All automorphisms of all Calkin algebras

Abstract

The Proper Forcing Axiom implies all automorphisms of every Calkin algebra associated with an infinite-dimensional complex Hilbert space and the ideal of compact operators are inner. As a means of the proof we introduce notions of metric ε1-trees and coherent families of Polish spaces and develop their theory parallel to the classical theory of trees of height !1 and coherent families indexed by a δ-directed ordering.