
TL;DR
This paper proves that under the Proper Forcing Axiom, all automorphisms of Calkin algebras are inner, introducing new concepts like Polish ω₁-trees and uniformization results.
Contribution
It introduces Polish ω₁-trees and coherent families of Polish spaces, providing a novel approach to analyze automorphisms of Calkin algebras under set-theoretic axioms.
Findings
All automorphisms of Calkin algebras are inner under the Proper Forcing Axiom.
Introduces Polish ω₁-trees and uniformization results for the study of automorphisms.
Provides a new framework for understanding automorphisms in operator 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 the notion of Polish -trees and cohrerent families of Polish spaces and prove some uniformization results.
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