All automorphisms of all Calkin algebras

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 $\omega_1$-trees and cohrerent families of Polish spaces and prove some uniformization results.