Constructing C*-diagonals in AH-algebras
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- by Xin Li and Ali I. Raad
- Trans. Amer. Math. Soc. 376 (2023), 8857-8875
- DOI: https://doi.org/10.1090/tran/9023
- Published electronically: September 12, 2023
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Abstract:
We construct Cartan subalgebras and hence groupoid models for classes of AH-algebras. Our results cover all AH-algebras whose building blocks have base spaces of dimension at most one as well as Villadsen algebras, and thus go beyond classifiable simple C*-algebras.References
- Selçuk Barlak and Xin Li, Cartan subalgebras and the UCT problem, Adv. Math. 316 (2017), 748–769. MR 3672919, DOI 10.1016/j.aim.2017.06.024
- Selçuk Barlak and Xin Li, Cartan subalgebras and the UCT problem, II, Math. Ann. 378 (2020), no. 1-2, 255–287. MR 4150918, DOI 10.1007/s00208-020-02022-4
- George A. Elliott, Guihua Gong, and Liangqing Li, Injectivity of the connecting maps in AH inductive limit systems, Canad. Math. Bull. 48 (2005), no. 1, 50–68. MR 2118763, DOI 10.4153/CMB-2005-005-9
- George A. Elliott and Zhuang Niu, On the classification of simple amenable $C^*$-algebras with finite decomposition rank, Operator algebras and their applications, Contemp. Math., vol. 671, Amer. Math. Soc., Providence, RI, 2016, pp. 117–125. MR 3546681, DOI 10.1090/conm/671/13506
- George A. Elliott, Guihua Gong, Huaxin Lin, and Zhuang Niu, Simple stably projectionless $\rm C^*$-algebras with generalized tracial rank one, J. Noncommut. Geom. 14 (2020), no. 1, 251–347. MR 4107516, DOI 10.4171/JNCG/367
- George A. Elliott, Guihua Gong, Huaxin Lin, and Zhuang Niu, The classification of simple separable KK-contractible $\rm C^*$-algebras with finite nuclear dimension, J. Geom. Phys. 158 (2020), 103861, 51. MR 4146445, DOI 10.1016/j.geomphys.2020.103861
- George A. Elliott, Toan M. Ho, and Andrew S. Toms, A class of simple $C^*$-algebras with stable rank one, J. Funct. Anal. 256 (2009), no. 2, 307–322. MR 2476944, DOI 10.1016/j.jfa.2008.08.001
- Julien Giol and David Kerr, Subshifts and perforation, J. Reine Angew. Math. 639 (2010), 107–119. MR 2608192, DOI 10.1515/CRELLE.2010.012
- Guihua Gong, On the classification of simple inductive limit $C^*$-algebras. I. The reduction theorem, Doc. Math. 7 (2002), 255–461. MR 2014489, DOI 10.4171/dm/127
- Guihua Gong, Chunlan Jiang, Liangqing Li, and Cornel Pasnicu, $A\Bbb T$ structure of $AH$ algebras with the ideal property and torsion free $K$-theory, J. Funct. Anal. 258 (2010), no. 6, 2119–2143. MR 2578465, DOI 10.1016/j.jfa.2009.11.016
- G. Gong and H. Lin, On classification of non-unital simple amenable C*-algebras, I, preprint, arXiv:1611.04440, 2016.
- Guihua Gong and Huaxin Lin, On classification of non-unital amenable simple $C^*$-algebras, II, J. Geom. Phys. 158 (2020), 103865, 102. MR 4157681, DOI 10.1016/j.geomphys.2020.103865
- Guihua Gong and Huaxin Lin, On classification of nonunital amenable simple $C^*$-algebras, III : The range and the reduction, Ann. K-Theory 7 (2022), no. 2, 279–384. MR 4486464, DOI 10.2140/akt.2022.7.279
- Guihua Gong, Huaxin Lin, and Zhuang Niu, A classification of finite simple amenable $\mathcal Z$-stable $C^\ast$-algebras, I: $C^\ast$-algebras with generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Can. 42 (2020), no. 3, 63–450 (English, with English and French summaries). MR 4215379
- Guihua Gong, Huaxin Lin, and Zhuang Niu, A classification of finite simple amenable $\mathcal Z$-stable $\textrm {C}^\ast$-algebras, II: $\textrm {C}^\ast$-algebras with rational generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Can. 42 (2020), no. 4, 451–539 (English, with English and French summaries). MR 4215380
- K. R. Goodearl, Notes on a class of simple $C^\ast$-algebras with real rank zero, Publ. Mat. 36 (1992), no. 2A, 637–654 (1993). MR 1209829, DOI 10.5565/PUBLMAT_{3}62A92_{2}3
- David Kerr, Dimension, comparison, and almost finiteness, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 11, 3697–3745. MR 4167017, DOI 10.4171/jems/995
- Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact $C^*$-algebras in the Cuntz algebra $\scr O_2$, J. Reine Angew. Math. 525 (2000), 17–53. MR 1780426, DOI 10.1515/crll.2000.065
- Alexander Kumjian, On $C^\ast$-diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008. MR 854149, DOI 10.4153/CJM-1986-048-0
- Bartosz Kosma Kwaśniewski and Ralf Meyer, Noncommutative Cartan $\rm C^*$-subalgebras, Trans. Amer. Math. Soc. 373 (2020), no. 12, 8697–8724. MR 4177273, DOI 10.1090/tran/8174
- Liangqing Li, Classification of simple $C^*$-algebras: inductive limits of matrix algebras over trees, Mem. Amer. Math. Soc. 127 (1997), no. 605, vii+123. MR 1376744, DOI 10.1090/memo/0605
- Xin Li, Continuous orbit equivalence rigidity, Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1543–1563. MR 3789176, DOI 10.1017/etds.2016.98
- Xin Li, Partial transformation groupoids attached to graphs and semigroups, Int. Math. Res. Not. IMRN 17 (2017), 5233–5259. MR 3694599, DOI 10.1093/imrn/rnw166
- Xin Li, Dynamic characterizations of quasi-isometry and applications to cohomology, Algebr. Geom. Topol. 18 (2018), no. 6, 3477–3535. MR 3868227, DOI 10.2140/agt.2018.18.3477
- Xin Li, Every classifiable simple $\rm C^*$-algebra has a Cartan subalgebra, Invent. Math. 219 (2020), no. 2, 653–699. MR 4054809, DOI 10.1007/s00222-019-00914-0
- Xin Li and Jean Renault, Cartan subalgebras in $\textrm {C}^*$-algebras. Existence and uniqueness, Trans. Amer. Math. Soc. 372 (2019), no. 3, 1985–2010. MR 3976582, DOI 10.1090/tran/7654
- X. Ma and J. Wu, Almost elementariness and fiberwise amenability for étale groupoids, Preprint, arXiv:2011.01182, 2020.
- Hiroki Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces, Proc. Lond. Math. Soc. (3) 104 (2012), no. 1, 27–56. MR 2876963, DOI 10.1112/plms/pdr029
- Zhuang Niu, Mean dimension and AH-algebras with diagonal maps, J. Funct. Anal. 266 (2014), no. 8, 4938–4994. MR 3177327, DOI 10.1016/j.jfa.2014.02.010
- N. Christopher Phillips, A classification theorem for nuclear purely infinite simple $C^*$-algebras, Doc. Math. 5 (2000), 49–114. MR 1745197, DOI 10.4171/dm/75
- Ali I. Raad, A generalization of Renault’s theorem for Cartan subalgebras, Proc. Amer. Math. Soc. 150 (2022), no. 11, 4801–4809. MR 4489313, DOI 10.1090/proc/16003
- A.I. Raad, Existence and uniqueness of inductive limit Cartan subalgebras in inductive limit $C^*$-algebras, PhD Thesis, available at https://theses.gla.ac.uk/82456/.
- Jean Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. MR 2460017, DOI 10.33232/BIMS.0061.29.63
- M. Rørdam, Classification of nuclear, simple $C^*$-algebras, Classification of nuclear $C^*$-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002, pp. 1–145. MR 1878882, DOI 10.1007/978-3-662-04825-2_{1}
- Klaus Thomsen, Homomorphisms between finite direct sums of circle algebras, Linear and Multilinear Algebra 32 (1992), no. 1, 33–50. MR 1198819, DOI 10.1080/03081089208818145
- Klaus Thomsen, On isomorphisms of inductive limit $C^*$-algebras, Proc. Amer. Math. Soc. 113 (1991), no. 4, 947–953. MR 1087472, DOI 10.1090/S0002-9939-1991-1087472-1
- Klaus Thomsen, On the embedding and diagonalization of matrices over $C(X)$, Math. Scand. 60 (1987), no. 2, 219–228. MR 914335, DOI 10.7146/math.scand.a-12181
- Klaus Thomsen, Diagonalization in inductive limits of circle algebras, J. Operator Theory 27 (1992), no. 2, 325–340. MR 1249649
- Aaron Tikuisis, Stuart White, and Wilhelm Winter, Quasidiagonality of nuclear $C^\ast$-algebras, Ann. of Math. (2) 185 (2017), no. 1, 229–284. MR 3583354, DOI 10.4007/annals.2017.185.1.4
- Andrew S. Toms, On the classification problem for nuclear $C^\ast$-algebras, Ann. of Math. (2) 167 (2008), no. 3, 1029–1044. MR 2415391, DOI 10.4007/annals.2008.167.1029
- Jesper Villadsen, Simple $C^*$-algebras with perforation, J. Funct. Anal. 154 (1998), no. 1, 110–116. MR 1616504, DOI 10.1006/jfan.1997.3168
- Jesper Villadsen, On the stable rank of simple $C^\ast$-algebras, J. Amer. Math. Soc. 12 (1999), no. 4, 1091–1102. MR 1691013, DOI 10.1090/S0894-0347-99-00314-8
Bibliographic Information
- Xin Li
- Affiliation: School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom
- ORCID: 0000-0002-2243-3742
- Email: xin.li@glasgow.ac.uk
- Ali I. Raad
- Affiliation: Department of Mathematics, KU Leuven, 200B Celestijnenlaan, 3001 Leuven, Belgium
- MR Author ID: 1527912
- ORCID: 0000-0001-8429-6272
- Email: ali.imadraad@kuleuven.be
- Received by editor(s): July 24, 2022
- Received by editor(s) in revised form: May 21, 2023, and July 13, 2023
- Published electronically: September 12, 2023
- Additional Notes: This project had received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 817597). The second author was supported by the Internal KU Leuven BOF project C14/19/088 and project G085020N funded by the Research Foundation Flanders (FWO)
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 8857-8875
- MSC (2020): Primary 46L05, 46L35; Secondary 22A22
- DOI: https://doi.org/10.1090/tran/9023
- MathSciNet review: 4669313