Logic and C*-algebras: set theoretical dichotomies in the theory of continuous quotients

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Given a nonunital C*-algebra A one constructs its corona algebra 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 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