Corona Rigidity
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915535952805888 |
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| author | Farah, Ilijas Ghasemi, Saeed Vaccaro, Andrea Vignati, Alessandro |
| author_facet | Farah, Ilijas Ghasemi, Saeed Vaccaro, Andrea Vignati, Alessandro |
| contents | We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal{P}(\mathbb{N})/\text{Fin}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of $\mathbb{N}$, while under the Continuum Hypothesis this rigidity fails and $\mathcal{P}(\mathbb{N})/\text{Fin}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, Čech-Stone remainders, and $\mathrm{C}^\ast$-algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_11618 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Corona Rigidity Farah, Ilijas Ghasemi, Saeed Vaccaro, Andrea Vignati, Alessandro Logic Operator Algebras 03E35, 03E50, 03E65, 03E57, 03E75, 03C50, 03C20, 03C98, 06E05, 46L05, 46L40, 54C05, 54D40, 03C66 We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal{P}(\mathbb{N})/\text{Fin}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of $\mathbb{N}$, while under the Continuum Hypothesis this rigidity fails and $\mathcal{P}(\mathbb{N})/\text{Fin}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, Čech-Stone remainders, and $\mathrm{C}^\ast$-algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible. |
| title | Corona Rigidity |
| topic | Logic Operator Algebras 03E35, 03E50, 03E65, 03E57, 03E75, 03C50, 03C20, 03C98, 06E05, 46L05, 46L40, 54C05, 54D40, 03C66 |
| url | https://arxiv.org/abs/2201.11618 |