Mathematical Research Letters

Volume 18 (2011)

Number 3

All automorphisms of all Calkin algebras

Pages: 489 – 503



Ilijas Farah (Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3 and Matematicki Institut, Kneza Mihaila 34, Belgrade, Serbia)


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.

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