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

Farah, Ilijas; McKenney, Paul

doi:10.1090/proc/13736

Cited by 8 publications

(9 citation statements)

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“…The following was proved by combining the ideas from [37, §4] and a stratification of the clopen algebra of βX with copies of P(N). Theorem 6.9 ( [53]). Assume OCA T and MA ℵ 1 .…”

Section: čEch-stone Remainders Of Zero-dimensional Spacesmentioning

confidence: 99%

Corona Rigidity

Farah¹,

Ghasemi²,

Vaccaro³

et al. 2022

Preprint

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In 1956 W. Rudin proved that the Continuum Hypothesis (CH) implies that the Čech-Stone remainder of N (with the discrete topology), βN \ N, has 2 c homeomorphisms. In 1979, Shelah described a forcing extension of the universe in which every autohomeomorphism of βN\N is the restriction of a continuous map of βN into itself. Since there are only c such maps, the conclusion contradicts Rudin's. Rudin's result is, by today's standards, trivial: By the Stone Duality, autohomeomorphisms of βN \ N correspond to automorphisms of the Boolean algebra P(N)/ Fin. This algebra is countably saturated hence CH implies that it is fully saturated. A standard back-and-forth argument produces a complete binary tree of height ℵ1 = c whose branches are distinct automorphisms. The fact that the theory of atomless Boolean algebras admits elimination of quantifiers is not even used in this argument.On the other hand, Shelah's result is, unlike most of the 1970s memorabilia, still as formidable as when it appeared. Extensions of Shelah's argument (nowadays facilitated by Forcing Axioms) show that this rigidity of P(N)/ Fin is shared by other similar quotient structures, and that is what this survey is about. Contents 1. Introduction: The general rigidity question.I. FARAH, S. GHASEMI, A. VACCARO, AND A. VIGNATI 4. Ulam-stability 21 4.1. From Borel to asymptotically additive 22 4.2. Asymptotic additivity 24 4.3. Boolean algebras 24 4.4. C * -algebras 26 4.5. Reduced products of C * -algebras 27 4.6. The Kadison-Kastler conjecture 28 5. Independence results, I. Nontrivial isomorphisms 29 5.1. Countable saturation 30 5.2. The failure of countable saturation 36 5.3. Stratifying Q(H) 39 5.4. Cohomology and automorphisms 40 5.5. Higher-dimensional Čech-Stone remainders 43 5.6. Other models for non-rigidity 44 6. Independence results, II. Rigidity 46 6.1. Shelah's proof 46 6.2. Quotients of P(N) and zero-dimensional remainders 49 6.3. Coronas and Forcing Axioms 52 6.4. Other models for rigidity 57 7. Endomorphisms 59 8. Larger Calkin algebras 62 9. Uniform Roe coronas 64 10. This paper was too short for. . .

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“…The following was proved by combining the ideas from [37, §4] and a stratification of the clopen algebra of βX with copies of P(N). Theorem 6.9 ( [53]). Assume OCA T and MA ℵ 1 .…”

Section: čEch-stone Remainders Of Zero-dimensional Spacesmentioning

confidence: 99%

Corona Rigidity

Farah¹,

Ghasemi²,

Vaccaro³

et al. 2022

Preprint

Self Cite

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“…Later on, Farah has proved that, under different set-theoretic assumptions, which imply in particular the negation of CH, all the automorphisms of the Calkin algebra are inner [32,33]. These results have later been generalized in [23,43,46,84] to other massive C*-algebras, which are obtained as corona algebras of separable C*-algebras.…”

Section: The Continuum Hypothesismentioning

confidence: 99%

“…Other examples are Calkin algebras and, more generally, corona algebras. Deep questions about such algebras have been recently addressed in [23,26,32,33,35,41,43,44,46,68,84]. Methods from model theory, set theory, and forcing are key components of this line of research.…”

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confidence: 99%

An Invitation to Model Theory and C*-Algebras

Lupini¹

2019

Bull. symb. log

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“…Later on, Farah has proved that, under different set-theoretic assumptions, which imply in particular the negation of CH, all the automorphisms of the Calkin algebra are inner [32,33]. These results have later been generalized in [23,43,46,83] to other massive C*-algebras, which are obtained as corona algebras of separable C*-algebras.…”

Section: The Effect Of the Continuum Hypothesismentioning

confidence: 99%

“…Other examples are Calkin algebras and, more generally, corona algebras. Deep questions about such algebras have been recently addressed in [23,26,32,33,35,41,43,44,46,67,83]. Methods from model theory, set theory, and forcing are key components of this line of research.…”

mentioning

confidence: 99%

An invitation to model theory and C*-algebras

Lupini¹

2017

Preprint

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We present an introductory survey to first order logic for metric structures and its applications to C*-algebras. IntroductionThis survey is designed as an introduction to the study of C*-algebras from the perspective of model theory for metric structures. The intended readership consists of anyone interested in learning about this subject, and naturally includes both logicians and operator algebraists. Considering this, we will not assume in these notes any previous knowledge of model theory, nor any in-depth knowledge of functional analysis, beyond a standard graduate-level course. A familiarity with C*-algebra theory and the classification programme [29] might be useful as a source of motivation and examples. Several facts from C*-algebra theory will be used, and detailed references provided. Most of the references will be to the comprehensive monographs [13,70].Logic for metric structures is a generalization of classical (or discrete) logic, suitable for applications to metric objects such as C*-algebras. The monograph [11] presents a quick but complete introduction to this subject, and explains how the fundamental results from classical model theory can be recast in the metric setting. The model-theoretic study of C*-algebras has been initiated in [38][39][40] where, in particular, it shown how C*-algebras fit into the framework of model theory for metric structures. The motivations behind this study are manifold. With no pretense of exhaustiveness, we attempt to illustrate some of them here.The most apparent contribution of first-order logic is to provide a syntactic counterpart to the semantic construction of ultraproducts and ultrapowers: the notion of formulas. Formulas allow one to express the fundamental properties of ultrapowers of C*-algebras (saturation) and diagonal embeddings into the ultrapower (elementarity). These general principles underpin most of the applications of the ultraproduct construction in C*-algebra theory, as they have appeared in various places in the literature under various names-Kirchberg's ε-test, reindexing arguments, etc. Isolating such general principles provides a valuable service of clarification and uniformization in the development of C*-algebra theory. In particular, this allows one to distinguish between, on one hand, what is just an instance of "general nonsense" and, on the other, what is a salient point where C*-algebras theory is crucially used.This abstract model-theoretic point of view also makes it easier to recognize analogies between different contexts. Furthermore, it provides a language to formalize such analogies as precise mathematical statements, rather than just intuitive ideas. For instance, this paradigm can be applied to some aspects of the equivariant theory of C*-algebras, which studies C*-algebras endowed with a group action (C*-dynamical systems). At least when the acting group is compact or discrete, C*-dynamical systems fit in the setting of first-order logic [47]. Adopting this perspective, one can naturally and effortlessly transfer ideas an...

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

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

Cited by 8 publications

References 30 publications

Corona Rigidity

Corona Rigidity

An Invitation to Model Theory and C*-Algebras

An invitation to model theory and C*-algebras

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