Trivial automorphisms

TL;DR

This paper demonstrates the relative consistency of all automorphisms of certain Boolean algebra quotients being trivial, using advanced forcing techniques, and explores implications for the structure of the Calkin algebra.

Contribution

It proves the consistency of trivial automorphisms for quotients over Borel ideals and analyzes the automorphism structure of the Calkin algebra under set-theoretic assumptions.

Findings

All automorphisms of $P()/Fin$ are trivial.

The Calkin algebra admits outer automorphisms.

The dominating number equals $\u00051$ and $2^{\u00051}>2^{0}$.

Abstract

We prove that the statement `For all Borel ideals I and J on $\omega$, every isomorphism between Boolean algebras $P(\omega)/I$ and $P(\omega)/J$ has a continuous representation' is relatively consistent with ZFC. In this model every isomorphism between $P(\omega)/I$ and any other quotient $P(\omega)/J$ over a Borel ideal is trivial for a number of Borel ideals I on $\omega$. We can also assure that the dominating number is equal to $\aleph_1$ and that $2^{\aleph_1}>2^{\aleph_0}$. Therefore the Calkin algebra has outer automorphisms while all automorphisms of $P(\omega)/Fin$ are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of suslin proper forcings.