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

Given a nonunital ${\mathrm{C}}^{\ast }$-algebra $A$ one constructs its corona algebra $\mathcal{M}(A)\mathrm{/}A$. 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 ${\mathrm{C}}^{\ast }$-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