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

Abstract

Given a nonunital ${\mathrm{C}}^{\ast }$-algebra $A$ one constructs its corona algebra $\mathcal{M}(A)/A$. This is the noncommutative analog of the Čech-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 ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-algebra is as rigid as possible, including only definable elements

This is the author's Ph.D. thesis, defended in April 2017 at York University, Toronto

References in corpus (3)

• Model theoretic forcing in analysis
• Reduced products of UHF algebras under forcing axioms
• The Countable Chain Condition for C*-Algebras