Subalgebras of simple AF-algebras

Christopher Schafhauser

doi:https://doi.org/10.4007/annals.2020.192.2.1

Abstract

It is shown that if $A$ is a separable, exact C$^*$-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then $A$ admits a trace-preserving embedding into a simple, unital AF-algebra with a unique trace. Modulo the UCT, this provides an abstract characterization of C$^*$-subalgebras of simple, unital AF-algebras.

As a consequence, for a countable, discrete, amenable group $G$ acting on a second countable, locally compact, Hausdorff space $X$, $C_0(X) \rtimes _r G$ embeds into a simple, unital AF-algebra if, and only if, $X$ admits a faithful, invariant, Borel, probability measure. Also, for any countable, discrete, amenable group $G$, the reduced group C$^*$-algebra $\mathrm {C}^*_r(G)$ admits a trace-preserving embedding into the universal UHF-algebra.

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@ARTICLE{Gabe:TWW, author = {Gabe, James}, title = {Quasidiagonal traces on exact {$\mathrm{C}^*$}-algebras}, journal = {J. Funct. Anal.}, fjournal = {Journal of Functional Analysis}, volume = {272}, year = {2017}, number = {3}, pages = {1104--1120}, issn = {0022-1236}, mrclass = {46L05 (46L35 46L80)}, mrnumber = {3579134}, mrreviewer = {Bachir Bekka}, doi = {10.1016/j.jfa.2016.10.022}, url = {https://doi.org/10.1016/j.jfa.2016.10.022}, zblnumber = {1369.46046}, }
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@MISC{GabeRuiz:AbsorbingCones, author = {Gabe, James and Ruiz, E.}, title = {Absorbing representations with respect to closed operator convex cones}, year = {2015}, arxiv = {1503.07799}, zblnumber = {}, }
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@MISC{GongLinNiu, author = {Gong, G. and Lin, H. and Niu, Z.}, title = {Classification of simple amenable {$\mathcal{Z}$}-stable {C}$^*$-algebras}, year = {2015}, arxiv = {1501.00135}, zblnumber = {}, }
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@ARTICLE{Haagerup:Quasitrace, author = {Haagerup, Uffe}, title = {Quasitraces on exact {$\mathrm{C}^*$}-algebras are traces}, journal = {C. R. Math. Acad. Sci. Soc. R. Can.}, fjournal = {Comptes Rendus Mathématiques de l'Académie des Sciences. La Société Royale du Canada. Mathematical Reports of the Academy of Science. The Royal Society of Canada}, volume = {36}, year = {2014}, number = {2-3}, pages = {67--92}, issn = {0706-1994}, mrclass = {46L35 (46L80)}, mrnumber = {3241179}, mrreviewer = {Francesc Perera}, zblnumber = {1325.46055}, url = {https://mr.math.ca/volume-issue/vol-36-2014/vol-36-2-2014/}, }
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@ARTICLE{HaagerupThorbjornsen, author = {Haagerup, Uffe and Thorbj{\o{r}}nsen, Steen}, title = {A new application of random matrices: {${\rm Ext}(\mathrm{C}^*_{\rm red}(F_2))$} is not a group}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {162}, year = {2005}, number = {2}, pages = {711--775}, issn = {0003-486X}, mrclass = {46L54 (15A52 46L35)}, mrnumber = {2183281}, mrreviewer = {Beno\^ıt Collins}, doi = {10.4007/annals.2005.162.711}, url = {https://doi.org/10.4007/annals.2005.162.711}, zblnumber = {1103.46032}, }
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@ARTICLE{Katsura:AFECuntzFlows, author = {Katsura, Takeshi}, title = {A{F}-embeddability of crossed products of {C}untz algebras}, journal = {J. Funct. Anal.}, fjournal = {Journal of Functional Analysis}, volume = {196}, year = {2002}, number = {2}, pages = {427--442}, issn = {0022-1236}, mrclass = {46L55 (46L05)}, mrnumber = {1943097}, mrreviewer = {Astrid An Huef}, doi = {10.1016/S0022-1236(02)00003-4}, url = {https://doi.org/10.1016/S0022-1236(02)00003-4}, zblnumber = {1024.46019}, }
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@ARTICLE{KerrNowak, author = {Kerr, David and Nowak, Piotr W.}, title = {Residually finite actions and crossed products}, journal = {Ergodic Theory Dynam. Systems}, fjournal = {Ergodic Theory and Dynamical Systems}, volume = {32}, year = {2012}, number = {5}, pages = {1585--1614}, issn = {0143-3857}, mrclass = {37A55 (22D40 46L55)}, mrnumber = {2974211}, mrreviewer = {Claire Anantharaman-Delaroche}, doi = {10.1017/S0143385711000575}, url = {https://doi.org/10.1017/S0143385711000575}, zblnumber = {1280.37009}, }
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@ARTICLE{Kirchberg:FactorizationProperty, author = {Kirchberg, Eberhard}, title = {Discrete groups with {K}azhdan's property {${\rm T}$} and factorization property are residually finite}, journal = {Math. Ann.}, fjournal = {Mathematische Annalen}, volume = {299}, year = {1994}, number = {3}, pages = {551--563}, issn = {0025-5831}, mrclass = {46L05 (22D10 22D25 43A65 46L35 46M07)}, mrnumber = {1282231}, mrreviewer = {Alain Valette}, doi = {10.1007/BF01459798}, url = {https://doi.org/10.1007/BF01459798}, zblnumber = {0805.46066}, }
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@ARTICLE{KirchbergRordam:TomsWinterConj, author = {Kirchberg, Eberhard and R{\o{r}}dam, Mikael}, title = {Central sequence {$\mathrm{C}^*$}-algebras and tensorial absorption of the {J}iang-{S}u algebra}, journal = {J. Reine Angew. Math.}, fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]}, volume = {695}, year = {2014}, pages = {175--214}, issn = {0075-4102}, mrclass = {46L35 (46C20)}, mrnumber = {3276157}, mrreviewer = {Aurelian Gheondea}, doi = {10.1515/crelle-2012-0118}, url = {https://doi.org/10.1515/crelle-2012-0118}, zblnumber = {1307.46046}, }
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@ARTICLE{KirchbergRordam:CentralSequenceAlgebras, author = {Kirchberg, Eberhard and R{\o{r}}dam, Mikael}, title = {When central sequence {$\mathrm{C}^*$}-algebras have characters}, journal = {Internat. J. Math.}, fjournal = {International Journal of Mathematics}, volume = {26}, year = {2015}, number = {7}, pages = {1550049, 32}, issn = {0129-167X}, mrclass = {46L35 (46L06)}, mrnumber = {3357038}, mrreviewer = {E. St\o rmer}, doi = {10.1142/S0129167X15500494}, url = {https://doi.org/10.1142/S0129167X15500494}, zblnumber = {1339.46055}, }
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@ARTICLE{KucerovskyNg, author = {Kucerovsky, Dan and Ng, P. W.}, title = {The corona factorization property and approximate unitary equivalence}, journal = {Houston J. Math.}, fjournal = {Houston Journal of Mathematics}, volume = {32}, year = {2006}, number = {2}, pages = {531--550}, issn = {0362-1588}, mrclass = {46L85 (46L05 46L80)}, mrnumber = {2219330}, mrreviewer = {Vladimir Manuilov}, zblnumber = {1111.46050}, }
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@ARTICLE{Lin:BrownPedersen, author = {Lin, Hua Xin}, title = {Exponential rank of {$\mathrm{C}^*$}-algebras with real rank zero and the {B}rown-{P}edersen conjectures}, journal = {J. Funct. Anal.}, fjournal = {Journal of Functional Analysis}, volume = {114}, year = {1993}, number = {1}, pages = {1--11}, issn = {0022-1236}, mrclass = {46L05 (19K56)}, mrnumber = {1220980}, mrreviewer = {Ronghui Ji}, doi = {10.1006/jfan.1993.1060}, url = {https://doi.org/10.1006/jfan.1993.1060}, zblnumber = {0812.46054}, }
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@ARTICLE{Lin:TAF, author = {Lin, Huaxin}, title = {Tracially {AF} {$\mathrm{C}^*$}-algebras}, journal = {Trans. Amer. Math. Soc.}, fjournal = {Transactions of the American Mathematical Society}, volume = {353}, year = {2001}, number = {2}, pages = {693--722}, issn = {0002-9947}, mrclass = {46L35 (46L05)}, mrnumber = {1804513}, mrreviewer = {Kengo Matsumoto}, doi = {10.1090/S0002-9947-00-02680-5}, url = {https://doi.org/10.1090/S0002-9947-00-02680-5}, zblnumber = {0964.46044}, }
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@ARTICLE{Lin:StableApproxEquivalence, author = {Lin, Huaxin}, title = {Stable approximate unitary equivalence of homomorphisms}, journal = {J. Operator Theory}, fjournal = {Journal of Operator Theory}, volume = {47}, year = {2002}, number = {2}, pages = {343--378}, issn = {0379-4024}, mrclass = {46L80 (46L05 46L35)}, mrnumber = {1911851}, mrreviewer = {Vladimir Manuilov}, zblnumber = {1029.46097}, url = {http://www.mathjournals.org/jot/2002-047-002/2002-047-002-006.html}, }
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@ARTICLE{Lin:TAFClassification, author = {Lin, Huaxin}, title = {Classification of simple {$\mathrm{C}^\ast$}-algebras of tracial topological rank zero}, journal = {Duke Math. J.}, fjournal = {Duke Mathematical Journal}, volume = {125}, year = {2004}, number = {1}, pages = {91--119}, issn = {0012-7094}, mrclass = {46L05 (46L35 46L80)}, mrnumber = {2097358}, mrreviewer = {Hiroyuki Osaka}, doi = {10.1215/S0012-7094-04-12514-X}, url = {https://doi.org/10.1215/S0012-7094-04-12514-X}, zblnumber = {1068.46032}, }
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@ARTICLE{Lin:AFEmbAHalg, author = {Lin, Huaxin}, title = {A{F}-embeddings of the crossed products of {AH}-algebras by finitely generated abelian groups}, journal = {International Mathematics Research Papers}, volume={2008}, month={01}, year = {2008}, number = {3}, doi = {10.1093/imrp/rpn007}, url = {https://doi.org/10.1093/imrp/rpn007}, note = {rpn007}, issn = {1687-3017}, mrclass = {46L55 (22D25 46L05 46L35)}, mrnumber = {2457848}, mrreviewer = {Francesc Perera}, zblnumber = {1166.46040}, }
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@ARTICLE{Luck:TraceConjecture, author = {Lück, Wolfgang}, title = {The relation between the {B}aum-{C}onnes conjecture and the trace conjecture}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {149}, year = {2002}, number = {1}, pages = {123--152}, issn = {0020-9910}, mrclass = {19K56 (46L85 58J22)}, mrnumber = {1914619}, mrreviewer = {Evgeniy V. Troitskiĭ}, doi = {10.1007/s002220200215}, url = {https://doi.org/10.1007/s002220200215}, zblnumber = {1035.19003}, }
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@ARTICLE{Matui:AFE, author = {Hiroki, Matui}, title = {A{F} embeddability of crossed products of {AT} algebras by the integers and its application}, journal = {J. Funct. Anal.}, fjournal = {Journal of Functional Analysis}, volume = {192}, year = {2002}, number = {2}, pages = {562--580}, issn = {0022-1236}, mrclass = {46L55 (46L05)}, mrnumber = {1923414}, mrreviewer = {Masaharu Kusuda}, doi = {10.1006/jfan.2001.3911}, url = {https://doi.org/10.1006/jfan.2001.3911}, zblnumber = {1021.46048}, }
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@ARTICLE{MatuiSato:ZStable, author = {Matui, Hiroki and Sato, Yasuhiko}, title = {Strict comparison and {$\mathcal{Z}$}-absorption of nuclear {$\mathrm{C}^*$}-algebras}, journal = {Acta Math.}, fjournal = {Acta Mathematica}, volume = {209}, year = {2012}, number = {1}, pages = {179--196}, issn = {0001-5962}, mrclass = {46L05}, mrnumber = {2979512}, mrreviewer = {Caleb Eckhardt}, doi = {10.1007/s11511-012-0084-4}, url = {https://doi.org/10.1007/s11511-012-0084-4}, zblnumber = {1277.46028}, }
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@ARTICLE{MatuiSato:DecompositionRank, author = {Matui, Hiroki and Sato, Yasuhiko}, title = {Decomposition rank of {UHF}-absorbing {$\mathrm{C}^*$}-algebras}, journal = {Duke Math. J.}, fjournal = {Duke Mathematical Journal}, volume = {163}, year = {2014}, number = {14}, pages = {2687--2708}, issn = {0012-7094}, mrclass = {46L06 (46L35 46L55)}, mrnumber = {3273581}, mrreviewer = {William Paschke}, doi = {10.1215/00127094-2826908}, url = {https://doi.org/10.1215/00127094-2826908}, zblnumber = {1317.46041}, }
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@ARTICLE{Mingo:KupiersTheorem, author = {Mingo, J. A.}, title = {{$K$}-theory and multipliers of stable {$\mathrm{C}^\ast$}-algebras}, journal = {Trans. Amer. Math. Soc.}, fjournal = {Transactions of the American Mathematical Society}, volume = {299}, year = {1987}, number = {1}, pages = {397--411}, issn = {0002-9947}, mrclass = {46L80 (18F25 19K56 46M20)}, mrnumber = {0869419}, mrreviewer = {Judith A. Packer}, doi = {10.2307/2000501}, url = {https://doi.org/10.2307/2000501}, zblnumber = {0616.46067}, }
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@ARTICLE{MvN:RingsOfOperatorsIV, author = {Murray, F. J. and von Neumann, J.}, title = {On rings of operators. {IV}}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {44}, year = {1943}, pages = {716--808}, issn = {0003-486X}, mrclass = {46.0X}, mrnumber = {0009096}, mrreviewer = {E. R. Lorch}, doi = {10.2307/1969107}, url = {https://doi.org/10.2307/1969107}, zblnumber = {0060.26903}, }
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@ARTICLE{OrtegaPereraRordamI, author = {Ortega, Eduard and Perera, Francesc and R{\o{r}}dam, Mikael}, title = {The {C}orona {F}actorization {P}roperty and refinement monoids}, journal = {Trans. Amer. Math. Soc.}, fjournal = {Transactions of the American Mathematical Society}, volume = {363}, year = {2011}, number = {9}, pages = {4505--4525}, issn = {0002-9947}, mrclass = {46L35 (06F05 46L80)}, mrnumber = {2806681}, mrreviewer = {Xiao Chun Fang}, doi = {10.1090/S0002-9947-2011-05480-2}, url = {https://doi.org/10.1090/S0002-9947-2011-05480-2}, zblnumber = {1235.46057}, }
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@ARTICLE{OrtegaPereraRordamII, author = {Ortega, Eduard and Perera, Francesc and R{\o{r}}dam, Mikael}, title = {The {C}orona {F}actorization {P}roperty, stability, and the {C}untz semigroup of a {$\mathrm{C}^\ast$}-algebra}, journal = {Int. Math. Res. Not. IMRN}, fjournal = {International Mathematics Research Notices. IMRN}, year = {2012}, number = {1}, pages = {34--66}, issn = {1073-7928}, mrclass = {46L35 (06F05 46L05 46L80)}, mrnumber = {2874927}, mrreviewer = {Aaron Tikuisis}, doi = {10.1093/imrn/rnr013}, url = {https://doi.org/10.1093/imrn/rnr013}, zblnumber = {1245.46049}, }
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@ARTICLE{Ozawa:Cone, author = {Ozawa, N.}, title = {Homotopy invariance of {AF}-embeddability}, journal = {Geom. Funct. Anal.}, fjournal = {Geometric and Functional Analysis}, volume = {13}, year = {2003}, number = {1}, pages = {216--222}, issn = {1016-443X}, mrclass = {46L05}, mrnumber = {1978495}, mrreviewer = {William Paschke}, doi = {10.1007/s000390300005}, url = {https://doi.org/10.1007/s000390300005}, zblnumber = {1028.46090}, }
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@ARTICLE{Ozawa:DixmierApproximation, author = {Ozawa, Narutaka}, title = {Dixmier approximation and symmetric amenability for {$\mathrm{C}^*$}-algebras}, journal = {J. Math. Sci. Univ. Tokyo}, fjournal = {The University of Tokyo. Journal of Mathematical Sciences}, volume = {20}, year = {2013}, number = {3}, pages = {349--374}, issn = {1340-5705}, mrclass = {46L05 (46L10)}, mrnumber = {3156986}, mrreviewer = {William Paschke}, zblnumber = {1300.46049}, url = {https://www.ms.u-tokyo.ac.jp/journal/abstract/jms200303.html}, }
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@ARTICLE{OzawaRordamSato, author = {Ozawa, Narutaka and R{\o{r}}dam, Mikael and Sato, Yasuhiko}, title = {Elementary amenable groups are quasidiagonal}, journal = {Geom. Funct. Anal.}, fjournal = {Geometric and Functional Analysis}, volume = {25}, year = {2015}, number = {1}, pages = {307--316}, issn = {1016-443X}, mrclass = {46L35 (43A07)}, mrnumber = {3320894}, mrreviewer = {Stefanos Orfanos}, doi = {10.1007/s00039-015-0315-x}, url = {https://doi.org/10.1007/s00039-015-0315-x}, zblnumber = {1335.46053}, }
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@ARTICLE{Pimsner:AFE, author = {Pimsner, Mihai V.}, title = {Embedding some transformation group {$C\sp{\ast} $}-algebras into {AF}-algebras}, journal = {Ergodic Theory Dynam. Systems}, fjournal = {Ergodic Theory and Dynamical Systems}, volume = {3}, year = {1983}, number = {4}, pages = {613--626}, issn = {0143-3857}, mrclass = {46L05 (57S05)}, mrnumber = {0753927}, mrreviewer = {G. A. Elliott}, doi = {10.1017/S0143385700002182}, url = {https://doi.org/10.1017/S0143385700002182}, zblnumber = {0582.46063}, }
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@ARTICLE{PimsnerVoiculescu, author = {Pimsner, Mihai V. and Voiculescu, D.}, title = {Imbedding the irrational rotation {$C\sp{\ast} $}-algebra into an {AF}-algebra}, journal = {J. Operator Theory}, fjournal = {Journal of Operator Theory}, volume = {4}, year = {1980}, number = {2}, pages = {201--210}, issn = {0379-4024}, mrclass = {46L35 (16A54)}, mrnumber = {0595412}, mrreviewer = {David Handelman}, zblnumber = {0525.46031}, url = {http://www.mathjournals.org/jot/1980-004-002/1980-004-002-003.html}, }
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Subalgebras of simple AF-algebras

Abstract

Authors