Homeomorphisms of Čech–Stone remainders: the zero-dimensional case

Farah, Ilijas; McKenney, Paul

doi:10.1090/proc/13736

Homeomorphisms of Čech–Stone remainders: the zero-dimensional case
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Proc. Amer. Math. Soc. 146 (2018), 2253-2262 Request permission

Abstract:

We prove, using a weakening of the Proper Forcing Axiom, that any homemomorphism between Čech–Stone remainders of any two locally compact, zero-dimensional Polish spaces is induced by a homeomorphism between their cocompact subspaces.

References

Uri Abraham, Matatyahu Rubin, and Saharon Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of $\aleph _1$-dense real order types, Ann. Pure Appl. Logic 29 (1985), no. 2, 123–206. MR 801036, DOI 10.1016/0168-0072(84)90024-1
L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of $C^*$-algebras and $K$-homology, Ann. of Math. (2) 105 (1977), no. 2, 265–324. MR 458196, DOI 10.2307/1970999
Samuel Coskey and Ilijas Farah, Automorphisms of corona algebras, and group cohomology, Trans. Amer. Math. Soc. 366 (2014), no. 7, 3611–3630. MR 3192609, DOI 10.1090/S0002-9947-2014-06146-1
Alan Dow, A non-trivial copy of $\beta \Bbb {N}\sbs \Bbb {N}$, Proc. Amer. Math. Soc. 142 (2014), no. 8, 2907–2913. MR 3209343, DOI 10.1090/S0002-9939-2014-11985-X
Alan Dow and Klaas Pieter Hart, $\omega ^*$ has (almost) no continuous images, Israel J. Math. 109 (1999), 29–39. MR 1679586, DOI 10.1007/BF02775024
Alan Dow and Klaas Pieter Hart, A universal continuum of weight $\aleph$, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1819–1838. MR 1707489, DOI 10.1090/S0002-9947-00-02601-5
Alan Dow and Klaas Pieter Hart, The measure algebra does not always embed, Fund. Math. 163 (2000), no. 2, 163–176. MR 1752102, DOI 10.4064/fm-163-2-163-176
Christopher J. Eagle and Alessandro Vignati, Saturation and elementary equivalence of $C^*$-algebras, J. Funct. Anal. 269 (2015), no. 8, 2631–2664. MR 3390013, DOI 10.1016/j.jfa.2015.04.013
Ilijas Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148 (2000), no. 702, xvi+177. MR 1711328, DOI 10.1090/memo/0702
Ilijas Farah, Rigidity conjectures, Logic Colloquium 2000, Lect. Notes Log., vol. 19, Assoc. Symbol. Logic, Urbana, IL, 2005, pp. 252–271. MR 2143881
Ilijas Farah, All automorphisms of the Calkin algebra are inner, Ann. of Math. (2) 173 (2011), no. 2, 619–661. MR 2776359, DOI 10.4007/annals.2011.173.2.1
Ilijas Farah and Bradd Hart, Countable saturation of corona algebras, C. R. Math. Acad. Sci. Soc. R. Can. 35 (2013), no. 2, 35–56 (English, with English and French summaries). MR 3114457
Ilijas Farah and Ilan Hirshberg, The Calkin algebra is not countably homogeneous, Proc. Amer. Math. Soc. 144 (2016), no. 12, 5351–5357. MR 3556277, DOI 10.1090/proc/13137
Ilijas Farah and Saharon Shelah, Rigidity of continuous quotients, J. Inst. Math. Jussieu 15 (2016), no. 1, 1–28. MR 3427592, DOI 10.1017/S1474748014000218
Saeed Ghasemi, Isomorphisms of quotients of FDD-algebras, Israel J. Math. 209 (2015), no. 2, 825–854. MR 3430261, DOI 10.1007/s11856-015-1238-9
Saeed Ghasemi, Reduced products of metric structures: a metric Feferman-Vaught theorem, J. Symb. Log. 81 (2016), no. 3, 856–875. MR 3569108, DOI 10.1017/jsl.2016.20
Klaas Pieter Hart, The Čech-Stone compactification of the real line, Recent progress in general topology (Prague, 1991) North-Holland, Amsterdam, 1992, pp. 317–352. MR 1229130, DOI 10.1016/0887-2333(92)90021-I
Kenneth Kunen, Set theory, Studies in Logic (London), vol. 34, College Publications, London, 2011. MR 2905394
P. McKenney. Reduced products of UHF algebras under forcing axioms. arXiv preprint arXiv:1303.5037, 2013.
P. McKenney and A. Vignati. Forcing axioms and coronas of nuclear $\textrm {C}^*$-algebras.
N. Christopher Phillips and Nik Weaver, The Calkin algebra has outer automorphisms, Duke Math. J. 139 (2007), no. 1, 185–202. MR 2322680, DOI 10.1215/S0012-7094-07-13915-2
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955, DOI 10.1007/978-3-662-21543-2
Saharon Shelah and Juris Steprāns, PFA implies all automorphisms are trivial, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1220–1225. MR 935111, DOI 10.1090/S0002-9939-1988-0935111-X
Stevo Todorčević, Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989. MR 980949, DOI 10.1090/conm/084
Jan van Mill, An introduction to $\beta \omega$, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 503–567. MR 776630
Boban Veli ković, Definable automorphisms of ${\scr P}(\omega )/\textrm {fin}$, Proc. Amer. Math. Soc. 96 (1986), no. 1, 130–135. MR 813825, DOI 10.1090/S0002-9939-1986-0813825-8
Boban Veli ković, $\textrm {OCA}$ and automorphisms of ${\scr P}(\omega )/\textrm {fin}$, Topology Appl. 49 (1993), no. 1, 1–13. MR 1202874, DOI 10.1016/0166-8641(93)90127-Y
A. Vignati. Logic and $\textrm {C}^*$-algebras: Set theoretical dichotomies in the theory of continuous quotients. PhD thesis, York University, 2017.
Alessandro Vignati, Nontrivial homeomorphisms of Čech-Stone remainders, Münster J. Math. 10 (2017), no. 1, 189–200. MR 3624107, DOI 10.17879/33249446858
Dan-Virgil Voiculescu, Countable degree-1 saturation of certain $C^*$-algebras which are coronas of Banach algebras, Groups Geom. Dyn. 8 (2014), no. 3, 985–1006. MR 3267530, DOI 10.4171/GGD/254
W. Hugh Woodin, Beyond $\utilde \Sigma ^2_1$ absoluteness, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 515–524. MR 1989202