All automorphisms of all Calkin algebras

Abstract

The Proper Forcing Axiom implies all automorphisms of every Calkin al- gebra associated with an innite-dimension al complex Hilbert space and the ideal of compact operators are inner. As a means of the proof we introduce notions of met- ric !1-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 -directed ordering. Fix an innite-dimension al complex Hilbert space H. Let B(H) be its alge- bra of bounded linear operators, K(H) its ideal of compact operators and C(H) = B(H)=K(H) the Calkin algebra. Answering a question rst asked by Brown{Douglas- Fillmore, in (10) and (5) it was proved that the existence of outer automorphisms of the Calkin algebra associated with a separable H is independent from ZFC. In the present paper we consider the existence of outer automorphisms of the Calkin algebra associated with an arbitrary complex, innite-dimension al Hilbert space. PFA stands for the Proper Forcing Axiom, MA for Martin's Axiom and TA stands for Todorcevic's Axiom (see e.g., (12) or (8) for PFA and TA and (7, Chapter II) for MA). It is well-known that both MA and TA are consequences of PFA. Theorem 1. MA and TA together imply all automorphisms of the Calkin algebra associated with Hilbert space with basis of cardinality@1 are inner.