Trivial automorphisms

Abstract

We prove that the statement `For all Borel ideals I and J on $ω$, every isomorphism between Boolean algebras $P(ω)/I$ and $P(ω)/J$ has a continuous representation' is relatively consistent with ZFC. In this model every isomorphism between $P(ω)/I$ and any other quotient $P(ω)/J$ over a Borel ideal is trivial for a number of Borel ideals I on $ω$. 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 $P(ω)/Fin$ are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of suslin proper forcings.