007/BF01445133
[25] Farsi C., J. Oper. Th. 30 pp 243– (1993)
[26] DOI: 10.1006/jfan.1993.1136 · Zbl 0799.46075 · doi:10.1006/jfan.1993.1136
[27] Farsi C., Canad. J. Math. 46 pp 1211– (1994) · Zbl 0811.46068 · doi:10.4153/CJM-1994-069-4
[28] Farsi C., Math. Scand. 75 pp 101– (1994)
[29] DOI: 10.1007/s002220000118 · Zbl 0988.19003 · doi:10.1007/s002220000118
[30] Hurder S., Ergod. Th. Dynam. Syst. 6 pp 541– (1986)
[31] V. F., Mem. Amer. Math. Soc. 28 pp 237– (1980)
[32] DOI: 10.1007/BF01404917 · Zbl 0647.46053 · doi:10.1007/BF01404917
[33] Kumjian A., C. R. Math. Rep. Acad. Sci. Canada 12 pp 87– (1990)
[34] DOI: 10.1090/S0002-9947-00-02680-5 · Zbl 0964.46044 · doi:10.1090/S0002-9947-00-02680-5
[35] DOI: 10.1112/plms/83.1.199 · Zbl 1015.46031 · doi:10.1112/plms/83.1.199
[36] DOI: 10.1215/S0012-7094-04-12514-X · Zbl 1068.46032 · doi:10.1215/S0012-7094-04-12514-X
[37] DOI: 10.2140/gt.2005.9.1639 · Zbl 1073.19004 · doi:10.2140/gt.2005.9.1639
[38] DOI: 10.1142/S0219199799000213 · Zbl 0959.58035 · doi:10.1142/S0219199799000213
[39] DOI: 10.1007/s002200000351 · Zbl 0982.58018 · doi:10.1007/s002200000351
[40] DOI: 10.1016/j.top.2005.07.001 · Zbl 1092.19004 · doi:10.1016/j.top.2005.07.001
[41] DOI: 10.2307/1994090 · doi:10.2307/1994090
[42] DOI: 10.2307/1994091 · doi:10.2307/1994091
[43] DOI: 10.2307/1997540 · Zbl 0366.22005 · doi:10.2307/1997540
[44] DOI: 10.1017/S0143385706000277 · Zbl 1119.46050 · doi:10.1017/S0143385706000277
[45] Packer J. A., Math. Proc. Cambridge Philos. Soc. 106 pp 293– (1989)
[46] DOI: 10.2307/2154478 · Zbl 0786.22009 · doi:10.2307/2154478
[47] DOI: 10.1007/BF00534138 · Zbl 0709.46034 · doi:10.1007/BF00534138
[48] Pimsner M., J. Oper. Th. 4 pp 93– (1980)
[49] Rie{\currency}el M. A., Proc. London Math. Soc. 47 (3) pp 285– (1983)
[50] Rie{\currency}el M. A., Canad. J. Math. 40 pp 257– (1988) · Zbl 0663.46073 · doi:10.4153/CJM-1988-012-9
[51] Rie{\currency}el M. A., Contemp. Math. 105 pp 191– (1990)
[52] DOI: 10.1215/S0012-7094-79-04602-7 · Zbl 0395.46049 · doi:10.1215/S0012-7094-79-04602-7
[53] DOI: 10.1215/S0012-7094-87-05524-4 · Zbl 0644.46051 · doi:10.1215/S0012-7094-87-05524-4
[54] DOI: 10.1112/blms/15.5.401 · Zbl 0561.57001 · doi:10.1112/blms/15.5.401
[55] DOI: 10.1007/BF01878451 · Zbl 0225.46068 · doi:10.1007/BF01878451
[56] DOI: 10.1023/A:1007744304422 · Zbl 0939.19001 · doi:10.1023/A:1007744304422
[57] Walters S. G., Canad. J. Math. 52 pp 633– (2000) · Zbl 0960.46045 · doi:10.4153/CJM-2000-028-9
[58] Walters S. G., Canad. J. Math. 53 pp 631– (2001) · Zbl 0973.46061 · doi:10.4153/CJM-2001-026-x
[59] Walters S. G., Math. 568 pp 139– (2004)
[60] Walters S. G., C. R. Math. Acad. Sci. Soc. Can. 26 pp 55– (2004)
[61] Watatani Y., Math. Japonica 26 pp 479– (1981)
[62] Zeller-Meier G., J. Math. Pures Appl. 47 (9) pp 101– (1968)
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