### 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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