Corona Rigidity

Abstract

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.

88 pages, updates on recent progress

References in corpus (15)

• Countable saturation of corona algebras
• Coarse geometry and topological phases
• Saturation and elementary equivalence of C*-algebras
• Automorphisms of corona algebras, and group cohomology
• Operator algebraic approach to inverse and stability theorems for amenable groups
• Uniform Roe algebras of uniformly locally finite metric spaces are rigid
• Forcing axioms and coronas of $C^{\ast }$-algebras
• Corona Rigidity
• Category forcings, $M{M}^{+++}$, and generic absoluteness for the theory of strong forcing axioms
• Separable Spaces of Continuous Functions as Calkin Algebras
• Index theorems for uniformly elliptic operators
• Ulam stability for some classes of C*-algebras
• Calkin algebra, Kazhdan's property (T), strongly self-absorbing C*-algebras
• Between reduced powers and ultrapowers
• Abstract classification theorems for amenable C*-algebras

Cited by in corpus (2)

• Corona Rigidity
• The weak Extension Principle