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On the Simple Inductive Limits of Splitting Interval Algebras with Dimension Drops

Published online by Cambridge University Press:  20 November 2018

Zhiqiang Li
Zhiqiang Li*
Affiliation:
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, China and The Fields Institute, Toronto, ON, M5T 3J1 email: [email protected]
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Abstract

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A $\text{K}$-theoretic classification is given of the simple inductive limits of finite direct sums of the type I ${{C}^{*}}$-algebras known as splitting interval algebras with dimension drops. (These are the subhomogeneous ${{C}^{*}}$-algebras, each having spectrum a finite union of points and an open interval, and torsion ${{K}_{1}}$-group.)

Keywords

46L0546L35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 3 , 01 June 2012 , pp. 544 - 572
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Blackadar, B., K-theory for operator algebras. Mathematical Sciences Research Institute Publications, 5, Springer-Verlag, New York, 1986.Google Scholar
[2] Dadarlat, M., Reduction to dimension three of local spectra of real rank zero C*-algebras. J. Reine Angew. Math. 460(1995), 189–212.Google Scholar
[3] Dadarlat, M. and Gong, G., A classification result for approximately homogeneous C*-algebras of real rank zero. Geom. Funct. Anal. 7(1997), no. 4, 646–711. http://dx.doi.org/10.1007/s000390050023 Google Scholar
[4] Dadarlat, M. and Nemethi, A., Shape theory and (connective) K-theory. J. Operator Theory 23(1990), no. 2, 207–291.Google Scholar
[5] Dadarlat, M., Nagy, G., Nemethi, A., and Pasnicu, C., Reduction of topological stable rank in inductive limits of C*-algebras. Pacific J. Math. 153(1992), no. 2, 267–276.Google Scholar
[6] Effros, E. G. and Kaminker, J., Homotopy continuity and shape theory for C*-algebras. In: Geometric methods in operator algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., 123, Longman Sci. Tech. Harlow, 1986, pp. 152–180.Google Scholar
[7] Elliott, G. A., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. J. Algebra 38(1976), no. 1, 29–44. http://dx.doi.org/10.1016/0021-8693(76)90242-8 Google Scholar
[8] Elliott, G. A., On the classification of C*-algebras of real rank zero. J. Reine Angew. Math. 443(1993), 179–219. http://dx.doi.org/10.1515/crll.1993.443.179 Google Scholar
[9] Elliott, G. A., A classification of certain simple C*-algebras. In: Quantum and non-commutative analysis (Kyoto, 1992), Math. Phys. Stud., 16, Kluwer Acad. Publ., Dordrecht, 1993, pp. 373–385.Google Scholar
[10] Elliott, G. A. and Thomsen, K., The state space of the K0-group of a simple separable C*-algebra. Geom. Funct. Anal. 4(1994), no. 5, 522–538. http://dx.doi.org/10.1007/BF01896406 Google Scholar
[11] Elliott, G. A., Gong, G., Lin, H., and Pasnicu, C., Abelian C*-subalgebras of C*-algebras of real rank zero and inductive limit C*-algebras. Duke Math. J. 85(1996), no. 3, 511–554. http://dx.doi.org/10.1215/S0012-7094-96-08520-8 Google Scholar
[12] Elliott, G. A. and Gong, G., On the classification of C*-algebras of real rank zero. II. Ann. of Math. (2) 144(1996), no. 3, 497–610. http://dx.doi.org/10.2307/2118565 Google Scholar
[13] 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. http://dx.doi.org/10.1007/s00222-006-0033-y Google Scholar
[14] Elliott, G. A., Gong, G., Jiang, X., and Su, H., A classification of simple limits of dimension drop C*-algebras. In: Operator algebras and their applications (Waterloo, ON, 1994/1995), Fields Inst. Commun., 13, American Mathematical Society, Providence, RI, 1997, pp. 125–143.Google Scholar
[15] Elliott, G. A. and Niu, Z., On tracial approximation. J. Funct. Anal. 254(2008), no. 2, 396–440. http://dx.doi.org/10.1016/j.jfa.2007.08.005 Google Scholar
[16] Gong, G., On inductive limits of matrix algebras over higher-dimensional spaces. I, II. Math. Scand. 80(1997), no. 1, 41–55, 56–100.Google Scholar
[17] Gong, G., Classification of C*-algebras of real rank zero and unsuspended E-equivalence types. J. Funct. Anal. 152(1998), no. 2, 281–329. http://dx.doi.org/10.1006/jfan.1997.3165 Google Scholar
[18] Gong, G., On the classification of simple inductive limit C*-algebras. I. The reduction theorem. Doc. Math. 7(2002), 255–461.Google Scholar
[19] Gong, G. and Lin, H., Almost multiplicative morphisms and K-theory. Internat. J. Math. 11(2000), no. 8, 983–1000. http://dx.doi.org/10.1142/S0129167X0000043X Google Scholar
[20] Jiang, X. and Su, H., A classification of simple limits of splitting interval algebras. J. Funct. Anal. 151(1997), no. 1, 50–76. http://dx.doi.org/10.1006/jfan.1997.3120 Google Scholar
[21] Jiang, X. and Su, H., On a simple unital projectionless C*-algebra. Amer. J. Math. 121(1999), no. 2, 359–413. http://dx.doi.org/10.1353/ajm.1999.0012 Google Scholar
[22] Kirchberg, E. and Winter, W., Covering dimension and quasidiagonality. Internat. J. Math. 15(2004), no. 1, 63–85. http://dx.doi.org/10.1142/S0129167X04002119 Google Scholar
[23] Li, L., Classification of simple C*-algebras: inductive limits of matrix algebras over trees. Mem. Amer. Math. Soc. 127(1997), no. 605.Google Scholar
[24] Li, L., Simple inductive limit C*-algebras: spectra and approximations by interval algebras. J. Reine Angew. Math. 507(1999), 57–79.Google Scholar
[25] Lin, H., Approximation by normal elements with finite spectra in C*-algebras of real rank zero. Pacific J. Math. 173(1996), no. 2, 443–489.Google Scholar
[26] Lin, H., Classification of simple tracially AF C*-algebras. Canad. J. Math. 53(2001), no. 1, 161–194. http://dx.doi.org/10.4153/CJM-2001-007-8 Google Scholar
[27] Lin, H., Classification of simple C*-algebras of tracial topological rank zero. Duke Math. J. 125(2004), no. 1, 91–119. http://dx.doi.org/10.1215/S0012-7094-04-12514-X Google Scholar
[28] Lin, H. and Niu, Z., Lifting KK -elements, asymptotic unitary equivalence and classification of simple C*-algebras. Adv. Math. 219(2008), no. 5, 1729–1769. http://dx.doi.org/10.1016/j.aim.2008.07.011 Google Scholar
[29] Nielsen, K. E. and Thomsen, K., Limits of circle algebras. Exposition. Math. 14(1996), no. 1, 17–56.Google Scholar
[30] Pasnicu, C., Shape equivalence, nonstable K-theory and AH algebras. Pacific J. Math. 192(2000), 159–182. http://dx.doi.org/10.2140/pjm.2000.192.159 Google Scholar
[31] Rosenberg, J. and Schochet, C., The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor. Duke Math. J. 55(1987), no. 2, 431–474. http://dx.doi.org/10.1215/S0012-7094-87-05524-4 Google Scholar
[32] Su, H., On the classification of C*-algebras of real rank zero: inductive limits of matrix algebras over non-Hausdorff graphs. Mem. Amer. Math. Soc. 114(1995), no. 547. (1995).Google Scholar
[33] Thomsen, K., Inductive limits of interval algebras: the tracial state space. Amer. J. Math. 116(1994), no. 3, 605–620. http://dx.doi.org/10.2307/2374993 Google Scholar
[34] Thomsen, K., Limits of certain subhomogeneous C*-algebras. Mém. Soc. Math. Fr. (N. S.) No. 71 (1997).Google Scholar
[35] Thomsen, K., From trace states to states on the K0-group of a simple C*-algebra. Bull. London Math. Soc. 28(1996), no. 1, 66–72. http://dx.doi.org/10.1112/blms/28.1.66 Google Scholar
[36] Tom, A. and Winter, W., Z-stable ASH algebras. Canad. J. Math. 60(2008), no. 3, 703–720. http://dx.doi.org/10.4153/CJM-2008-031-6 Google Scholar
[37] Winter, W., Decomposition rank of subhomogeneous C*-algebras. Proc. London Math. Soc. (3) 89(2004), no. 2, 427–456. http://dx.doi.org/10.1112/S0024611504014716 Google Scholar
[38] Winter, W., Decomposition rank and Z-stability. Invent. Math. 179(2010), no. 2, 229–301. http://dx.doi.org/10.1007/s00222-009-0216-4Google Scholar