Trivial automorphisms

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.

Thoroughly revised version

Cited by in corpus (1)

• Automorphisms of corona algebras, and group cohomology