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Tensor products and regularity properties of Cuntz semigroups
About this Title
Ramon Antoine, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain, Francesc Perera, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain and Hannes Thiel, Mathematisches Institut, Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 251, Number 1199
ISBNs: 978-1-4704-2797-9 (print); 978-1-4704-4282-8 (online)
DOI: https://doi.org/10.1090/memo/1199
Published electronically: November 7, 2017
Keywords: Cuntz semigroup,
tensor product,
continuous poset,
$C^*$-algebra
MSC: Primary 06B35, 06F05, 15A69, 46L05.; Secondary 06B30, 06F25, 13J25, 16W80, 16Y60, 18B35, 18D20, 19K14, 46L06, 46M15, 54F05.
Table of Contents
Chapters
- 1. Introduction
- 2. Pre-completed Cuntz semigroups
- 3. Completed Cuntz semigroups
- 4. Additional axioms
- 5. Structure of \texorpdfstring{$\mathrm {Cu}$}Cu-semigroups
- 6. Bimorphisms and tensor products
- 7. \texorpdfstring{$\mathrm {Cu}$}Cu-semirings and \texorpdfstring{$\mathrm {Cu}$}Cu-semimodules
- 8. Structure of \texorpdfstring{$\mathrm {Cu}$}Cu-semirings
- 9. Concluding remarks and open problems
- A. Monoidal and enriched categories
- B. Partially ordered monoids, groups and rings
Abstract
The Cuntz semigroup of a $C^*$-algebra is an important invariant in the structure and classification theory of $C^*$-algebras. It captures more information than $K$-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups.
Given a $C^*$-algebra $A$, its (concrete) Cuntz semigroup $\mathrm {Cu}(A)$ is an object in the category $\mathrm {Cu}$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu (2008). To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $\mathrm {Cu}$-semigroups.
We establish the existence of tensor products in the category $\mathrm {Cu}$ and study the basic properties of this construction. We show that $\mathrm {Cu}$ is a symmetric, monoidal category and relate $\mathrm {Cu}(A\otimes B)$ with $\mathrm {Cu}(A)\otimes _{\mathrm {Cu}}\mathrm {Cu}(B)$ for certain classes of $C^*$-algebras.
As a main tool for our approach we introduce the category $\mathrm {W}$ of pre-completed Cuntz semigroups. We show that $\mathrm {Cu}$ is a full, reflective subcategory of $\mathrm {W}$. One can then easily deduce properties of $\mathrm {Cu}$ from respective properties of $\mathrm {W}$, for example the existence of tensor products and inductive limits. The advantage is that constructions in $\mathrm {W}$ are much easier since the objects are purely algebraic.
For every (local) $C^*$-algebra $A$, the classical Cuntz semigroup $W(A)$ together with a natural auxiliary relation is an object of $\mathrm {W}$. This defines a functor from $C^*$-algebras to $\mathrm {W}$ which preserves inductive limits. We deduce that the assignment $A\mapsto \mathrm {Cu}(A)$ defines a functor from $C^*$-algebras to $\mathrm {Cu}$ which preserves inductive limits. This generalizes a result from Coward, Elliott, and Ivanescu (2008).
We also develop a theory of $\mathrm {Cu}$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing $C^*$-algebra has a natural product giving it the structure of a $\mathrm {Cu}$-semiring. For $C^*$-algebras, it is an important regularity property to tensorially absorb a strongly self-absorbing $C^*$-algebra. Accordingly, it is of particular interest to analyse the tensor products of $\mathrm {Cu}$-semigroups with the $\mathrm {Cu}$-semiring of a strongly self-absorbing $C^*$-algebra. This leads us to define ‘solid’ $\mathrm {Cu}$-semirings (adopting the terminology from solid rings), as those $\mathrm {Cu}$-semirings $S$ for which the product induces an isomorphism between $S\otimes _{\mathrm {Cu}} S$ and $S$. This can be considered as an analog of being strongly self-absorbing for $\mathrm {Cu}$-semirings. As it turns out, if a strongly self-absorbing $C^*$-algebra satisfies the UCT, then its $\mathrm {Cu}$-semiring is solid. We prove a classification theorem for solid $\mathrm {Cu}$-semirings. This raises the question of whether the Cuntz semiring of every strongly self-absorbing $C^*$-algebra is solid.
If $R$ is a solid $\mathrm {Cu}$-semiring, then a $\mathrm {Cu}$-semigroup $S$ is a semimodule over $R$ if and only if $R\otimes _{\mathrm {Cu}}S$ is isomorphic to $S$. Thus, analogous to the case for $C^*$-algebras, we can think of semimodules over $R$ as $\mathrm {Cu}$-semigroups that tensorially absorb $R$. We give explicit characterizations when a $\mathrm {Cu}$-semigroup is such a semimodule for the cases that $R$ is the $\mathrm {Cu}$-semiring of a strongly self-absorbing $C^*$-algebra satisfying the UCT. For instance, we show that a $\mathrm {Cu}$-semigroup $S$ tensorially absorbs the $\mathrm {Cu}$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.
- Ramon Antoine, Joan Bosa, and Francesc Perera, Completions of monoids with applications to the Cuntz semigroup, Internat. J. Math. 22 (2011), no. 6, 837–861. MR 2812090, DOI 10.1142/S0129167X11007057
- Ramon Antoine, Joan Bosa, and Francesc Perera, The Cuntz semigroup of continuous fields, Indiana Univ. Math. J. 62 (2013), no. 4, 1105–1131. MR 3179685, DOI 10.1512/iumj.2013.62.5071
- Ramon Antoine, Joan Bosa, Francesc Perera, and Henning Petzka, Geometric structure of dimension functions of certain continuous fields, J. Funct. Anal. 266 (2014), no. 4, 2403–2423. MR 3150165, DOI 10.1016/j.jfa.2013.09.013
- Ramon Antoine, Marius Dadarlat, Francesc Perera, and Luis Santiago, Recovering the Elliott invariant from the Cuntz semigroup, Trans. Amer. Math. Soc. 366 (2014), no. 6, 2907–2922. MR 3180735, DOI 10.1090/S0002-9947-2014-05833-9
- P. Ara, K. R. Goodearl, K. C. O’Meara, and E. Pardo, Separative cancellation for projective modules over exchange rings, Israel J. Math. 105 (1998), 105–137. MR 1639739, DOI 10.1007/BF02780325
- G. Aranda Pino, K. R. Goodearl, F. Perera, and M. Siles Molina, Non-simple purely infinite rings, Amer. J. Math. 132 (2010), no. 3, 563–610. MR 2666902, DOI 10.1353/ajm.0.0119
- Ramon Antoine, Francesc Perera, and Luis Santiago, Pullbacks, $C(X)$-algebras, and their Cuntz semigroup, J. Funct. Anal. 260 (2011), no. 10, 2844–2880. MR 2774057, DOI 10.1016/j.jfa.2011.02.016
- Pere Ara, Francesc Perera, and Andrew S. Toms, $K$-theory for operator algebras. Classification of $C^*$-algebras, Aspects of operator algebras and applications, Contemp. Math., vol. 534, Amer. Math. Soc., Providence, RI, 2011, pp. 1–71. MR 2767222, DOI 10.1090/conm/534/10521
- J. Bosa, N. P. Brown, Y. Sato, A. Tikuisis, S. White, W. Winter, Covering dimension of \ca{s} and $2$-coloured classification, to appear in Mem. Amer. Math. Soc. (arXiv:1506.03974 [math.OA]).
- Nathanial P. Brown and Alin Ciuperca, Isomorphism of Hilbert modules over stably finite $C^*$-algebras, J. Funct. Anal. 257 (2009), no. 1, 332–339. MR 2523343, DOI 10.1016/j.jfa.2008.12.004
- Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR 463928
- Bruce Blackadar and David Handelman, Dimension functions and traces on $C^{\ast }$-algebras, J. Functional Analysis 45 (1982), no. 3, 297–340. MR 650185, DOI 10.1016/0022-1236(82)90009-X
- A. K. Bousfield and D. M. Kan, The core of a ring, J. Pure Appl. Algebra 2 (1972), 73–81. MR 308107, DOI 10.1016/0022-4049(72)90023-0
- Etienne Blanchard and Eberhard Kirchberg, Non-simple purely infinite $C^*$-algebras: the Hausdorff case, J. Funct. Anal. 207 (2004), no. 2, 461–513. MR 2032998, DOI 10.1016/j.jfa.2003.06.008
- Bruce Blackadar, Comparison theory for simple $C^*$-algebras, Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., vol. 135, Cambridge Univ. Press, Cambridge, 1988, pp. 21–54. MR 996438
- Bruce Blackadar, Rational $C^*$-algebras and nonstable $K$-theory, Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), 1990, pp. 285–316. MR 1065831, DOI 10.1216/rmjm/1181073108
- Bruce Blackadar, $K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031
- B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. Theory of $C^*$-algebras and von Neumann algebras; Operator Algebras and Non-commutative Geometry, III. MR 2188261
- Bernhard Banaschewski and Evelyn Nelson, Tensor products and bimorphisms, Canad. Math. Bull. 19 (1976), no. 4, 385–402. MR 442059, DOI 10.4153/CMB-1976-060-2
- Francis Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. Basic category theory. MR 1291599
- Nathanial P. Brown, Francesc Perera, and Andrew S. Toms, The Cuntz semigroup, the Elliott conjecture, and dimension functions on $C^*$-algebras, J. Reine Angew. Math. 621 (2008), 191–211. MR 2431254, DOI 10.1515/CRELLE.2008.062
- Bruce Blackadar and Mikael Rørdam, Extending states on preordered semigroups and the existence of quasitraces on $C^*$-algebras, J. Algebra 152 (1992), no. 1, 240–247. MR 1190414, DOI 10.1016/0021-8693(92)90098-7
- R. A. Bowshell and P. Schultz, Unital rings whose additive endomorphisms commute, Math. Ann. 228 (1977), no. 3, 197–214. MR 498691, DOI 10.1007/BF01420290
- Nathanial P. Brown and Andrew S. Toms, Three applications of the Cuntz semigroup, Int. Math. Res. Not. IMRN 19 (2007), Art. ID rnm068, 14. MR 2359541, DOI 10.1093/imrn/rnm068
- J. Bosa, G. Tornetta, and J. Zacharias, A bivariant theory for the Cuntz semigroup, preprint, (arXiv: 1602.02043 [math.OA]), 2016.
- Kristofer T. Coward, George A. Elliott, and Cristian Ivanescu, The Cuntz semigroup as an invariant for $C^*$-algebras, J. Reine Angew. Math. 623 (2008), 161–193. MR 2458043, DOI 10.1515/CRELLE.2008.075
- Alin Ciuperca, Some properties of the Cuntz semigroup and an isomorphism theorem for a certain class of non-simple C*-algebras, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–University of Toronto (Canada). MR 2712681
- Joachim Cuntz, Ralf Meyer, and Jonathan M. Rosenberg, Topological and bivariant $K$-theory, Oberwolfach Seminars, vol. 36, Birkhäuser Verlag, Basel, 2007. MR 2340673
- Alin Ciuperca, Leonel Robert, and Luis Santiago, The Cuntz semigroup of ideals and quotients and a generalized Kasparov stabilization theorem, J. Operator Theory 64 (2010), no. 1, 155–169. MR 2669433
- Joachim Cuntz, Dimension functions on simple $C^*$-algebras, Math. Ann. 233 (1978), no. 2, 145–153. MR 467332, DOI 10.1007/BF01421922
- Marius Dadarlat, Ilan Hirshberg, Andrew S. Toms, and Wilhelm Winter, The Jiang-Su algebra does not always embed, Math. Res. Lett. 16 (2009), no. 1, 23–26. MR 2480557, DOI 10.4310/MRL.2009.v16.n1.a3
- Eduardo J. Dubuc and Horacio Porta, Convenient categories of topological algebras, and their duality theory, J. Pure Appl. Algebra 1 (1971), no. 3, 281–316. MR 301079, DOI 10.1016/0022-4049(71)90023-5
- Marius Dadarlat and Mikael Rørdam, Strongly self-absorbing $C^*$-algebras which contain a nontrivial projection, Münster J. Math. 2 (2009), 35–44. MR 2545606
- Warren Dicks and W. Stephenson, Epimorphs and dominions of Dedekind domains, J. London Math. Soc. (2) 29 (1984), no. 2, 224–228. MR 744090, DOI 10.1112/jlms/s2-29.2.224
- Marius Dadarlat and Andrew S. Toms, Ranks of operators in simple $C^*$-algebras, J. Funct. Anal. 259 (2010), no. 5, 1209–1229. MR 2652186, DOI 10.1016/j.jfa.2010.03.022
- G. A. Elliott, G. Gong, H. Lin, and Z. Niu, On the classification of simple amenable \ca{s} with finite decomposition rank, II, preprint (arXiv:1507.03437 [math.OA]), 2015
- Edward G. Effros, David E. Handelman, and Chao Liang Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980), no. 2, 385–407. MR 564479, DOI 10.2307/2374244
- M. Engbers, Decomposition of simple Cuntz semigroups, Ph.D Thesis, 2014.
- George A. Elliott and Mikael Rørdam, Perturbation of Hausdorff moment sequences, and an application to the theory of $C^*$-algebras of real rank zero, Operator Algebras: The Abel Symposium 2004, Abel Symp., vol. 1, Springer, Berlin, 2006, pp. 97–115. MR 2265045, DOI 10.1007/978-3-540-34197-0_{5}
- George A. Elliott, Leonel Robert, and Luis Santiago, The cone of lower semicontinuous traces on a $C^*$-algebra, Amer. J. Math. 133 (2011), no. 4, 969–1005. MR 2823868, DOI 10.1353/ajm.2011.0027
- George A. Elliott and Andrew S. Toms, Regularity properties in the classification program for separable amenable $C^*$-algebras, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 2, 229–245. MR 2383304, DOI 10.1090/S0273-0979-08-01199-3
- Ronald Fulp, Tensor and torsion products of semigroups, Pacific J. Math. 32 (1970), 685–696. MR 272928
- E. Gardella and L. Santiago, The equivariant Cuntz semigroup, to appear in Proc. Lon. Math. Soc. (arXiv:1506.07572 [math.OA]), 2015.
- K. R. Goodearl and D. Handelman, Rank functions and $K_{O}$ of regular rings, J. Pure Appl. Algebra 7 (1976), no. 2, 195–216. MR 389965, DOI 10.1016/0022-4049(76)90032-3
- K. R. Goodearl and D. E. Handelman, Stenosis in dimension groups and AF $C^{\ast }$-algebras, J. Reine Angew. Math. 332 (1982), 1–98. MR 656856, DOI 10.1515/crll.1982.332.1
- G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous lattices and domains, Encyclopedia of Mathematics and its Applications, vol. 93, Cambridge University Press, Cambridge, 2003. MR 1975381
- Guihua Gong, Xinhui Jiang, and Hongbing Su, Obstructions to $\scr Z$-stability for unital simple $C^*$-algebras, Canad. Math. Bull. 43 (2000), no. 4, 418–426. MR 1793944, DOI 10.4153/CMB-2000-050-1
- G. Gong, H. Lin, and Z. Niu, Classification of finite simple amenable ${\mathcal Z}$-stable \ca{s}, preprint (arXiv:1501.00135 [math.OA]), 2015.
- Jonathan S. Golan, Semirings and their applications, Kluwer Academic Publishers, Dordrecht, 1999. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science [Longman Sci. Tech., Harlow, 1992; MR1163371 (93b:16085)]. MR 1746739
- K. R. Goodearl, Partially ordered abelian groups with interpolation, Mathematical Surveys and Monographs, vol. 20, American Mathematical Society, Providence, RI, 1986. MR 845783
- K. R. Goodearl, $K_0$ of multiplier algebras of $C^\ast$-algebras with real rank zero, $K$-Theory 10 (1996), no. 5, 419–489. MR 1404412, DOI 10.1007/BF00536600
- Pierre-Antoine Grillet, The tensor product of commutative semigroups, Trans. Amer. Math. Soc. 138 (1969), 281–293. MR 237688, DOI 10.1090/S0002-9947-1969-0237688-1
- Javier J. Gutiérrez, On solid and rigid monoids in monoidal categories, Appl. Categ. Structures 23 (2015), no. 4, 575–589. MR 3367132, DOI 10.1007/s10485-014-9370-y
- Uffe Haagerup, Quasitraces on exact $C^*$-algebras are traces, C. R. Math. Acad. Sci. Soc. R. Can. 36 (2014), no. 2-3, 67–92 (English, with English and French summaries). MR 3241179
- David Handelman, Simple Archimedean dimension groups, Proc. Amer. Math. Soc. 141 (2013), no. 11, 3787–3792. MR 3091768, DOI 10.1090/S0002-9939-2013-11672-2
- Bhishan Jacelon, A simple, monotracial, stably projectionless $C^\ast$-algebra, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 365–383. MR 3046276, DOI 10.1112/jlms/jds049
- Xinhui Jiang and Hongbing Su, On a simple unital projectionless $C^*$-algebra, Amer. J. Math. 121 (1999), no. 2, 359–413. MR 1680321
- G. M. Kelly, Basic concepts of enriched category theory, Repr. Theory Appl. Categ. 10 (2005), vi+137. Reprint of the 1982 original [Cambridge Univ. Press, Cambridge; MR0651714]. MR 2177301
- Klaus Keimel and Jimmie D. Lawson, D-completions and the $d$-topology, Ann. Pure Appl. Logic 159 (2009), no. 3, 292–306. MR 2522623, DOI 10.1016/j.apal.2008.06.019
- 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
- Eberhard Kirchberg and Mikael Rørdam, Non-simple purely infinite $C^\ast$-algebras, Amer. J. Math. 122 (2000), no. 3, 637–666. MR 1759891
- Eberhard Kirchberg and Mikael Rørdam, Central sequence $C^*$-algebras and tensorial absorption of the Jiang-Su algebra, J. Reine Angew. Math. 695 (2014), 175–214. MR 3276157, DOI 10.1515/crelle-2012-0118
- Jimmie Lawson, The round ideal completion via sobrification, Proceedings of the 12th Summer Conference on General Topology and its Applications (North Bay, ON, 1997), 1997, pp. 261–274. MR 1718887
- Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798
- R. Meyer, Analytic cyclic cohomology., Münster: Univ. Münster, Fachbereich Mathematik und Informatik, 1999 (English).
- Hiroki Matui and Yasuhiko Sato, Strict comparison and $\scr {Z}$-absorption of nuclear $C^*$-algebras, Acta Math. 209 (2012), no. 1, 179–196. MR 2979512, DOI 10.1007/s11511-012-0084-4
- Hiroki Matui and Yasuhiko Sato, Decomposition rank of UHF-absorbing $\mathrm {C}^*$-algebras, Duke Math. J. 163 (2014), no. 14, 2687–2708. MR 3273581, DOI 10.1215/00127094-2826908
- Eduard Ortega, Francesc Perera, and Mikael Rørdam, The corona factorization property and refinement monoids, Trans. Amer. Math. Soc. 363 (2011), no. 9, 4505–4525. MR 2806681, DOI 10.1090/S0002-9947-2011-05480-2
- Eduard Ortega, Francesc Perera, and Mikael Rørdam, The corona factorization property, stability, and the Cuntz semigroup of a $C^\ast$-algebra, Int. Math. Res. Not. IMRN 1 (2012), 34–66. MR 2874927, DOI 10.1093/imrn/rnr013
- Eduard Ortega, Mikael Rørdam, and Hannes Thiel, The Cuntz semigroup and comparison of open projections, J. Funct. Anal. 260 (2011), no. 12, 3474–3493. MR 2781968, DOI 10.1016/j.jfa.2011.02.017
- Theodore W. Palmer, Banach algebras and the general theory of $*$-algebras. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 79, Cambridge University Press, Cambridge, 2001. $*$-algebras. MR 1819503
- Francesc Perera, The structure of positive elements for $C^*$-algebras with real rank zero, Internat. J. Math. 8 (1997), no. 3, 383–405. MR 1454480, DOI 10.1142/S0129167X97000196
- Francesc Perera, Ideal structure of multiplier algebras of simple $C^*$-algebras with real rank zero, Canad. J. Math. 53 (2001), no. 3, 592–630. MR 1827822, DOI 10.4153/CJM-2001-025-2
- Francesc Perera and Mikael Rørdam, AF-embeddings into $C^*$-algebras of real rank zero, J. Funct. Anal. 217 (2004), no. 1, 142–170. MR 2097610, DOI 10.1016/j.jfa.2004.05.001
- Francesc Perera and Andrew S. Toms, Recasting the Elliott conjecture, Math. Ann. 338 (2007), no. 3, 669–702. MR 2317934, DOI 10.1007/s00208-007-0093-3
- Leonel Robert, Classification of inductive limits of 1-dimensional NCCW complexes, Adv. Math. 231 (2012), no. 5, 2802–2836. MR 2970466, DOI 10.1016/j.aim.2012.07.010
- Leonel Robert, The cone of functionals on the Cuntz semigroup, Math. Scand. 113 (2013), no. 2, 161–186. MR 3145179, DOI 10.7146/math.scand.a-15568
- Leonel Robert, The Cuntz semigroup of some spaces of dimension at most two, C. R. Math. Acad. Sci. Soc. R. Can. 35 (2013), no. 1, 22–32 (English, with English and French summaries). MR 3098039
- Mikael Rørdam, On the structure of simple $C^*$-algebras tensored with a UHF-algebra. II, J. Funct. Anal. 107 (1992), no. 2, 255–269. MR 1172023, DOI 10.1016/0022-1236(92)90106-S
- 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}
- Mikael Rørdam, A simple $C^*$-algebra with a finite and an infinite projection, Acta Math. 191 (2003), no. 1, 109–142. MR 2020420, DOI 10.1007/BF02392697
- Mikael Rørdam, The stable and the real rank of $\scr Z$-absorbing $C^*$-algebras, Internat. J. Math. 15 (2004), no. 10, 1065–1084. MR 2106263, DOI 10.1142/S0129167X04002661
- Leonel Robert and Mikael Rørdam, Divisibility properties for $C^*$-algebras, Proc. Lond. Math. Soc. (3) 106 (2013), no. 6, 1330–1370. MR 3072284, DOI 10.1112/plms/pds082
- Leonel Robert and Luis Santiago, Classification of $C^\ast$-homomorphisms from $C_0(0,1]$ to a $C^\ast$-algebra, J. Funct. Anal. 258 (2010), no. 3, 869–892. MR 2558180, DOI 10.1016/j.jfa.2009.06.025
- Leonel Robert and Aaron Tikuisis, Nuclear dimension and $\mathcal {Z}$-stability of non-simple $\rm C^*$-algebras, Trans. Amer. Math. Soc. 369 (2017), no. 7, 4631–4670. MR 3632545, DOI 10.1090/S0002-9947-2016-06842-7
- Mikael Rørdam and Wilhelm Winter, The Jiang-Su algebra revisited, J. Reine Angew. Math. 642 (2010), 129–155. MR 2658184, DOI 10.1515/CRELLE.2010.039
- Y. Sato, Trace spaces of simple nuclear \ca{s} with finite-dimensional extreme boundary, preprint (arXiv:1209.3000 [math.OA]), 2012.
- Yasuhiko Sato, Stuart White, and Wilhelm Winter, Nuclear dimension and $\mathcal {Z}$-stability, Invent. Math. 202 (2015), no. 2, 893–921. MR 3418247, DOI 10.1007/s00222-015-0580-1
- Aaron Tikuisis, The Cuntz semigroup of continuous functions into certain simple $C^\ast$-algebras, Internat. J. Math. 22 (2011), no. 8, 1051–1087. MR 2826555, DOI 10.1142/S0129167X11007136
- Andrew Toms, On the independence of $K$-theory and stable rank for simple $C^*$-algebras, J. Reine Angew. Math. 578 (2005), 185–199. MR 2113894, DOI 10.1515/crll.2005.2005.578.185
- 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
- Andrew S. Toms and Wilhelm Winter, Strongly self-absorbing $C^*$-algebras, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3999–4029. MR 2302521, DOI 10.1090/S0002-9947-07-04173-6
- Andrew S. Toms and Wilhelm Winter, $\scr Z$-stable ASH algebras, Canad. J. Math. 60 (2008), no. 3, 703–720. MR 2414961, DOI 10.4153/CJM-2008-031-6
- Andrew S. Toms and Wilhelm Winter, The Elliott conjecture for Villadsen algebras of the first type, J. Funct. Anal. 256 (2009), no. 5, 1311–1340. MR 2490221, DOI 10.1016/j.jfa.2008.12.015
- Andrew S. Toms, Stuart White, and Wilhelm Winter, $\mathcal {Z}$-stability and finite-dimensional tracial boundaries, Int. Math. Res. Not. IMRN 10 (2015), 2702–2727. MR 3352253, DOI 10.1093/imrn/rnu001
- 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
- Friedrich Wehrung, Injective positively ordered monoids. I, II, J. Pure Appl. Algebra 83 (1992), no. 1, 43–82, 83–100. MR 1190444, DOI 10.1016/0022-4049(92)90104-N
- Friedrich Wehrung, Restricted injectivity, transfer property and decompositions of separative positively ordered monoids, Comm. Algebra 22 (1994), no. 5, 1747–1781. MR 1264740, DOI 10.1080/00927879408824934
- Friedrich Wehrung, Tensor products of structures with interpolation, Pacific J. Math. 176 (1996), no. 1, 267–285. MR 1433994
- Friedrich Wehrung, Embedding simple commutative monoids into simple refinement monoids, Semigroup Forum 56 (1998), no. 1, 104–129. MR 1490558, DOI 10.1007/s00233-002-7008-0
- Wilhelm Winter, Strongly self-absorbing $C^*$-algebras are $\scr Z$-stable, J. Noncommut. Geom. 5 (2011), no. 2, 253–264. MR 2784504, DOI 10.4171/JNCG/74
- Wilhelm Winter, Nuclear dimension and $\scr {Z}$-stability of pure $\rm C^*$-algebras, Invent. Math. 187 (2012), no. 2, 259–342. MR 2885621, DOI 10.1007/s00222-011-0334-7
- Wilhelm Winter and Joachim Zacharias, Completely positive maps of order zero, Münster J. Math. 2 (2009), 311–324. MR 2545617