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RIGIDITY OF CONTINUOUS QUOTIENTS

Part of: Selfadjoint operator algebras Model theory Set theory

Published online by Cambridge University Press:  21 July 2014

Ilijas Farah  and
Saharon Shelah
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
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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.

Keywords

Reduced productscountable saturationasymptotic sequence algebraProper Forcing Axiomcontinuum hypothesis

MSC classification

Primary: 03C65: Models of other mathematical theories 03E35: Consistency and independence results 03E50: Continuum hypothesis and Martin's axiom
Secondary: 03E57: Generic absoluteness and forcing axioms 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 1 , January 2016 , pp. 1 - 28
Copyright
© Cambridge University Press 2014 

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