On the Range of Certain ASH Algebras of Real Rank Zero

Abstract

In this paper, the authors consider the range of a certain class of ASH algebras in [An, Q., Elliott, G. A., Li, Z. and Liu, Z., The classification of certain ASH C*-algebras of real rank zero, J. Topol. Anal., 14(1), 2022, 183–202], which is under the scheme of the Elliott program in the setting of real rank zero C^*-algebras. As a reduction theorem, they prove that all these ASH algebras are still the AD algebras studied in [Dadarlat, M. and Loring, T. A., Classifying C ^*-algebras via ordered, mod-p K-theory, Math. Ann., 305, 1996, 601–616].

Keywords

Classification / AD algebra / Range / Reduction

References

[1] An Q, Elliott G A. On the KK-theory of Elliott-Thomsen algebras. J. Operator Theory, 2017, 78(2): 435-472

[2] An Q, Elliott G A, Li Z, Liu Z. The classification of certain ASH C*-algebras of real rank zero. J. Topol. Anal., 2022, 14(1): 183-202

[3] Blackadar B. The homotopy lifting theorem for semiprojective C*-algebras. Math. Scand., 2016, 118(2): 291-302

[4] Bratteli O. Inductive limits of finite-dimensional C*-algebras. Trans. Amer. Math. Soc., 1972, 171: 195-234

[5] Dadarlat M, Gong G. A classification result for approximately homogeneous C*-algebras of real rank zero. Geom. Funct. Anal., 1997, 7(4): 646-711

[6] Dadarlat M, Loring T A. Classifying C*-algebras via ordered, mod-p K-theory. Math. Ann., 1996, 305: 601-616

[7] Dixmier J. On some C*-algebras considered by Glimm. J. Funct. Anal., 1967, 1: 182-203

[8] Effros E, Handelman D, Shen C L. Dimension groups and their affine representations. Amer. J. Math., 1980, 102: 385-407

[9] Eilers S. A complete invariant for AD algebras with bounded torsion in K ₁. J. Funct. Anal., 1996, 139(2): 325-348

[10] Eilers S, Loring T A, Pedersen G K. Fragility of subhomogeneous C*-algebras with one-dimensional spectrum. Bull. London Math. Soc., 1999, 31(3): 337-344

[11] Elliott G A. On the classification of inductive limits of sequence of semisimple finite-dimensional algebras. J. Algebra, 1976, 38: 29-44

[12] Elliott G A. On the classification of C*-algebras of real rank zero. J. Reine Angew. Math., 1993, 443: 179-219

[13] Elliott, G. A., Gong, G., Lin, H. and Niu, Z., On the classification of simple amenable C*-algebras with finite decomposition rank II, 2015, arXiv: 1507.03437v2.

[14] Elliott G A, Gong G, Su H. On the Classification of C*-algebras of Real Rank Zero, IV, 1998, Providence, RI: Amer. Math. Soc.

[15] Glimm J. On a certain class of operator algebras. Trans. Amer. Math. Soc., 1960, 95: 318-340

[16] Gong G. On the classification of simple inductive limit C*-algebras. I. The reduction theorem. Doc. Math., 2002, 7: 255-461

[17] Gong G, Lin H, Niu Z. A classification of finite simple amenable $\cal{Z}$-stable C*-algebras, I: C*-algebras with generalized tracial rank one. C. R. Math. Acad. Sci. Soc. R. Can., 2020, 42(3): 63-450

[18] Gong G, Lin H, Niu Z. A classification of finite simple amenable $\cal{Z}$-stable C*-algebras, II: C*-algebras with rational generalized tracial rank one. C. R. Math. Acad. Sci. Soc. R. Can., 2020, 42(4): 451-539

[19] Jiang X, Su H. On a simple unital projectionless C*-algebra. Amer. J. Math., 1999, 121(2): 359-413

[20] Li Z. On the simple inductive limits of splitting interval algebras with dimension drops. Canad. J. Math., 2012, 64(3): 544-572

[21] Liu Z. A decomposition theorem for real rank zero inductive limits of 1-dimensional non-commutative CW complexes. J. Topol. Anal., 2019, 11(1): 181-204

[22] Loring T A. C*-algebras generated by stable relations. J. Funct. Anal., 1993, 112: 159-201

[23] Mygind J. Classification of certain simple C*-algebras with torsion in K1. Canad. J. Math., 2001, 53(6): 1223-1308

[24] Thomsen K. Limits of certain subhomogeneous C*-algebras. Mem. Soc. Math. Fr. (N.S.), 1997, 71: 125

[25] Tikuisis A, White S, Winter W. Quasidiagonality of nuclear C*-algebras. Ann. Math., 2017, 185(1): 229-284

[26] Zhang S. A Riesz decomposition property and ideal structure of multiplier algebras. J. Operator Theory, 1990, 24(2): 209-225