Logic and C*-Algebras: Set Theoretical Dichotomies in the Theory of Continuous Quotients

Given a nonunital C*-algebra A one constructs its corona algebra M(A)/A. This is the noncommutative analog of the Cech-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 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 C*-algebra is as rigid as possible, including only definable elements.