Mathematical Research Letters
Volume 18 (2011)
All automorphisms of all Calkin algebras
Pages: 489 – 503
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 $\omega_1$-trees and coherent families of Polish spaces and develop their theory parallel to the classical theory of trees of height $\omega_1$ and coherent families indexed by a $\sigma$-directed ordering.