Trivial automorphisms

by Farah and Shelah. [FaSh:987]

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 a model of this statement we have that for a number of Borel ideals I on~ omega every isomorphism between P (omega)/ I and any other quotient P (omega)/ J over a Borel ideal is trivial. We can also assure that in this model the dominating number,

d, is equal to aleph_1 and that 2^{aleph_1} is arbitrarily large. In this model Calkin algebra has outer automorphisms while all automorphisms of P (omega)/{mathop {Fin}} are trivial.

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