Quasidiagonality of nuclear C$^{*}$-algebras

Abstract

We prove that faithful traces on separable and nuclear $\mathrm{C}^*$-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear $\mathrm{C}^*$-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms–Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal $\mathrm{C}^*$-algebras.

  • [APT:Contemp] Go to document P. Ara, F. Perera, and A. S. Toms, "$K$-theory for operator algebras. Classification of $C^*$-algebras," in Aspects of Operator Algebras and Applications, Providence, RI: Amer. Math. Soc., 2011, vol. 534, pp. 1-71.
    @INCOLLECTION{APT:Contemp, mrkey = {2767222},
      author = {Ara, Pere and Perera, Francesc and Toms, Andrew S.},
      mrclass = {46L35 (19K14 46L05 46L80)},
      series = {Contemp. Math.},
      address = {Providence, RI},
      publisher = {Amer. Math. Soc.},
      doi = {10.1090/conm/534/10521},
      zblnumber = {1219.46053},
      volume = {534},
      mrnumber = {2767222},
      booktitle = {Aspects of Operator Algebras and Applications},
      mrreviewer = {Vladimir Manuilov},
      title = {{$K$}-theory for operator algebras. {C}lassification of {$C\sp *$}-algebras},
      pages = {1--71},
      year = {2011},
      }
  • [B:LMSnotes] Go to document B. Blackadar, "Comparison theory for simple $C^*$-algebras," in Operator Algebras and Applications, Vol. 1, Cambridge: Cambridge Univ. Press, 1988, vol. 135, pp. 21-54.
    @INCOLLECTION{B:LMSnotes, mrkey = {0996438},
      author = {Blackadar, Bruce},
      mrclass = {46L05 (19K14 46L80)},
      series = {London Math. Soc. Lecture Note Ser.},
      address = {Cambridge},
      publisher = {Cambridge Univ. Press},
      zblnumber = {0706.46043},
      volume = {135},
      mrnumber = {0996438},
      booktitle = {Operator Algebras and Applications, {V}ol. 1},
      mrreviewer = {Victor Nistor},
      title = {Comparison theory for simple {$C\sp *$}-algebras},
      pages = {21--54},
      year = {1988},
      doi = {10.1007/978-1-4613-9572-0},
      }
  • [B:KThy] B. Blackadar, $K$-Theory for Operator Algebras, Second ed., Cambridge: Cambridge Univ. Press, 1998, vol. 5.
    @BOOK{B:KThy, mrkey = {1656031},
      author = {Blackadar, Bruce},
      mrclass = {46L80 (19Kxx 58G12)},
      series = {Math. Sci. Res. Inst. Publ.},
      edition = {Second},
      isbn = {0-521-63532-2},
      address = {Cambridge},
      publisher = {Cambridge Univ. Press},
      zblnumber = {0913.46054},
      volume = {5},
      mrnumber = {1656031},
      title = {{$K$}-Theory for Operator Algebras},
      year = {1998},
      pages = {xx+300},
      }
  • [BlaHan:dimtraces] Go to document B. Blackadar and D. Handelman, "Dimension functions and traces on $C^{\ast} $-algebras," J. Funct. Anal., vol. 45, iss. 3, pp. 297-340, 1982.
    @ARTICLE{BlaHan:dimtraces, mrkey = {0650185},
      number = {3},
      issn = {0022-1236},
      author = {Blackadar, Bruce and Handelman, David},
      mrclass = {46L05},
      doi = {10.1016/0022-1236(82)90009-X},
      journal = {J. Funct. Anal.},
      zblnumber = {0513.46047},
      volume = {45},
      mrnumber = {0650185},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {J. W. Bunce},
      coden = {JFUAAW},
      title = {Dimension functions and traces on {$C\sp{\ast} $}-algebras},
      year = {1982},
      pages = {297--340},
      }
  • [BlaKir:MathAnn] Go to document B. Blackadar and E. Kirchberg, "Generalized inductive limits of finite-dimensional $C^*$-algebras," Math. Ann., vol. 307, iss. 3, pp. 343-380, 1997.
    @ARTICLE{BlaKir:MathAnn, mrkey = {1437044},
      number = {3},
      issn = {0025-5831},
      author = {Blackadar, Bruce and Kirchberg, Eberhard},
      mrclass = {46L05 (46L35)},
      doi = {10.1007/s002080050039},
      journal = {Math. Ann.},
      zblnumber = {0874.46036},
      volume = {307},
      mrnumber = {1437044},
      fjournal = {Mathematische Annalen},
      mrreviewer = {Terry A. Loring},
      coden = {MAANA},
      title = {Generalized inductive limits of finite-dimensional {$C\sp *$}-algebras},
      year = {1997},
      pages = {343--380},
      }
  • [BBSTWW:arXiv] J. Bosa, N. Brown, Y. Sato, A. Tikuisis, S. White, and W. Winter, Covering dimension of $\mathrm{C}^*$-algebras and $2$-coloured classification, 2015.
    @MISC{BBSTWW:arXiv,
      author = {Bosa, Joan and Brown, Nathanial~P. and Sato, Yasuhiko and Tikuisis, Aaron and White, Stuart and Winter, Wilhelm},
      arxiv = {1506.03974v3},
      note = {{\em Mem. Amer. Math. Soc.},
      to appear},
      title = {Covering dimension of {$\mathrm{C}^*$}-algebras and {$2$}-coloured classification},
      year = {2015},
      }
  • [B:PJM] Go to document L. G. Brown, "Stable isomorphism of hereditary subalgebras of $C\sp*$-algebras," Pacific J. Math., vol. 71, iss. 2, pp. 335-348, 1977.
    @ARTICLE{B:PJM, mrkey = {0454645},
      number = {2},
      issn = {0030-8730},
      author = {Brown, Lawrence G.},
      mrclass = {46L05},
      journal = {Pacific J. Math.},
      zblnumber = {0362.46042},
      volume = {71},
      mrnumber = {0454645},
      fjournal = {Pacific Journal of Mathematics},
      mrreviewer = {Jun Tomiyama},
      title = {Stable isomorphism of hereditary subalgebras of {$C\sp*$}-algebras},
      year = {1977},
      pages = {335--348},
      doi = {10.2140/pjm.1977.71.335},
      }
  • [B:OAGR] L. G. Brown, "The universal coefficient theorem for ${ Ext}$ and quasidiagonality," in Operator Algebras and Group Representations, Vol. I, Boston: Pitman, 1984, vol. 17, pp. 60-64.
    @INCOLLECTION{B:OAGR, mrkey = {0731763},
      author = {Brown, Lawrence G.},
      mrclass = {46L80 (46L05 46M20)},
      series = {Monogr. Stud. Math.},
      address = {Boston},
      publisher = {Pitman},
      volume = {17},
      mrnumber = {0731763},
      booktitle = {Operator Algebras and Group Representations, {V}ol. {I}},
      mrreviewer = {Claude Schochet},
      venue = {{N}eptun, 1980},
      title = {The universal coefficient theorem for {${\rm Ext}$} and quasidiagonality},
      pages = {60--64},
      year = {1984},
      zblnumber = {0548.46055},
      }
  • [Bro:survey] N. P. Brown, "On quasidiagonal $C^*$-algebras," in Operator Algebras and Applications, Tokyo: Math. Soc. Japan, 2004, vol. 38, pp. 19-64.
    @INCOLLECTION{Bro:survey, mrkey = {2059800},
      author = {Brown, Nathanial P.},
      mrclass = {46L05},
      series = {Adv. Stud. Pure Math.},
      address = {Tokyo},
      publisher = {Math. Soc. Japan},
      zblnumber = {1065.46033},
      volume = {38},
      mrnumber = {2059800},
      booktitle = {Operator Algebras and Applications},
      title = {On quasidiagonal {$C\sp *$}-algebras},
      pages = {19--64},
      year = {2004},
      }
  • [B:MAMS] Go to document N. P. Brown, "Invariant means and finite representation theory of $C^*$-algebras," Mem. Amer. Math. Soc., vol. 184, iss. 865, p. viii, 2006.
    @ARTICLE{B:MAMS, mrkey = {2263412},
      number = {865},
      issn = {0065-9266},
      author = {Brown, Nathanial P.},
      mrclass = {46L05 (46L35)},
      doi = {10.1090/memo/0865},
      journal = {Mem. Amer. Math. Soc.},
      zblnumber = {1111.46030},
      volume = {184},
      mrnumber = {2263412},
      fjournal = {Memoirs of the American Mathematical Society},
      mrreviewer = {Narutaka Ozawa},
      coden = {MAMCAU},
      title = {Invariant means and finite representation theory of {$C\sp *$}-algebras},
      year = {2006},
      pages = {viii+105},
      }
  • [BrOz] Go to document N. P. Brown and N. Ozawa, $C^*$-Algebras and Finite-Dimensional Approximations, Providence, RI: Amer. Math. Soc., 2008.
    @BOOK{BrOz, mrkey = {2391387},
      number = {88},
      author = {Brown, Nathanial P. and Ozawa, Narutaka},
      mrclass = {46L05 (43A07 46-02 46L10)},
      series = {Grad. Stud. Math.},
      isbn = {978-0-8218-4381-9; 0-8218-4381-8},
      address = {Providence, RI},
      publisher = {Amer. Math. Soc.},
      doi = {10.1090/gsm/088},
      zblnumber = {1160.46001},
      mrnumber = {2391387},
      mrreviewer = {Mikael Rørdam},
      title = {{$C\sp *$}-Algebras and Finite-Dimensional Approximations},
      year = {2008},
      pages = {xvi+509},
      }
  • [BW:CRMASSRC] Go to document N. P. Brown and W. Winter, "Quasitraces are traces: a short proof of the finite-nuclear-dimension case," C. R. Math. Acad. Sci. Soc. R. Can., vol. 33, iss. 2, pp. 44-49, 2011.
    @ARTICLE{BW:CRMASSRC, mrkey = {2828512},
      number = {2},
      issn = {0706-1994},
      author = {Brown, Nathanial P. and Winter, Wilhelm},
      mrclass = {46L05},
      journal = {C. R. Math. Acad. Sci. Soc. R. Can.},
      zblnumber = {1236.46048},
      volume = {33},
      mrnumber = {2828512},
      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},
      mrreviewer = {K. T. Ravindran},
      title = {Quasitraces are traces: a short proof of the finite-nuclear-dimension case},
      year = {2011},
      pages = {44--49},
      url = {https://mr.math.ca/article/quasitraces-are-traces-a-short-proof-of-the-finite-nuclear-dimension-case/},
      }
  • [CDE:JFA] Go to document J. R. Carrión, M. Dadarlat, and C. Eckhardt, "On groups with quasidiagonal $C^*$-algebras," J. Funct. Anal., vol. 265, iss. 1, pp. 135-152, 2013.
    @ARTICLE{CDE:JFA, mrkey = {3049883},
      number = {1},
      issn = {0022-1236},
      author = {Carri{ó}n, Jos{é} R. and Dadarlat, Marius and Eckhardt, Caleb},
      mrclass = {46L05 (22D25)},
      doi = {10.1016/j.jfa.2013.04.004},
      journal = {J. Funct. Anal.},
      zblnumber = {1287.46043},
      volume = {265},
      mrnumber = {3049883},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {Tomasz Kania},
      title = {On groups with quasidiagonal {$C\sp *$}-algebras},
      year = {2013},
      pages = {135--152},
      }
  • [CE:Ann] Go to document M. D. Choi and E. G. Effros, "The completely positive lifting problem for $C\sp*$-algebras," Ann. of Math., vol. 104, iss. 3, pp. 585-609, 1976.
    @ARTICLE{CE:Ann, mrkey = {0417795},
      number = {3},
      issn = {0003-486X},
      author = {Choi, Man Duen and Effros, Edward G.},
      mrclass = {46L05},
      doi = {10.2307/1970968},
      journal = {Ann. of Math.},
      zblnumber = {0361.46067},
      volume = {104},
      mrnumber = {0417795},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {Maurice J. Dupre},
      title = {The completely positive lifting problem for {$C\sp*$}-algebras},
      year = {1976},
      pages = {585--609},
      }
  • [CE:IMRN] Go to document A. Ciuperca and G. A. Elliott, "A remark on invariants for $C^*$-algebras of stable rank one," Int. Math. Res. Not., vol. 2008, iss. 5, p. I, 2008.
    @ARTICLE{CE:IMRN, mrkey = {2418289},
      number = {5},
      issn = {1073-7928},
      author = {Ciuperca, Alin and Elliott, George A.},
      mrclass = {46L35 (46L05 46L80)},
      doi = {10.1093/imrn/rnm158},
      journal = {Int. Math. Res. Not.},
      zblnumber = {1159.46036},
      volume = {2008},
      mrnumber = {2418289},
      fjournal = {International Mathematics Research Notices. IMRN},
      mrreviewer = {Michael Frank},
      title = {A remark on invariants for {$C\sp *$}-algebras of stable rank one},
      year = {2008},
      pages = {Art. ID rnm 158, 33},
      }
  • [C:Ann] Go to document A. Connes, "Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not=1$," Ann. of Math., vol. 104, iss. 1, pp. 73-115, 1976.
    @ARTICLE{C:Ann, mrkey = {0454659},
      number = {1},
      issn = {0003-486X},
      author = {Connes, A.},
      mrclass = {46L10},
      doi = {10.2307/1971057},
      journal = {Ann. of Math.},
      zblnumber = {0343.46042},
      volume = {104},
      mrnumber = {0454659},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {Francois Combes},
      title = {Classification of injective factors. {C}ases {$II\sb{1},
     $} {$II\sb{\infty },
     $} {$III\sb{\lambda },
     $} {$\lambda \not=1$}},
      year = {1976},
      pages = {73--115},
      }
  • [CEI:Crelle] Go to document K. T. Coward, G. A. Elliott, and C. Ivanescu, "The Cuntz semigroup as an invariant for $C^*$-algebras," J. Reine Angew. Math., vol. 623, pp. 161-193, 2008.
    @ARTICLE{CEI:Crelle, mrkey = {2458043},
      issn = {0075-4102},
      author = {Coward, Kristofer T. and Elliott, George A. and Ivanescu, Cristian},
      mrclass = {46L35 (19K14 46L05 46L08 46L80 46M15)},
      doi = {10.1515/CRELLE.2008.075},
      journal = {J. Reine Angew. Math.},
      zblnumber = {1161.46029},
      volume = {623},
      mrnumber = {2458043},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      mrreviewer = {Francesc Perera},
      coden = {JRMAA8},
      title = {The {C}untz semigroup as an invariant for {$C\sp *$}-algebras},
      year = {2008},
      pages = {161--193},
      }
  • [C:MA] Go to document J. Cuntz, "Dimension functions on simple $C\sp*$-algebras," Math. Ann., vol. 233, iss. 2, pp. 145-153, 1978.
    @ARTICLE{C:MA, mrkey = {0467332},
      number = {2},
      issn = {0025-5831},
      author = {Cuntz, Joachim},
      mrclass = {46L05},
      doi = {10.1007/BF01421922},
      journal = {Math. Ann.},
      zblnumber = {0354.46043},
      volume = {233},
      mrnumber = {0467332},
      fjournal = {Mathematische Annalen},
      mrreviewer = {J. W. Bunce},
      title = {Dimension functions on simple {$C\sp*$}-algebras},
      year = {1978},
      pages = {145--153},
      }
  • [D:KThy] Go to document M. Dadarlat, "Approximately unitarily equivalent morphisms and inductive limit $C^\ast$-algebras," $K$-Theory, vol. 9, iss. 2, pp. 117-137, 1995.
    @ARTICLE{D:KThy, mrkey = {1340842},
      number = {2},
      issn = {0920-3036},
      author = {Dadarlat, Marius},
      mrclass = {46L05 (19K56 46L35 46L80)},
      doi = {10.1007/BF00961456},
      journal = {$K$-Theory},
      zblnumber = {0828.46055},
      volume = {9},
      mrnumber = {1340842},
      fjournal = {$K$-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of $K$-Theory in the Mathematical Sciences},
      mrreviewer = {Hua Xin Lin},
      coden = {KTHEEO},
      title = {Approximately unitarily equivalent morphisms and inductive limit {$C\sp \ast$}-algebras},
      year = {1995},
      pages = {117--137},
      }
  • [D:JFA97] Go to document M. Dadarlat, "Quasidiagonal morphisms and homotopy," J. Funct. Anal., vol. 151, iss. 1, pp. 213-233, 1997.
    @ARTICLE{D:JFA97, mrkey = {1487776},
      number = {1},
      issn = {0022-1236},
      author = {Dadarlat, Marius},
      mrclass = {46L05 (46L85)},
      doi = {10.1006/jfan.1997.3145},
      journal = {J. Funct. Anal.},
      zblnumber = {0992.46042},
      volume = {151},
      mrnumber = {1487776},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {Hiroshi Takai},
      coden = {JFUAAW},
      title = {Quasidiagonal morphisms and homotopy},
      year = {1997},
      pages = {213--233},
      }
  • [Dad:IJM] Go to document M. Dadarlat, "Morphisms of simple tracially AF algebras," Internat. J. Math., vol. 15, iss. 9, pp. 919-957, 2004.
    @ARTICLE{Dad:IJM, mrkey = {2106154},
      number = {9},
      issn = {0129-167X},
      author = {Dadarlat, Marius},
      mrclass = {46L05 (19K14 19K35 46L35 46L80)},
      doi = {10.1142/S0129167X04002636},
      journal = {Internat. J. Math.},
      zblnumber = {1071.46036},
      volume = {15},
      mrnumber = {2106154},
      fjournal = {International Journal of Mathematics},
      mrreviewer = {Xiao Chun Fang},
      title = {Morphisms of simple tracially {AF} algebras},
      year = {2004},
      pages = {919--957},
      }
  • [DE:PLMS] Go to document M. Dadarlat and S. Eilers, "On the classification of nuclear $C^*$-algebras," Proc. London Math. Soc., vol. 85, iss. 1, pp. 168-210, 2002.
    @ARTICLE{DE:PLMS, mrkey = {1901373},
      number = {1},
      issn = {0024-6115},
      author = {Dadarlat, Marius and Eilers, Søren},
      mrclass = {19K35 (46L05 46L35 46L80)},
      doi = {10.1112/S0024611502013679},
      journal = {Proc. London Math. Soc.},
      zblnumber = {1031.46070},
      volume = {85},
      mrnumber = {1901373},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      mrreviewer = {Emmanuel C. Germain},
      coden = {PLMTAL},
      title = {On the classification of nuclear {$C\sp *$}-algebras},
      year = {2002},
      pages = {168--210},
      }
  • [DL:DMJ] Go to document M. Dadarlat and T. A. Loring, "A universal multicoefficient theorem for the Kasparov groups," Duke Math. J., vol. 84, iss. 2, pp. 355-377, 1996.
    @ARTICLE{DL:DMJ, mrkey = {1404333},
      number = {2},
      issn = {0012-7094},
      author = {Dadarlat, Marius and Loring, Terry A.},
      mrclass = {46L80 (19K35 46L35)},
      doi = {10.1215/S0012-7094-96-08412-4},
      journal = {Duke Math. J.},
      zblnumber = {0881.46048},
      volume = {84},
      mrnumber = {1404333},
      fjournal = {Duke Mathematical Journal},
      mrreviewer = {Kevin McClanahan},
      coden = {DUMJAO},
      title = {A universal multicoefficient theorem for the {K}asparov groups},
      year = {1996},
      pages = {355--377},
      }
  • [DP:JFA] Go to document M. Dadarlat and C. Pasnicu, "Continuous fields of Kirchberg $C^*$-algebras," J. Funct. Anal., vol. 226, iss. 2, pp. 429-451, 2005.
    @ARTICLE{DP:JFA, mrkey = {2160103},
      number = {2},
      issn = {0022-1236},
      author = {Dadarlat, Marius and Pasnicu, Cornel},
      mrclass = {46L35 (46L05 46L80)},
      doi = {10.1016/j.jfa.2005.01.005},
      journal = {J. Funct. Anal.},
      zblnumber = {1083.46035},
      volume = {226},
      mrnumber = {2160103},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {Emmanuel C. Germain},
      coden = {JFUAAW},
      title = {Continuous fields of {K}irchberg {$C\sp *$}-algebras},
      year = {2005},
      pages = {429--451},
      }
  • [Dixmier:C*Book] J. Dixmier, $C\sp*$-algebras, Amsterdam: North-Holland Publishing Co., 1977, vol. 15.
    @BOOK{Dixmier:C*Book, mrkey = {0458185},
      author = {Dixmier, Jacques},
      mrclass = {46L05},
      series = {North-Holland Math. Lib.},
      isbn = {0-7204-0762-1},
      address = {Amsterdam},
      publisher = {North-Holland Publishing Co.},
      volume = {15},
      mrnumber = {0458185},
      note = {translated from the French by Francis Jellett},
      title = {{$C\sp*$}-algebras},
      year = {1977},
      pages = {xiii+492},
      zblnumber = {0372.46058},
      }
  • [EHS:AJM] Go to document E. G. Effros, D. E. Handelman, and C. L. Shen, "Dimension groups and their affine representations," Amer. J. Math., vol. 102, iss. 2, pp. 385-407, 1980.
    @ARTICLE{EHS:AJM, mrkey = {0564479},
      number = {2},
      issn = {0002-9327},
      author = {Effros, Edward G. and Handelman, David E. and Shen, Chao Liang},
      mrclass = {46L55 (22A26 58G12 82A15)},
      doi = {10.2307/2374244},
      journal = {Amer. J. Math.},
      zblnumber = {0457.46047},
      volume = {102},
      mrnumber = {0564479},
      fjournal = {American Journal of Mathematics},
      mrreviewer = {Alain Connes},
      coden = {AJMAAN},
      title = {Dimension groups and their affine representations},
      year = {1980},
      pages = {385--407},
      }
  • [EffRos:PJM] Go to document E. G. Effros and J. Rosenberg, "$C^{\ast} $-algebras with approximately inner flip," Pacific J. Math., vol. 77, iss. 2, pp. 417-443, 1978.
    @ARTICLE{EffRos:PJM, mrkey = {0510932},
      number = {2},
      issn = {0030-8730},
      author = {Effros, Edward G. and Rosenberg, Jonathan},
      mrclass = {46L05},
      journal = {Pacific J. Math.},
      zblnumber = {0412.46052},
      volume = {77},
      mrnumber = {0510932},
      fjournal = {Pacific Journal of Mathematics},
      mrreviewer = {Trond Digernes},
      coden = {PJMAAI},
      title = {{$C\sp{\ast} $}-algebras with approximately inner flip},
      year = {1978},
      pages = {417--443},
      doi = {10.2140/pjm.1978.77.417},
      }
  • [E:JA] Go to document G. A. Elliott, "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras," J. Algebra, vol. 38, iss. 1, pp. 29-44, 1976.
    @ARTICLE{E:JA, mrkey = {0397420},
      number = {1},
      issn = {0021-8693},
      author = {Elliott, George A.},
      mrclass = {46L05 (16A46)},
      doi = {10.1016/0021-8693(76)90242-8},
      journal = {J. Algebra},
      zblnumber = {0323.46063},
      volume = {38},
      mrnumber = {0397420},
      fjournal = {Journal of Algebra},
      mrreviewer = {Horst Behncke},
      title = {On the classification of inductive limits of sequences of semisimple finite-dimensional algebras},
      year = {1976},
      pages = {29--44},
      }
  • [E:CMS] G. A. Elliott, "An invariant for simple $C^*$-algebras," in Canadian Mathematical Society 1945–1995, Vol. 3, Ottawa, ON: Canadian Math. Soc., 1996, pp. 61-90.
    @INCOLLECTION{E:CMS, mrkey = {1661611},
      author = {Elliott, George A.},
      mrclass = {46L05 (46L35)},
      address = {Ottawa, ON},
      publisher = {Canadian Math. Soc.},
      zblnumber = {1206.46046},
      mrnumber = {1661611},
      booktitle = {Canadian {M}athematical {S}ociety 1945--1995, {V}ol. 3},
      mrreviewer = {Claire Anantharaman-Delaroche},
      title = {An invariant for simple {$C\sp *$}-algebras},
      pages = {61--90},
      year = {1996},
      }
  • [EG:Ann] Go to document G. A. Elliott and G. Gong, "On the classification of $C^*$-algebras of real rank zero. II," Ann. of Math., vol. 144, iss. 3, pp. 497-610, 1996.
    @ARTICLE{EG:Ann, mrkey = {1426886},
      number = {3},
      issn = {0003-486X},
      author = {Elliott, George A. and Gong, Guihua},
      mrclass = {46L05 (19K56 46L80)},
      doi = {10.2307/2118565},
      journal = {Ann. of Math.},
      zblnumber = {0867.46041},
      volume = {144},
      mrnumber = {1426886},
      fjournal = {Annals of Mathematics. Second Series},
      coden = {ANMAAH},
      title = {On the classification of {$C\sp *$}-algebras of real rank zero. {II}},
      year = {1996},
      pages = {497--610},
      }
  • [EGLN:arXiv] G. A. Elliott, G. Gong, H. Lin, and Z. Niu, On the classification of simple amenable $\mathrm C^*$-algebras with finite decomposition rank, II, 2015.
    @MISC{EGLN:arXiv,
      author = {Elliott, George A. and Gong, Guihua and Lin, Huaxin and Niu, Zhuang},
      arxiv = {1507.03437v2},
      title = {On the classification of simple amenable {$\mathrm C^*$}-algebras with finite decomposition rank, {II}},
      year = {2015},
      }
  • [EGLP:DMJ] Go to document G. A. Elliott, G. Gong, H. Lin, and C. Pasnicu, "Abelian $C^*$-subalgebras of $C^*$-algebras of real rank zero and inductive limit $C^*$-algebras," Duke Math. J., vol. 85, iss. 3, pp. 511-554, 1996.
    @ARTICLE{EGLP:DMJ, mrkey = {1422356},
      number = {3},
      issn = {0012-7094},
      author = {Elliott, George A. and Gong, Guihua and Lin, Huaxin and Pasnicu, Cornel},
      mrclass = {46L35 (46L45 46L80)},
      doi = {10.1215/S0012-7094-96-08520-8},
      journal = {Duke Math. J.},
      zblnumber = {0869.46030},
      volume = {85},
      mrnumber = {1422356},
      fjournal = {Duke Mathematical Journal},
      mrreviewer = {George Popescu},
      coden = {DUMJAO},
      title = {Abelian {$C\sp *$}-subalgebras of {$C\sp *$}-algebras of real rank zero and inductive limit {$C\sp *$}-algebras},
      year = {1996},
      pages = {511--554},
      }
  • [EN:arXiv] G. A. Elliott and Z. Niu, "On the classification of simple unital $\mathrm{C}^*$-algebras with finite decomposition rank," in Operator Algebras and Their Applications: A Tribute to Richard V. Kadison, Providence, RI: Amer. Math. Soc., 2016, vol. 671, pp. 117-126.
    @INCOLLECTION{EN:arXiv, volume = {671},
      author = {Elliott, George A. and Niu, Zhuang},
      series = {Contemp. Math.},
      booktitle = {Operator Algebras and Their Applications: A Tribute to Richard V. Kadison},
      address = {Providence, RI},
      title = {On the classification of simple unital $\mathrm{C}^*$-algebras with finite decomposition rank},
      publisher = {Amer. Math. Soc.},
      year = {2016},
      pages = {117--126},
      }
  • [ERS:AJM] Go to document G. A. Elliott, L. Robert, and L. Santiago, "The cone of lower semicontinuous traces on a $C^*$-algebra," Amer. J. Math., vol. 133, iss. 4, pp. 969-1005, 2011.
    @ARTICLE{ERS:AJM, mrkey = {2823868},
      number = {4},
      issn = {0002-9327},
      author = {Elliott, George A. and Robert, Leonel and Santiago, Luis},
      mrclass = {46L30},
      doi = {10.1353/ajm.2011.0027},
      journal = {Amer. J. Math.},
      zblnumber = {1236.46052},
      volume = {133},
      mrnumber = {2823868},
      fjournal = {American Journal of Mathematics},
      coden = {AJMAAN},
      title = {The cone of lower semicontinuous traces on a {$C\sp *$}-algebra},
      year = {2011},
      pages = {969--1005},
      }
  • [G:arXiv] J. Gabe, Quasidiagonal traces on exact $\mathrm{C}^*$-algebras, 2015.
    @MISC{G:arXiv,
      author = {Gabe, James},
      arxiv = {1511.02760v1},
      title = {Quasidiagonal traces on exact $\mathrm{C}^*$-algebras},
      year = {2015},
      }
  • [GLN:arXiv] G. Gong, H. Lin, and Z. Niu, Classification of finite simple amenable $\mathcal Z$-stable $\mathrm{C}^*$-algebras, 2015.
    @MISC{GLN:arXiv,
      author = {Gong, Guihua and Lin, Huaxin and Niu, Zhuang},
      arxiv = {1501.00135v4},
      title = {Classification of finite simple amenable {$\mathcal Z$}-stable {$\mathrm{C}^*$}-algebras},
      year = {2015},
      }
  • [Haa:quasitraces] U. Haagerup, "Quasitraces on exact $C^*$-algebras are traces," C. R. Math. Acad. Sci. Soc. R. Can., vol. 36, iss. 2-3, pp. 67-92, 2014.
    @ARTICLE{Haa:quasitraces, mrkey = {3241179},
      number = {2-3},
      issn = {0706-1994},
      author = {Haagerup, Uffe},
      mrclass = {46L35 (46L80)},
      journal = {C. R. Math. Acad. Sci. Soc. R. Can.},
      zblnumber = {1325.46055},
      volume = {36},
      mrnumber = {3241179},
      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},
      mrreviewer = {Francesc Perera},
      title = {Quasitraces on exact {$C\sp *$}-algebras are traces},
      year = {2014},
      pages = {67--92},
      note={circulated in manuscript form in 1991},
      }
  • [Hadwin:JOT] D. Hadwin, "Strongly quasidiagonal $C^*$-algebras," J. Operator Theory, vol. 18, iss. 1, pp. 3-18, 1987.
    @ARTICLE{Hadwin:JOT, mrkey = {0912809},
      number = {1},
      issn = {0379-4024},
      author = {Hadwin, Don},
      mrclass = {46L05 (46L55 46M05 47A66 47D25)},
      journal = {J. Operator Theory},
      zblnumber = {0644.46036},
      volume = {18},
      mrnumber = {0912809},
      note = {with an appendix by Jonathan Rosenberg},
      fjournal = {Journal of Operator Theory},
      mrreviewer = {Judith A. Packer},
      title = {Strongly quasidiagonal {$C\sp *$}-algebras},
      year = {1987},
      pages = {3--18},
      }
  • [Hal:BAMS] Go to document P. R. Halmos, "Ten problems in Hilbert space," Bull. Amer. Math. Soc., vol. 76, pp. 887-933, 1970.
    @ARTICLE{Hal:BAMS, mrkey = {0270173},
      issn = {0002-9904},
      author = {Halmos, P. R.},
      mrclass = {47.00 (46.00)},
      doi = {10.1090/S0002-9904-1970-12502-2},
      journal = {Bull. Amer. Math. Soc.},
      zblnumber = {0204.15001},
      volume = {76},
      mrnumber = {0270173},
      fjournal = {Bulletin of the American Mathematical Society},
      mrreviewer = {R. G. Douglas},
      title = {Ten problems in {H}ilbert space},
      year = {1970},
      pages = {887--933},
      }
  • [JST:arXiv] Go to document B. Jacelon, K. R. Strung, and A. S. Toms, "Unitary orbits of self-adjoint operators in simple $\mathcal{Z}$-stable $ C^*$-algebras," J. Funct. Anal., vol. 269, iss. 10, pp. 3304-3315, 2015.
    @ARTICLE{JST:arXiv, mrkey = {3401619},
      number = {10},
      issn = {0022-1236},
      author = {Jacelon, Bhishan and Strung, Karen R. and Toms, Andrew S.},
      mrclass = {46L05 (46L35)},
      doi = {10.1016/j.jfa.2015.08.019},
      journal = {J. Funct. Anal.},
      zblnumber = {1342.46051},
      volume = {269},
      mrnumber = {3401619},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {William Paschke},
      title = {Unitary orbits of self-adjoint operators in simple {$\mathcal{Z}$}-stable {$\rm C\sp *$}-algebras},
      year = {2015},
      pages = {3304--3315},
      }
  • [JS:AJM] Go to document X. Jiang and H. Su, "On a simple unital projectionless $C^*$-algebra," Amer. J. Math., vol. 121, iss. 2, pp. 359-413, 1999.
    @ARTICLE{JS:AJM, mrkey = {1680321},
      number = {2},
      issn = {0002-9327},
      author = {Jiang, Xinhui and Su, Hongbing},
      mrclass = {46L35 (19K35 46L80)},
      journal = {Amer. J. Math.},
      zblnumber = {0923.46069},
      volume = {121},
      mrnumber = {1680321},
      fjournal = {American Journal of Mathematics},
      mrreviewer = {Vicumpriya S. Perera},
      coden = {AJMAAN},
      title = {On a simple unital projectionless {$C\sp *$}-algebra},
      year = {1999},
      pages = {359--413},
      doi = {10.1353/ajm.1999.0012},
      }
  • [Kir:ICM] E. Kirchberg, "Exact ${ C}^*$-algebras, tensor products, and the classification of purely infinite algebras," in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Basel, 1995, pp. 943-954.
    @INPROCEEDINGS{Kir:ICM, mrkey = {1403994},
      author = {Kirchberg, Eberhard},
      mrclass = {46L05 (46L35 46M05 46M15)},
      address = {Basel},
      publisher = {Birkhäuser},
      zblnumber = {0897.46057},
      mrnumber = {1403994},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2},
      venue = {{Z}ürich, 1994},
      mrreviewer = {Robert S. Doran},
      title = {Exact {${\rm C}\sp *$}-algebras, tensor products, and the classification of purely infinite algebras},
      pages = {943--954},
      year = {1995},
      }
  • [K:FI] E. Kirchberg, "On the existence of traces on exact stably projectionless simple $C^\ast$-algebras," in Operator Algebras and their Applications, Providence, RI: Amer. Math. Soc., 1997, vol. 13, pp. 171-172.
    @INCOLLECTION{K:FI, mrkey = {1424961},
      author = {Kirchberg, Eberhard},
      mrclass = {46L05 (46L80)},
      series = {Fields Inst. Commun.},
      address = {Providence, RI},
      publisher = {Amer. Math. Soc.},
      zblnumber = {0894.46043},
      volume = {13},
      mrnumber = {1424961},
      booktitle = {Operator Algebras and their Applications},
      mrreviewer = {Sze-Kai Tsui},
      venue = {{W}aterloo, {ON},
      1994/1995},
      title = {On the existence of traces on exact stably projectionless simple {$C\sp \ast$}-algebras},
      pages = {171--172},
      year = {1997},
      }
  • [K:Abel] Go to document E. Kirchberg, "Central sequences in $C^*$-algebras and strongly purely infinite algebras," in Operator Algebras: The Abel Symposium 2004, New York: Springer-Verlag, 2006, vol. 1, pp. 175-231.
    @INCOLLECTION{K:Abel, mrkey = {2265050},
      author = {Kirchberg, Eberhard},
      mrclass = {46L05},
      series = {Abel Symp.},
      address = {New York},
      publisher = {Springer-Verlag},
      doi = {10.1007/978-3-540-34197-0_10},
      zblnumber = {1118.46054},
      volume = {1},
      mrnumber = {2265050},
      booktitle = {Operator {A}lgebras: {T}he {A}bel {S}ymposium 2004},
      mrreviewer = {Wilhelm Winter},
      title = {Central sequences in {$C\sp *$}-algebras and strongly purely infinite algebras},
      pages = {175--231},
      year = {2006},
      }
  • [KR:Crelle] Go to document E. Kirchberg and M. Rørdam, "Central sequence $C^*$-algebras and tensorial absorption of the Jiang-Su algebra," J. Reine Angew. Math., vol. 695, pp. 175-214, 2014.
    @ARTICLE{KR:Crelle, mrkey = {3276157},
      issn = {0075-4102},
      author = {Kirchberg, Eberhard and Rørdam, Mikael},
      mrclass = {46L35 (46C20)},
      doi = {10.1515/crelle-2012-0118},
      journal = {J. Reine Angew. Math.},
      zblnumber = {1307.46046},
      volume = {695},
      mrnumber = {3276157},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      mrreviewer = {Aurelian Gheondea},
      title = {Central sequence {$C\sp *$}-algebras and tensorial absorption of the {J}iang-{S}u algebra},
      year = {2014},
      pages = {175--214},
      }
  • [KirWin:dr] Go to document E. Kirchberg and W. Winter, "Covering dimension and quasidiagonality," Internat. J. Math., vol. 15, iss. 1, pp. 63-85, 2004.
    @ARTICLE{KirWin:dr, mrkey = {2039212},
      number = {1},
      issn = {0129-167X},
      author = {Kirchberg, Eberhard and Winter, Wilhelm},
      mrclass = {46L85 (46L53)},
      doi = {10.1142/S0129167X04002119},
      journal = {Internat. J. Math.},
      zblnumber = {1065.46053},
      volume = {15},
      mrnumber = {2039212},
      fjournal = {International Journal of Mathematics},
      mrreviewer = {William Paschke},
      title = {Covering dimension and quasidiagonality},
      year = {2004},
      pages = {63--85},
      }
  • [Lin:JOT] H. Lin, "Stable approximate unitary equivalence of homomorphisms," J. Operator Theory, vol. 47, iss. 2, pp. 343-378, 2002.
    @ARTICLE{Lin:JOT, mrkey = {1911851},
      number = {2},
      issn = {0379-4024},
      author = {Lin, Huaxin},
      mrclass = {46L80 (46L05 46L35)},
      journal = {J. Operator Theory},
      zblnumber = {1029.46097},
      volume = {47},
      mrnumber = {1911851},
      fjournal = {Journal of Operator Theory},
      mrreviewer = {Vladimir Manuilov},
      title = {Stable approximate unitary equivalence of homomorphisms},
      year = {2002},
      pages = {343--378},
      }
  • [Lin:Duke] Go to document H. Lin, "Classification of simple $C^\ast$-algebras of tracial topological rank zero," Duke Math. J., vol. 125, iss. 1, pp. 91-119, 2004.
    @ARTICLE{Lin:Duke, mrkey = {2097358},
      number = {1},
      issn = {0012-7094},
      author = {Lin, Huaxin},
      mrclass = {46L05 (46L35 46L80)},
      doi = {10.1215/S0012-7094-04-12514-X},
      journal = {Duke Math. J.},
      zblnumber = {1068.46032},
      volume = {125},
      mrnumber = {2097358},
      fjournal = {Duke Mathematical Journal},
      mrreviewer = {Hiroyuki Osaka},
      coden = {DUMJAO},
      title = {Classification of simple {$C\sp \ast$}-algebras of tracial topological rank zero},
      year = {2004},
      pages = {91--119},
      }
  • [Lin:apprUCT] Go to document H. Lin, "An approximate universal coefficient theorem," Trans. Amer. Math. Soc., vol. 357, iss. 8, pp. 3375-3405, 2005.
    @ARTICLE{Lin:apprUCT, mrkey = {2135753},
      number = {8},
      issn = {0002-9947},
      author = {Lin, Huaxin},
      mrclass = {46L05 (19K35 46L35 46L80)},
      doi = {10.1090/S0002-9947-05-03696-2},
      journal = {Trans. Amer. Math. Soc.},
      zblnumber = {1068.46039},
      volume = {357},
      mrnumber = {2135753},
      fjournal = {Transactions of the American Mathematical Society},
      mrreviewer = {Efton Park},
      coden = {TAMTAM},
      title = {An approximate universal coefficient theorem},
      year = {2005},
      pages = {3375--3405},
      }
  • [LN:Adv] Go to document H. Lin and Z. Niu, "Lifting $KK$-elements, asymptotic unitary equivalence and classification of simple $C^\ast$-algebras," Adv. Math., vol. 219, iss. 5, pp. 1729-1769, 2008.
    @ARTICLE{LN:Adv, mrkey = {2458153},
      number = {5},
      issn = {0001-8708},
      author = {Lin, Huaxin and Niu, Zhuang},
      mrclass = {46L35 (46L80)},
      doi = {10.1016/j.aim.2008.07.011},
      journal = {Adv. Math.},
      zblnumber = {1162.46033},
      volume = {219},
      mrnumber = {2458153},
      fjournal = {Advances in Mathematics},
      mrreviewer = {Andrew S. Toms},
      coden = {ADMTA4},
      title = {Lifting {$KK$}-elements, asymptotic unitary equivalence and classification of simple {$C\sp \ast$}-algebras},
      year = {2008},
      pages = {1729--1769},
      }
  • [L:Book] T. A. Loring, Lifting Solutions to Perturbing Problems in $C^*$-algebras, Providence, RI: Amer. Math. Soc., 1997, vol. 8.
    @BOOK{L:Book, mrkey = {1420863},
      author = {Loring, Terry A.},
      mrclass = {46L85 (46J99 46L05 47A62)},
      series = {Fields Inst. Monogr.},
      isbn = {0-8218-0602-5},
      address = {Providence, RI},
      publisher = {Amer. Math. Soc.},
      zblnumber = {1155.46310},
      volume = {8},
      mrnumber = {1420863},
      mrreviewer = {Hiroshi Takai},
      title = {Lifting Solutions to Perturbing Problems in {$C\sp *$}-algebras},
      year = {1997},
      pages = {x+165},
      }
  • [MS:Acta] Go to document H. Matui and Y. Sato, "Strict comparison and $\mathcal{Z}$-absorption of nuclear $C^*$-algebras," Acta Math., vol. 209, iss. 1, pp. 179-196, 2012.
    @ARTICLE{MS:Acta, mrkey = {2979512},
      number = {1},
      issn = {0001-5962},
      author = {Matui, Hiroki and Sato, Yasuhiko},
      mrclass = {46L05},
      doi = {10.1007/s11511-012-0084-4},
      journal = {Acta Math.},
      zblnumber = {1277.46028},
      volume = {209},
      mrnumber = {2979512},
      fjournal = {Acta Mathematica},
      mrreviewer = {Caleb Eckhardt},
      coden = {ACMAA8},
      title = {Strict comparison and {$\mathcal{Z}$}-absorption of nuclear {$C\sp *$}-algebras},
      year = {2012},
      pages = {179--196},
      }
  • [MS:DMJ] Go to document H. Matui and Y. Sato, "Decomposition rank of UHF-absorbing $\mathrm{C}^*$-algebras," Duke Math. J., vol. 163, iss. 14, pp. 2687-2708, 2014.
    @ARTICLE{MS:DMJ, mrkey = {3273581},
      number = {14},
      issn = {0012-7094},
      author = {Matui, Hiroki and Sato, Yasuhiko},
      mrclass = {46L06 (46L35 46L55)},
      doi = {10.1215/00127094-2826908},
      journal = {Duke Math. J.},
      zblnumber = {1317.46041},
      volume = {163},
      mrnumber = {3273581},
      fjournal = {Duke Mathematical Journal},
      mrreviewer = {William Paschke},
      title = {Decomposition rank of {UHF}-absorbing {$\mathrm{C}\sp *$}-algebras},
      year = {2014},
      pages = {2687--2708},
      }
  • [MS:AJM] Go to document H. Matui and Y. Sato, "$\mathcal Z$-stability of crossed products by strongly outer actions II," Amer. J. Math., vol. 136, iss. 6, pp. 1441-1496, 2014.
    @ARTICLE{MS:AJM, mrkey = {3282978},
      number = {6},
      issn = {0002-9327},
      author = {Matui, Hiroki and Sato, Yasuhiko},
      mrclass = {46L55},
      doi = {10.1353/ajm.2014.0043},
      journal = {Amer. J. Math.},
      zblnumber = {1317.46042},
      volume = {136},
      mrnumber = {3282978},
      fjournal = {American Journal of Mathematics},
      title = {{$\mathcal Z$}-stability of crossed products by strongly outer actions {II}},
      year = {2014},
      pages = {1441--1496},
      }
  • [O:GAFA] Go to document N. Ozawa, "Homotopy invariance of AF-embeddability," Geom. Funct. Anal., vol. 13, iss. 1, pp. 216-222, 2003.
    @ARTICLE{O:GAFA, mrkey = {1978495},
      number = {1},
      issn = {1016-443X},
      author = {Ozawa, N.},
      mrclass = {46L05},
      doi = {10.1007/s000390300005},
      journal = {Geom. Funct. Anal.},
      zblnumber = {1028.46090},
      volume = {13},
      mrnumber = {1978495},
      fjournal = {Geometric and Functional Analysis},
      mrreviewer = {William Paschke},
      coden = {GFANFB},
      title = {Homotopy invariance of {AF}-embeddability},
      year = {2003},
      pages = {216--222},
      }
  • [O:JMSUT] Go to document N. Ozawa, "Dixmier approximation and symmetric amenability for $ C^*$-algebras," J. Math. Sci. Univ. Tokyo, vol. 20, iss. 3, pp. 349-374, 2013.
    @ARTICLE{O:JMSUT, mrkey = {3156986},
      number = {3},
      issn = {1340-5705},
      author = {Ozawa, Narutaka},
      mrclass = {46L05 (46L10)},
      journal = {J. Math. Sci. Univ. Tokyo},
      zblnumber = {1300.46049},
      volume = {20},
      mrnumber = {3156986},
      fjournal = {The University of Tokyo. Journal of Mathematical Sciences},
      mrreviewer = {William Paschke},
      title = {Dixmier approximation and symmetric amenability for {$\rm C\sp *$}-algebras},
      year = {2013},
      pages = {349--374},
      url={http://www.ms.u-tokyo.ac.jp/journal/pdf/jms200303.pdf},
     }
  • [ORS:GAFA] Go to document N. Ozawa, M. Rørdam, and Y. Sato, "Elementary amenable groups are quasidiagonal," Geom. Funct. Anal., vol. 25, iss. 1, pp. 307-316, 2015.
    @ARTICLE{ORS:GAFA, mrkey = {3320894},
      number = {1},
      issn = {1016-443X},
      author = {Ozawa, Narutaka and Rørdam, Mikael and Sato, Yasuhiko},
      mrclass = {46L35 (43A07)},
      doi = {10.1007/s00039-015-0315-x},
      journal = {Geom. Funct. Anal.},
      zblnumber = {1335.46053},
      volume = {25},
      mrnumber = {3320894},
      fjournal = {Geometric and Functional Analysis},
      mrreviewer = {Stefanos Orfanos},
      title = {Elementary amenable groups are quasidiagonal},
      year = {2015},
      pages = {307--316},
      }
  • [Popa:PJM] Go to document S. Popa, "On local finite-dimensional approximation of $C^*$-algebras," Pacific J. Math., vol. 181, iss. 1, pp. 141-158, 1997.
    @ARTICLE{Popa:PJM, mrkey = {1491038},
      number = {1},
      issn = {0030-8730},
      author = {Popa, Sorin},
      mrclass = {46L05},
      doi = {10.2140/pjm.1997.181.141},
      journal = {Pacific J. Math.},
      zblnumber = {0907.46046},
      volume = {181},
      mrnumber = {1491038},
      fjournal = {Pacific Journal of Mathematics},
      mrreviewer = {Sze-Kai Tsui},
      coden = {PJMAAI},
      title = {On local finite-dimensional approximation of {$C\sp *$}-algebras},
      year = {1997},
      pages = {141--158},
      }
  • [R:Adv] Go to document L. Robert, "Classification of inductive limits of 1-dimensional NCCW complexes," Adv. Math., vol. 231, iss. 5, pp. 2802-2836, 2012.
    @ARTICLE{R:Adv, mrkey = {2970466},
      number = {5},
      issn = {0001-8708},
      author = {Robert, Leonel},
      mrclass = {46L35},
      doi = {10.1016/j.aim.2012.07.010},
      journal = {Adv. Math.},
      zblnumber = {1268.46041},
      volume = {231},
      mrnumber = {2970466},
      fjournal = {Advances in Mathematics},
      mrreviewer = {Changguo Wei},
      title = {Classification of inductive limits of 1-dimensional {NCCW} complexes},
      year = {2012},
      pages = {2802--2836},
      }
  • [RS:JFA] Go to document L. Robert and L. Santiago, "Classification of $C^\ast$-homomorphisms from $C_0(0,1]$ to a $C^\ast$-algebra," J. Funct. Anal., vol. 258, iss. 3, pp. 869-892, 2010.
    @ARTICLE{RS:JFA, mrkey = {2558180},
      number = {3},
      issn = {0022-1236},
      author = {Robert, Leonel and Santiago, Luis},
      mrclass = {46L05},
      doi = {10.1016/j.jfa.2009.06.025},
      journal = {J. Funct. Anal.},
      zblnumber = {1192.46057},
      volume = {258},
      mrnumber = {2558180},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {Valentin Deaconu},
      coden = {JFUAAW},
      title = {Classification of {$C\sp \ast$}-homomorphisms from {$C\sb 0(0,1]$} to a {$C\sp \ast$}-algebra},
      year = {2010},
      pages = {869--892},
      }
  • [R:JFA2] Go to document M. Rørdam, "On the structure of simple $C^*$-algebras tensored with a UHF-algebra. II," J. Funct. Anal., vol. 107, iss. 2, pp. 255-269, 1992.
    @ARTICLE{R:JFA2, mrkey = {1172023},
      number = {2},
      issn = {0022-1236},
      author = {Rørdam, Mikael},
      mrclass = {46L05 (46L85)},
      doi = {10.1016/0022-1236(92)90106-S},
      journal = {J. Funct. Anal.},
      zblnumber = {0810.46067},
      volume = {107},
      mrnumber = {1172023},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {Mahmood Khoshkam},
      coden = {JFUAAW},
      title = {On the structure of simple {$C\sp *$}-algebras tensored with a {UHF}-algebra. {II}},
      year = {1992},
      pages = {255--269},
      }
  • [R:Book] Go to document M. Rørdam, "Classification of nuclear, simple $C^*$-algebras," in Classification of Nuclear $C^*$-Algebras. Entropy in Operator Algebras, New York: Springer-Verlag, 2002, vol. 126, pp. 1-145.
    @INCOLLECTION{R:Book, mrkey = {1878882},
      author = {Rørdam, Mikael},
      mrclass = {46L05 (19K56 46L35 46L80)},
      series = {Encyclopaedia Math. Sci.},
      address = {New York},
      publisher = {Springer-Verlag},
      doi = {10.1007/978-3-662-04825-2_1},
      zblnumber = {1016.46037},
      volume = {126},
      mrnumber = {1878882},
      booktitle = {Classification of Nuclear {$C\sp *$}-Algebras. {E}ntropy in Operator Algebras},
      mrreviewer = {Judith A. Packer},
      title = {Classification of nuclear, simple {$C\sp *$}-algebras},
      pages = {1--145},
      year = {2002},
      }
  • [R:IJM] Go to document M. Rørdam, "The stable and the real rank of $\mathcal Z$-absorbing $C^*$-algebras," Internat. J. Math., vol. 15, iss. 10, pp. 1065-1084, 2004.
    @ARTICLE{R:IJM, mrkey = {2106263},
      number = {10},
      issn = {0129-167X},
      author = {Rørdam, Mikael},
      mrclass = {46L35 (19K14 46L05 46L06)},
      doi = {10.1142/S0129167X04002661},
      journal = {Internat. J. Math.},
      zblnumber = {1077.46054},
      volume = {15},
      mrnumber = {2106263},
      fjournal = {International Journal of Mathematics},
      mrreviewer = {Nadia S. Larsen},
      title = {The stable and the real rank of {$\mathcal Z$}-absorbing {$C\sp *$}-algebras},
      year = {2004},
      pages = {1065--1084},
      }
  • [RS:DMJ] Go to document J. Rosenberg and C. Schochet, "The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized $K$-functor," Duke Math. J., vol. 55, iss. 2, pp. 431-474, 1987.
    @ARTICLE{RS:DMJ, mrkey = {0894590},
      number = {2},
      issn = {0012-7094},
      author = {Rosenberg, Jonathan and Schochet, Claude},
      mrclass = {46L80 (19K33 46M20 58G12)},
      doi = {10.1215/S0012-7094-87-05524-4},
      journal = {Duke Math. J.},
      zblnumber = {0644.46051},
      volume = {55},
      mrnumber = {0894590},
      fjournal = {Duke Mathematical Journal},
      mrreviewer = {Thierry Fack},
      coden = {DUMJAO},
      title = {The {K}ünneth theorem and the universal coefficient theorem for {K}asparov's generalized {$K$}-functor},
      year = {1987},
      pages = {431--474},
      }
  • [S:Preprint2] Y. Sato, Trace spaces of simple nuclear $\mathrm C^*$-algebras with finite-dimensional extreme boundary, 2012.
    @MISC{S:Preprint2,
      author = {Sato, Yasuhiko},
      arxiv = {1209.3000v1},
      title = {Trace spaces of simple nuclear {$\mathrm C^*$}-algebras with finite-dimensional extreme boundary},
      year = {2012},
      }
  • [SWW:Invent] Go to document Y. Sato, S. White, and W. Winter, "Nuclear dimension and $\mathcal{Z}$-stability," Invent. Math., vol. 202, iss. 2, pp. 893-921, 2015.
    @ARTICLE{SWW:Invent, mrkey = {3418247},
      number = {2},
      issn = {0020-9910},
      author = {Sato, Yasuhiko and White, Stuart and Winter, Wilhelm},
      mrclass = {46L35 (45L05)},
      doi = {10.1007/s00222-015-0580-1},
      journal = {Invent. Math.},
      zblnumber = {06514601},
      volume = {202},
      mrnumber = {3418247},
      fjournal = {Inventiones Mathematicae},
      mrreviewer = {Aaron Tikuisis},
      title = {Nuclear dimension and {$\mathcal{Z}$}-stability},
      year = {2015},
      pages = {893--921},
      }
  • [Sch:PJM] Go to document C. Schochet, "Topological methods for $C^{\ast} $-algebras. IV. Mod $p$ homology," Pacific J. Math., vol. 114, iss. 2, pp. 447-468, 1984.
    @ARTICLE{Sch:PJM, mrkey = {0757511},
      number = {2},
      issn = {0030-8730},
      author = {Schochet, Claude},
      mrclass = {46L80 (19K33 46M20 55N99)},
      journal = {Pacific J. Math.},
      zblnumber = {0491.46062},
      volume = {114},
      mrnumber = {0757511},
      fjournal = {Pacific Journal of Mathematics},
      mrreviewer = {Vern Paulsen},
      coden = {PJMAAI},
      title = {Topological methods for {$C\sp{\ast} $}-algebras. {IV}. {M}od {$p$} homology},
      year = {1984},
      pages = {447--468},
      doi = {10.2140/pjm.1984.114.447},
      }
  • [S:KThy] Go to document G. Skandalis, "Une notion de nucléarité en $K$-théorie (d’après J. Cuntz)," $K$-Theory, vol. 1, iss. 6, pp. 549-573, 1988.
    @ARTICLE{S:KThy, mrkey = {0953916},
      number = {6},
      issn = {0920-3036},
      author = {Skandalis, Georges},
      mrclass = {46L80 (19K33 46L05 46M20)},
      doi = {10.1007/BF00533786},
      journal = {$K$-Theory},
      zblnumber = {0653.46065},
      volume = {1},
      mrnumber = {0953916},
      fjournal = {$K$-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of $K$-Theory in the Mathematical Sciences},
      mrreviewer = {J. A. Mingo},
      coden = {KTHEEO},
      title = {Une notion de nucléarité en {$K$}-théorie (d'après {J}. {C}untz)},
      year = {1988},
      pages = {549--573},
      }
  • [Thayer] Go to document T. F. Javier, "Quasi-diagonal $C\sp*$-algebras," J. Functional Analysis, vol. 25, iss. 1, pp. 50-57, 1977.
    @ARTICLE{Thayer, mrkey = {0448098},
      number = {1},
      author = {Thayer F. Javier},
      mrclass = {46L05},
      journal = {J. Functional Analysis},
      zblnumber = {0352.46037},
      volume = {25},
      mrnumber = {0448098},
      mrreviewer = {Ole A. Nielsen},
      title = {Quasi-diagonal {$C\sp*$}-algebras},
      year = {1977},
      pages = {50--57},
      doi = {10.1016/0022-1236(77)90031-3},
      }
  • [T:MA] Go to document A. Tikuisis, "Nuclear dimension, $\mathcal {Z}$-stability, and algebraic simplicity for stably projectionless $C^\ast$-algebras," Math. Ann., vol. 358, iss. 3-4, pp. 729-778, 2014.
    @ARTICLE{T:MA, mrkey = {3175139},
      number = {3-4},
      issn = {0025-5831},
      author = {Tikuisis, Aaron},
      mrclass = {46L35 (46L05 46L80)},
      doi = {10.1007/s00208-013-0951-0},
      journal = {Math. Ann.},
      zblnumber = {1319.46043},
      volume = {358},
      mrnumber = {3175139},
      fjournal = {Mathematische Annalen},
      mrreviewer = {Joav Orovitz},
      title = {Nuclear dimension, {$\mathcal {Z}$}-stability, and algebraic simplicity for stably projectionless {$C\sp \ast$}-algebras},
      year = {2014},
      pages = {729--778},
      }
  • [TWW:IMRN] Go to document A. S. Toms, S. White, and W. Winter, "$\mathcal{Z}$-stability and finite-dimensional tracial boundaries," Int. Math. Res. Not., vol. 2015, iss. 10, pp. 2702-2727, 2015.
    @ARTICLE{TWW:IMRN, mrkey = {3352253},
      number = {10},
      issn = {1073-7928},
      author = {Toms, Andrew S. and White, Stuart and Winter, Wilhelm},
      mrclass = {46L35 (46L40)},
      doi = {10.1093/imrn/rnu001},
      journal = {Int. Math. Res. Not.},
      zblnumber = {1335.46054},
      volume = {2015},
      mrnumber = {3352253},
      fjournal = {International Mathematics Research Notices. IMRN},
      mrreviewer = {C. J. K. Batty},
      title = {{$\mathcal{Z}$}-stability and finite-dimensional tracial boundaries},
      year = {2015},
      pages = {2702--2727},
      }
  • [TW:TAMS] Go to document A. S. Toms and W. Winter, "Strongly self-absorbing $C^*$-algebras," Trans. Amer. Math. Soc., vol. 359, iss. 8, pp. 3999-4029, 2007.
    @ARTICLE{TW:TAMS, mrkey = {2302521},
      number = {8},
      issn = {0002-9947},
      author = {Toms, Andrew S. and Winter, Wilhelm},
      mrclass = {46L05 (46L35)},
      doi = {10.1090/S0002-9947-07-04173-6},
      journal = {Trans. Amer. Math. Soc.},
      zblnumber = {1120.46046},
      volume = {359},
      mrnumber = {2302521},
      fjournal = {Transactions of the American Mathematical Society},
      mrreviewer = {Hua Xin Lin},
      coden = {TAMTAM},
      title = {Strongly self-absorbing {$C\sp *$}-algebras},
      year = {2007},
      pages = {3999--4029},
      }
  • [TW:JFA] Go to document A. S. Toms and W. Winter, "The Elliott conjecture for Villadsen algebras of the first type," J. Funct. Anal., vol. 256, iss. 5, pp. 1311-1340, 2009.
    @ARTICLE{TW:JFA, mrkey = {2490221},
      number = {5},
      issn = {0022-1236},
      author = {Toms, Andrew S. and Winter, Wilhelm},
      mrclass = {46L05 (46L35)},
      doi = {10.1016/j.jfa.2008.12.015},
      journal = {J. Funct. Anal.},
      zblnumber = {1184.46061},
      volume = {256},
      mrnumber = {2490221},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {Xiao Chun Fang},
      coden = {JFUAAW},
      title = {The {E}lliott conjecture for {V}illadsen algebras of the first type},
      year = {2009},
      pages = {1311--1340},
      }
  • [Tu:KT] Go to document J. Tu, "La conjecture de Baum-Connes pour les feuilletages moyennables," $K$-Theory, vol. 17, iss. 3, pp. 215-264, 1999.
    @ARTICLE{Tu:KT, mrkey = {1703305},
      number = {3},
      issn = {0920-3036},
      author = {Tu, Jean-Louis},
      mrclass = {19K35 (46L85 58J22)},
      doi = {10.1023/A:1007744304422},
      journal = {$K$-Theory},
      zblnumber = {0939.19001},
      volume = {17},
      mrnumber = {1703305},
      fjournal = {$K$-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of $K$-Theory in the Mathematical Sciences},
      mrreviewer = {Hiroshi Takai},
      coden = {KTHEEO},
      title = {La conjecture de {B}aum-{C}onnes pour les feuilletages moyennables},
      year = {1999},
      pages = {215--264},
      }
  • [Voi:Duke] Go to document D. Voiculescu, "A note on quasi-diagonal $C^*$-algebras and homotopy," Duke Math. J., vol. 62, iss. 2, pp. 267-271, 1991.
    @ARTICLE{Voi:Duke, mrkey = {1104525},
      number = {2},
      issn = {0012-7094},
      author = {Voiculescu, Dan},
      mrclass = {46L05 (46L85)},
      doi = {10.1215/S0012-7094-91-06211-3},
      journal = {Duke Math. J.},
      zblnumber = {0833.46055},
      volume = {62},
      mrnumber = {1104525},
      fjournal = {Duke Mathematical Journal},
      mrreviewer = {Hiroshi Takai},
      coden = {DUMJAO},
      title = {A note on quasi-diagonal {$C\sp *$}-algebras and homotopy},
      year = {1991},
      pages = {267--271},
      }
  • [V:IEOT] Go to document D. Voiculescu, "Around quasidiagonal operators," Integral Equations Operator Theory, vol. 17, iss. 1, pp. 137-149, 1993.
    @ARTICLE{V:IEOT, mrkey = {1220578},
      number = {1},
      issn = {0378-620X},
      author = {Voiculescu, Dan},
      mrclass = {47A66 (46L80 46L85)},
      doi = {10.1007/BF01322551},
      journal = {Integral Equations Operator Theory},
      zblnumber = {1048.47501},
      volume = {17},
      mrnumber = {1220578},
      fjournal = {Integral Equations and Operator Theory},
      mrreviewer = {William Paschke},
      title = {Around quasidiagonal operators},
      year = {1993},
      pages = {137--149},
      }
  • [Win:AJM] Go to document W. Winter, "Classifying crossed product $ C^*$-algebras," Amer. J. Math., vol. 138, iss. 3, pp. 793-820, 2016.
    @ARTICLE{Win:AJM, mrkey = {3506386},
      number = {3},
      issn = {0002-9327},
      author = {Winter, Wilhelm},
      mrclass = {47C10 (37A55)},
      doi = {10.1353/ajm.2016.0029},
      journal = {Amer. J. Math.},
      zblnumber = {06603608},
      volume = {138},
      mrnumber = {3506386},
      fjournal = {American Journal of Mathematics},
      coden = {AJMAAN},
      title = {Classifying crossed product {$\rm C\sp *$}-algebras},
      year = {2016},
      pages = {793--820},
      }
  • [W:PLMS] Go to document W. Winter, "Decomposition rank of subhomogeneous $C^*$-algebras," Proc. London Math. Soc., vol. 89, iss. 2, pp. 427-456, 2004.
    @ARTICLE{W:PLMS, mrkey = {2078703},
      number = {2},
      issn = {0024-6115},
      author = {Winter, Wilhelm},
      mrclass = {46L05 (46L85)},
      doi = {10.1112/S0024611504014716},
      journal = {Proc. London Math. Soc.},
      zblnumber = {1081.46049},
      volume = {89},
      mrnumber = {2078703},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      mrreviewer = {William Paschke},
      coden = {PLMTAL},
      title = {Decomposition rank of subhomogeneous {$C\sp *$}-algebras},
      year = {2004},
      pages = {427--456},
      }
  • [W:JNCG] Go to document W. Winter, "Strongly self-absorbing $C^*$-algebras are $\mathcal Z$-stable," J. Noncommut. Geom., vol. 5, iss. 2, pp. 253-264, 2011.
    @ARTICLE{W:JNCG, mrkey = {2784504},
      number = {2},
      issn = {1661-6952},
      author = {Winter, Wilhelm},
      mrclass = {46L35 (46L05 46L80)},
      doi = {10.4171/JNCG/74},
      journal = {J. Noncommut. Geom.},
      zblnumber = {1227.46041},
      volume = {5},
      mrnumber = {2784504},
      fjournal = {Journal of Noncommutative Geometry},
      mrreviewer = {Camillo Trapani},
      title = {Strongly self-absorbing {$C\sp *$}-algebras are {$\mathcal Z$}-stable},
      year = {2011},
      pages = {253--264},
      }
  • [W:Invent2] Go to document W. Winter, "Nuclear dimension and $\mathcal{Z}$-stability of pure $ C^*$-algebras," Invent. Math., vol. 187, iss. 2, pp. 259-342, 2012.
    @ARTICLE{W:Invent2, mrkey = {2885621},
      number = {2},
      issn = {0020-9910},
      author = {Winter, Wilhelm},
      mrclass = {46L85 (46L35)},
      doi = {10.1007/s00222-011-0334-7},
      journal = {Invent. Math.},
      zblnumber = {1280.46041},
      volume = {187},
      mrnumber = {2885621},
      fjournal = {Inventiones Mathematicae},
      coden = {INVMBH},
      title = {Nuclear dimension and {$\mathcal{Z}$}-stability of pure {$\rm C\sp *$}-algebras},
      year = {2012},
      pages = {259--342},
      }
  • [Win:localizing] W. Winter, "Localizing the Elliott conjecture at strongly self-absorbing $C^*$-algebras," J. Reine Angew. Math., vol. 692, pp. 193-231, 2014.
    @ARTICLE{Win:localizing, mrkey = {3274552},
      issn = {0075-4102},
      author = {Winter, Wilhelm},
      mrclass = {46L35 (46L80)},
      journal = {J. Reine Angew. Math.},
      zblnumber = {1327.46058},
      volume = {692},
      mrnumber = {3274552},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      mrreviewer = {Aaron Tikuisis},
      title = {Localizing the {E}lliott conjecture at strongly self-absorbing {$C\sp *$}-algebras},
      year = {2014},
      pages = {193--231},
      }
  • [WZ:MJM] Go to document W. Winter and J. Zacharias, "Completely positive maps of order zero," Münster J. Math., vol. 2, pp. 311-324, 2009.
    @ARTICLE{WZ:MJM, mrkey = {2545617},
      issn = {1867-5778},
      author = {Winter, Wilhelm and Zacharias, Joachim},
      mrclass = {46L05},
      journal = {Münster J. Math.},
      zblnumber = {1190.46042},
      volume = {2},
      mrnumber = {2545617},
      fjournal = {Münster Journal of Mathematics},
      mrreviewer = {William Paschke},
      title = {Completely positive maps of order zero},
      year = {2009},
      pages = {311--324},
      url = {http://wwwmath.uni-muenster.de/mjm/vol_2/mjm_vol_2_14.pdf},
      }
  • [WinZac:dimnuc] Go to document W. Winter and J. Zacharias, "The nuclear dimension of $C^\ast$-algebras," Adv. Math., vol. 224, iss. 2, pp. 461-498, 2010.
    @ARTICLE{WinZac:dimnuc, mrkey = {2609012},
      number = {2},
      issn = {0001-8708},
      author = {Winter, Wilhelm and Zacharias, Joachim},
      mrclass = {46L05},
      doi = {10.1016/j.aim.2009.12.005},
      journal = {Adv. Math.},
      zblnumber = {1201.46056},
      volume = {224},
      mrnumber = {2609012},
      fjournal = {Advances in Mathematics},
      mrreviewer = {William Paschke},
      coden = {ADMTA4},
      title = {The nuclear dimension of {$C\sp \ast$}-algebras},
      year = {2010},
      pages = {461--498},
      }
  • [Z:JFA] Go to document W. Zhang, "Tracial state space with non-compact extreme boundary," J. Funct. Anal., vol. 267, iss. 8, pp. 2884-2906, 2014.
    @ARTICLE{Z:JFA, mrkey = {3255477},
      number = {8},
      issn = {0022-1236},
      author = {Zhang, Wei},
      mrclass = {46L30 (46L05)},
      doi = {10.1016/j.jfa.2014.08.013},
      journal = {J. Funct. Anal.},
      zblnumber = {1311.46057},
      volume = {267},
      mrnumber = {3255477},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {Igor Klep},
      title = {Tracial state space with non-compact extreme boundary},
      year = {2014},
      pages = {2884--2906},
      }

Authors

Aaron Tikuisis

University of Aberdeen, Scotland

Stuart White

University of Glasgow, Glasgow, Scotland and Mathematisches Institut der WWU Münster, Münster, Germany

Wilhelm Winter

Mathematisches Institut der WWU Münster, Münster, Germany