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RIGIDITY OF CONTINUOUS QUOTIENTS

Published online by Cambridge University Press:  21 July 2014

Ilijas Farah
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3 Matematicki Institut, Kneza Mihaila 35, Belgrade, Serbia (ifarah@mathstat.yorku.ca) URL: http://www.math.yorku.ca/∼ifarah
Saharon Shelah
Affiliation:
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, Jerusalem 91904, Israel (shelah@math.huji.ac.il) URL: http://shelah.logic.at/ Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA

Abstract

We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion that the Stone–Čech remainder of the half-line has only trivial automorphisms is independent from ZFC (Zermelo-Fraenkel axiomatization of set theory with the Axiom of Choice). Consistency of this statement follows from the Proper Forcing Axiom, and this is the first known example of a connected space with this property.

Type
Research Article
Copyright
© Cambridge University Press 2014 

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References

Akemann, C. A., Pedersen, G. K. and Tomiyama, J., Multipliers of C*-algebras, J. Funct. Anal. 13(3) (1973), 277301.CrossRefGoogle Scholar
Ben Yaacov, I., Berenstein, A., Henson, C.W. and Usvyatsov, A., Model theory for metric structures, in Model Theory with Applications to Algebra and Analysis, Vol. II, (ed. Chatzidakis, Z. et al. ), London Math. Soc. Lecture Notes Series, Volume 350, pp. 315427 (Cambridge University Press, 2008).Google Scholar
Blackadar, B., Operator algebras, Encyclopaedia of Mathematical Sciences, Volume 122 (Springer-Verlag, Berlin, 2006) Theory of $C^{\ast }$-algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III.Google Scholar
Chang , C. C. and Keisler, H. J., Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, Volume 73 (North-Holland Publishing Co., Amsterdam, 1990).Google Scholar
Choi, Y., Farah, I. and Ozawa, N., A nonseparable amenable operator algebra which is not isomorphic to a C -algebra, Forum Math. Sigma 2 (2014), 12 pages.CrossRefGoogle Scholar
Coskey, S. and Farah, I., Automorphisms of corona algebras, and group cohomology, Trans. Amer. Math. Soc. 366(7) (2014), 36113630.CrossRefGoogle Scholar
Dow , A. and Hart, K. P., 𝜔 has (almost) no continuous images, Israel J. Math. 109 (1999), 2939.CrossRefGoogle Scholar
Dow, A. and Hart, K. P., A universal continuum of weight , Trans. Amer. Math. Soc. 353(5) (2001), 18191838.CrossRefGoogle Scholar
Eagle , C. and Vignati, A., Saturation of C*-algebras, Preprint, arXiv:1406.4875, 2014.Google Scholar
Farah, I., Cauchy nets and open colorings, Publ. Inst. Math. (Beograd) (N.S.) 64(78) (1998), 146152.Google Scholar
Farah, I., Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148(702) (2000).Google Scholar
Farah, I., Dimension phenomena associated with 𝛽ℕ-spaces, Top. Appl. 125 (2002), 279297.CrossRefGoogle Scholar
Farah, I., How many Boolean algebras O(ℕ)∕I are there? Illinois J. Math. 46 (2003), 9991033.Google Scholar
Farah, I., Rigidity conjectures, in Logic Colloquium 2000, Lect. Notes Log., Volume 19, pp. 252271 (Assoc. Symbol. Logic, Urbana, IL, 2005).Google Scholar
Farah, I., All automorphisms of the Calkin algebra are inner, Ann. of Math. 173 (2011), 619661.CrossRefGoogle Scholar
Farah, I. and Hart, B., Countable saturation of corona algebras, C. R. Math. Rep. Acad. Sci. Canada 35 (2013), 3556.Google Scholar
Farah, I., Hart, B. and Sherman, D., Model theory of operator algebras I: stability, Bull. Lond. Math. Soc. 45 (2013), 825838.CrossRefGoogle Scholar
Farah, I., Hart, B. and Sherman, D., Model theory of operator algebras II: model theory, Israel J. Math. arXiv:1004.0741, to appear.Google Scholar
Farah, I. and McKenney, P., Homeomorphisms of Čech–Stone remainders: the zero-dimensional case, Preprint, 2012.Google Scholar
Farah, I. and Shelah, S., A dichotomy for the number of ultrapowers, J. Math. Log. 10 (2010), 4581.CrossRefGoogle Scholar
Feferman, S. and Vaught, R., The first order properties of products of algebraic systems, Fund. Math. 47(1) (1959), 57103.CrossRefGoogle Scholar
Ghasemi, S., Isomorphisms of quotients of FDD-algebras, Israel J. Math. (2013), Preprint, arXiv:1310.1353.Google Scholar
Hart, B., Continuous model theory and its applications, Course notes, 2012 (available at http://www.math.mcmaster.ca/∼bradd/courses/math712/index.html).Google Scholar
Hart, K. P., The Čech-Stone compactification of the real line, in Recent progress in general topology (Prague, 1991), pp. 317352 (North-Holland, Amsterdam, 1992).Google Scholar
Hodges, W., Model theory, Encyclopedia of Mathematics and its Applications, Volume 42 (Cambridge university press, 1993).CrossRefGoogle Scholar
Just, W., Repercussions on a problem of Erdös and Ulam about density ideals, Canad. J. Math. 42 (1990), 902914.CrossRefGoogle Scholar
Just, W. and Krawczyk, A., On certain Boolean algebras O(𝜔)∕I, Trans. Amer. Math. Soc. 285 (1984), 411429.Google Scholar
Kirchberg, E. and Rørdam, M., Central sequence C*-algebras and tensorial absorption of the Jiang–Su algebra, J. Reine Aangew. Math. (Crelle’s Journal) (2012), arXiv:1209.5311.Google Scholar
Lopes, V.C., Reduced products and sheaves of metric structures, Math. Log. Q. 59(3) (2013), 219229.CrossRefGoogle Scholar
Marker, D., Model theory, Graduate Texts in Mathematics, Volume 217 (Springer-Verlag, New York, 2002).Google Scholar
McKenney, P., Reduced products of UHF algebras under forcing axioms, Preprint, 2013 (arXiv:1303.5037).Google Scholar
van Mill, J., An introduction to 𝛽𝜔, in Handbook of Set-theoretic topology, (ed. Kunen, K. and Vaughan, J.), pp. 503560 (North-Holland, 1984).CrossRefGoogle Scholar
Phillips, N.C. and Weaver, N., The Calkin algebra has outer automorphisms, Duke Math. J. 139 (2007), 185202.CrossRefGoogle Scholar
Shelah, S., Proper forcing, Lecture Notes in Mathematics, Volume 940 (Springer, 1982).CrossRefGoogle Scholar
Shelah, S. and Steprāns, J., PFA implies all automorphisms are trivial, Proc. Amer. Math. Soc. 104 (1988), 12201225.CrossRefGoogle Scholar
Solecki, S., Analytic ideals, Bull. Symbolic Logic 2 (1996), 339348.CrossRefGoogle Scholar
Veličković, B., OCA and automorphisms of O(𝜔)∕Fin, Top. Appl. 49 (1992), 113.CrossRefGoogle Scholar
Voiculescu, Dan-Virgil, Countable degree-1 saturation of certain C*-algebras which are coronas of Banach algebras, 2013 (arXiv:1310.4862).Google Scholar
Woodin, W.H., Beyond Σ12 absoluteness, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pp. 515524 (Higher Ed. Press, Beijing, 2002).Google Scholar
Zamora-Aviles, B., Gaps in the poset of projections in the Calkin algebra, Israel J. Math., to appear.Google Scholar
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