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Corona Rigidity

Ilijas Farah, Saeed Ghasemi, Andrea Vaccaro, Alessandro Vignati

TL;DR

This survey develops a unifying framework for rigidity of Borel quotient structures across Boolean algebras and $ ext{C}^*$-algebras, highlighting the recurring dichotomy that CH enables many nontrivial automorphisms while forcing axioms enforce topological and algebraic triviality of liftings. Central to the analysis are liftings, notions of topological versus algebraic triviality, and Ulam-stability linking approximate morphisms to genuine ones, connected through reduced products, corona algebras, and Čech–Stone remainders. The text synthesizes model-theoretic saturation results (under CH) with forcing-axiom techniques (e.g., OCA, MA, PFA) to explain when nontrivial automorphisms exist or are eliminated, and extends these ideas to large coronas, uniform Roe coronas, and Higson coronas. The work underscores deep interactions among set theory, model theory, and operator algebras, identifies major open problems, and outlines a meta-theory for rigidity applicable across commutative and noncommutative contexts, including endomorphisms and extension theory. Overall, it provides a broad, interoperable framework to study when quotient structures admit Borel liftings and when such liftings are forced to be algebraically (and thus structurally) trivial, with wide implications for dynamics, classification, and coarse geometry.

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.

Corona Rigidity

TL;DR

This survey develops a unifying framework for rigidity of Borel quotient structures across Boolean algebras and -algebras, highlighting the recurring dichotomy that CH enables many nontrivial automorphisms while forcing axioms enforce topological and algebraic triviality of liftings. Central to the analysis are liftings, notions of topological versus algebraic triviality, and Ulam-stability linking approximate morphisms to genuine ones, connected through reduced products, corona algebras, and Čech–Stone remainders. The text synthesizes model-theoretic saturation results (under CH) with forcing-axiom techniques (e.g., OCA, MA, PFA) to explain when nontrivial automorphisms exist or are eliminated, and extends these ideas to large coronas, uniform Roe coronas, and Higson coronas. The work underscores deep interactions among set theory, model theory, and operator algebras, identifies major open problems, and outlines a meta-theory for rigidity applicable across commutative and noncommutative contexts, including endomorphisms and extension theory. Overall, it provides a broad, interoperable framework to study when quotient structures admit Borel liftings and when such liftings are forced to be algebraically (and thus structurally) trivial, with wide implications for dynamics, classification, and coarse geometry.

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 , 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 , while under the Continuum Hypothesis this rigidity fails and 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 -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.
Paper Structure (77 sections, 64 theorems, 69 equations, 2 figures)

This paper contains 77 sections, 64 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The general rigidity question
  3. The abelian case: Boolean algebras and Čech--Stone remainders
  4. Borel ideals in $\mathcal{P}({\mathbb N})$.
  5. Čech--Stone remainders
  6. The origins of rigidity questions in $\mathrm{C}^*$-algebras
  7. Preliminaries
  8. Examples I: Abelian $\mathrm{C}^*$-algebras
  9. Examples II: General $\mathrm{C}^*$-algebras
  10. Multiplier and corona algebras
  11. Algebraically trivial automorphisms
  12. The BDF Question
  13. Sakai's Question
  14. Ulam-stability
  15. Approximate homomorphisms
  16. ...and 62 more sections

Key Result

Theorem 3.6

An ideal is $F_\sigma$ if and only if it is of the form $\mathop{\mathrm{Fin}}\nolimits(\mu)$ for a lower semicontinuous submeasure $\mu$. An analytic ideal is a P-ideal if and only if it is of the form $\mathop{\mathrm{Exh}}\nolimits(\mu)$ for a lower semicontinuous submeasure $\mu$. In this case, is a complete metric on $\mathcal{P}({\mathbb N})/{\mathcal{I}}$.

Figures (2)

  • Figure 1: A lifting $\Phi_*$ of $\Phi$.
  • Figure 2: With $E_j=[n(j),n(j+1))$, for every $a$ in $\mathcal{F}[\mathbf E]$ its support (i.e., the set $\{\{m,n\}: q_naq_m\neq 0\}$) is included in the union of the solid line square and the dashed line square regions.

Theorems & Definitions (127)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.7
  • Example 2.8
  • Conjecture 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 117 more