The Global Glimm Property

Thiel, Hannes; Vilalta, Eduard

doi:10.1090/tran/8880

The Global Glimm Property
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by Hannes Thiel and Eduard Vilalta PDF

Trans. Amer. Math. Soc. 376 (2023), 4713-4744

Abstract:

It is known that a $C^*$-algebra with the Global Glimm Property is nowhere scattered (it has no elementary ideal-quotients), and the Global Glimm Problem asks if the converse holds. We provide a new approach to this long-standing problem by showing that a $C^*$-algebra has the Global Glimm Property if and only if it is nowhere scattered and its Cuntz semigroup is ideal-filtered (the Cuntz classes generating a given ideal are downward directed) and has property (V) (a weak form of being sup-semilattice ordered).

We show that ideal-filteredness and property (V) are automatic for $C^*$-algebras that have stable rank one or real rank zero, thereby recovering the solutions to the Global Glimm Problem in these cases. We also use our approach to solve the Global Glimm Problem for new classes of $C^*$-algebras.

References

P. Ara and E. Pardo, Refinement monoids with weak comparability and applications to regular rings and $C^*$-algebras, Proc. Amer. Math. Soc. 124 (1996), no. 3, 715–720. MR 1301484, DOI 10.1090/S0002-9939-96-03059-6
Ramon Antoine, Francesc Perera, Leonel Robert, and Hannes Thiel, Edwards’ condition for quasitraces on $C^*$-algebras, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 2, 525–547. MR 4241290, DOI 10.1017/prm.2020.26
Ramon Antoine, Francesc Perera, Leonel Robert, and Hannes Thiel, C*-algebras of stable rank one and their Cuntz semigroups, Duke Math. J. 171 (2022), no. 1, 33–99. MR 4364731, DOI 10.1215/00127094-2021-0009
Ramon Antoine, Francesc Perera, and Hannes Thiel, Tensor products and regularity properties of Cuntz semigroups, Mem. Amer. Math. Soc. 251 (2018), no. 1199, viii+191. MR 3756921, DOI 10.1090/memo/1199
Ramon Antoine, Francesc Perera, and Hannes Thiel, Abstract bivariant Cuntz semigroups, Int. Math. Res. Not. IMRN 17 (2020), 5342–5386. MR 4146340, DOI 10.1093/imrn/rny143
Ramon Antoine, Francesc Perera, and Hannes Thiel, Abstract bivariant Cuntz semigroups II, Forum Math. 32 (2020), no. 1, 45–62. MR 4048453, DOI 10.1515/forum-2018-0285
Ramon Antoine, Francesc Perera, and Hannes Thiel, Cuntz semigroups of ultraproduct $C^*$-algebras, J. Lond. Math. Soc. (2) 102 (2020), no. 3, 994–1029. MR 4186119, DOI 10.1112/jlms.12343
Etienne Blanchard and Eberhard Kirchberg, Global Glimm halving for $C^*$-bundles, J. Operator Theory 52 (2004), no. 2, 385–420. MR 2120237
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
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, DOI 10.1007/3-540-28517-2
Lawrence G. Brown and Gert K. Pedersen, Ideal structure and $C^\ast$-algebras of low rank, Math. Scand. 100 (2007), no. 1, 5–33. MR 2327718, DOI 10.7146/math.scand.a-15014
Lawrence G. Brown and Gert K. Pedersen, Limits and $C^\ast$-algebras of low rank or dimension, J. Operator Theory 61 (2009), no. 2, 381–417. MR 2501012, DOI 10.1140/epjc/s10052-009-1024-0
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
Abhinav Chand and Leonel Robert, Simplicity, bounded normal generation, and automatic continuity of groups of unitaries, Preprint, arXiv:2108.09796 [math.OA], 2021.
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
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
James Glimm, Type I $C^{\ast }$-algebras, Ann. of Math. (2) 73 (1961), 572–612. MR 124756, DOI 10.2307/1970319
Eberhard Kirchberg and Mikael Rørdam, Non-simple purely infinite $C^\ast$-algebras, Amer. J. Math. 122 (2000), no. 3, 637–666. MR 1759891, DOI 10.1353/ajm.2000.0021
Eberhard Kirchberg and Mikael Rørdam, Infinite non-simple $C^*$-algebras: absorbing the Cuntz algebras $\scr O_\infty$, Adv. Math. 167 (2002), no. 2, 195–264. MR 1906257, DOI 10.1006/aima.2001.2041
Céline Moreira Dos Santos, A refinement monoid whose maximal antisymmetric quotient is not a refinement monoid, Semigroup Forum 65 (2002), no. 2, 249–263. MR 1911728, DOI 10.1007/s002330010104
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
Cornel Pasnicu and N. Christopher Phillips, The weak ideal property and topological dimension zero, Canad. J. Math. 69 (2017), no. 6, 1385–1421. MR 3715016, DOI 10.4153/CJM-2017-012-4
Cornel Pasnicu and Mikael Rørdam, Purely infinite $C^*$-algebras of real rank zero, J. Reine Angew. Math. 613 (2007), 51–73. MR 2377129, DOI 10.1515/CRELLE.2007.091
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, Advances in the theory of unitary rank and regular approximation, Ann. of Math. (2) 128 (1988), no. 1, 153–172. MR 951510, DOI 10.2307/1971465
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
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
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
Hannes Thiel, The Cuntz semigroup, Lecture notes, 2017, hannesthiel.org/publications.
Hannes Thiel, The generator rank of subhomogeneous $C^*$-algebras, Canad. J. Math. (2020), To appear, DOI: 10.4153/S0008414X22000268, Preprint, arXiv:2006.03624 [math.OA].
Hannes Thiel, Ranks of operators in simple $C^*$-algebras with stable rank one, Comm. Math. Phys. 377 (2020), no. 1, 37–76. MR 4107924, DOI 10.1007/s00220-019-03491-8
Aaron Peter Tikuisis and Andrew Toms, On the structure of Cuntz semigroups in (possibly) nonunital $\rm C^*$-algebras, Canad. Math. Bull. 58 (2015), no. 2, 402–414. MR 3334936, DOI 10.4153/CMB-2014-040-5
Hannes Thiel and Eduard Vilalta, Covering dimension of Cuntz semigroups II, Internat. J. Math. 32 (2021), no. 14, Paper No. 2150100, 27. MR 4369753, DOI 10.1142/S0129167X21501007
Hannes Thiel and Eduard Vilalta, Nowhere scattered $C^*$-algebras, J. Noncommut. Geom. (2021), To appear, Preprint, arXiv:2112.09877 [math.OA].
Hannes Thiel and Eduard Vilalta, Covering dimension of Cuntz semigroups, Adv. Math. 394 (2022), Paper No. 108016, 44. MR 4355725, DOI 10.1016/j.aim.2021.108016
Hannes Thiel and Eduard Vilalta, Soft $C^*$-algebras, 2023, In preparation.
Eduard Vilalta, The Cuntz semigroup of unital commutative AI-algebras, Canad. J. Math. (2021), To appear, DOI: 10.4153/S0008414X22000542, Preprint, arXiv:2104.08165 [math.OA].