Logic and $\mathrm{C}^*$-algebras: set theoretical dichotomies in the theory of continuous quotients

TL;DR

This paper explores how set-theoretic assumptions like CH and Forcing Axioms influence the automorphism groups of corona $ ext{C}^*$-algebras, revealing a dichotomy between rigidity and nondefinability.

Contribution

It demonstrates a set-theoretic dichotomy in the automorphism groups of corona $ ext{C}^*$-algebras, linking logical axioms to algebraic symmetries.

Findings

Under CH, corona $ ext{C}^*$-algebras have large, nondefinable automorphism groups.

With Forcing Axioms, automorphism groups are highly rigid, containing only definable automorphisms.

The work connects set theory and operator algebras, showing how axioms influence algebraic automorphisms.

Abstract

Given a nonunital $\mathrm{C}^*$-algebra $A$ one constructs its corona algebra $\mathcal M(A)/A$. This is the noncommutative analog of the \v{C}ech-Stone remainder of a topological space. We analyze the two faces of these algebras: the first one is given assuming CH, and the other one arises when Forcing Axioms are assumed. In their first face, corona $\mathrm{C}^*$-algebras have a large group of automorphisms that includes nondefinable ones. The second face is the Forcing Axiom one; here the automorphism group of a corona $\mathrm{C}^*$-algebra is as rigid as possible, including only definable elements