No Access

Robinson forcing and the quasidiagonality problem

Isaac Goldbring

Department of Mathematics, University of California, Irvine, 340 Rowland Hall (Bldg.# 400), Irvine, CA 92697-3875, USA

E-mail Address: isaac@math.uci.edu

Search for more papers by this author

and

Thomas Sinclair

Mathematics Department, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA

E-mail Address: tsincla@purdue.edu

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0129167X17500082Cited by:3 (Source: Crossref)

Abstract

We introduce weakenings of two of the more prominent open problems in the classification of C*-algebras, namely the quasidiagonality problem and the UCT problem. We show that a positive solution of the conjunction of the two weaker problems implies a positive solution of the original quasidiagonality problem as well as allows us to give a local, finitary criteria for the MF problem, which asks whether every stably finite C*-algebra is MF.

Keywords:

AMSC: 46L35, 03C25, 03C98

References

1. S. Barlak and G. Szabo, Sequentially split ∗-homomorphisms between C $^{*}$ -algebras, preprint (2015), arXiv:1510.04555. Google Scholar
2. I. Ben Yaacov and J. Iovino, Model-theoretic forcing in analysis, Ann. Pure Appl. Logic 158 (2009) 163–174. Crossref, Web of Science, Google Scholar
3. B. Blackadar, K-Theory for Operator Algebras, MSRI Publications, Vol. 5 (Cambridge University Press, Cambridge, 1998). Google Scholar
4. B. Blackadar and D. Handelman, Dimension functions and traces on $C^{*}$ -algebras, J. Funct. Anal. 45 (1982) 297–340. Crossref, Web of Science, Google Scholar
5. B. Blackadar and E. Kirchberg, Generalized inductive limits of finite dimensional $C^{*}$ -algebras, Math. Ann. 307 (1997) 343–380. Crossref, Web of Science, Google Scholar
6. N. P. Brown and N. Ozawa, C∗-Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics, Vol. 88 (American Mathematical Society, Providence, RI, 2008). Crossref, Google Scholar
7. K. Carlson, E. Cheung, I. Farah, A. Gerhardt-Bourke, B. Hart, L. Mezuman, N. Sequeira and A. Sherman, Omitting types and AF algebras, Arch. Math. Logic 53 (2014) 157–169. Crossref, Web of Science, Google Scholar
8. A. Connes, Classification of injective factors. Cases II₁, II $_{\infty}$ , III $_{λ}$ , $λ \neq 1$ , Ann. Math. (2) 104 (1976) 73–115. Crossref, Web of Science, Google Scholar
9. E. G. Effros, D. E. Handelman and C. L. Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980) 385–407. Crossref, Web of Science, Google Scholar
10. E. G. Effros and J. Rosenberg, $C^{*}$ -algebras with approximately inner flip, Pacific. J. Math. 77 (1978) 417–443. Crossref, Web of Science, Google Scholar
11. I. Farah, I. Goldbring and B. Hart, Existentially closed II₁ factors, Fund. Math. 233 (2016) 173–196. Web of Science, Google Scholar
12. I. Farah, B. Hart, M. Lupini, L. Robert, A. P. Tikuisis, A. Vignati and W. Winter, Model theory of nuclear $C^{*}$ -algebras, preprint (2016), arXiv:1602.08072. Google Scholar
13. I. Farah and M. Magidor, Omitting types in logic of metric structures, preprint (2014), arXiv:1411.2987. Google Scholar
14. E. Gardella, Crossed products by compact group actions with the Rokhlin property, to appear in J. Noncommutative Geom. Google Scholar
15. I. Goldbring and T. Sinclair, On Kirchberg’s embedding problem, J. Funct. Anal. 269 (2015) 155–198. Crossref, Web of Science, Google Scholar
16. I. Goldbring and T. Sinclair, Omitting types in operator systems, to appear in Indiana Univ. Math. J. Google Scholar
17. U. Haagerup, Quasitraces on exact $C^{*}$ -algebras are traces, C. R. Math. Acad. Sci. Soc. R. Can. 36 (2014) 67–92. Google Scholar
18. D. Handelman, Homomorphisms of $C^{*}$ algebras into finite ${A W}^{*}$ algebras, Michigan Math. J. 28 (1981) 229–240. Crossref, Web of Science, Google Scholar
19. E. Kirchberg and M. Rordam, Central sequence $C^{*}$ -algebras and tensorial absorption of the Jiang–Su algebra, J. Reine Angew. Math. 695 (2014) 175–214. Crossref, Web of Science, Google Scholar
20. G. Pisier, Introduction to Operator Space Theory, London Mathematical Society Lecture Notes Series, Vol. 294 (Cambridge University Press, Cambridge, 2003). Crossref, Google Scholar
21. Y. Sato, S. White and W. Winter, Nuclear dimension and $𝒵$ -stability, Invent. Math. 202 (2015) 893–921. Crossref, Web of Science, Google Scholar
22. A. Tikuisis, S. White and W. Winter, Quasidiagonality of nuclear $C^{*}$ -algebras, to appear in Ann. Math. Google Scholar
23. A. S. Toms and W. Winter, Strongly self-absorbing $C^{*}$ -algebras, Trans. Amer. Math. Soc. 359 (2007) 3999–4029. Crossref, Web of Science, Google Scholar
24. W. Winter, QDQ vs. UCT, preprint (2016), arXiv:1603.01493. Google Scholar