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Trivial automorphisms (1112.3571v2)
Published 15 Dec 2011 in math.LO
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.