# Sh:987

• Farah, I., & Shelah, S. (2014). Trivial automorphisms. Israel J. Math., 201(2), 701–728.
• Abstract:
We prove that the statement ‘For all Borel ideals {\mathcal I} and {\mathcal J} on \omega, every isomorphism between Boolean algebras {\mathcal P}(\omega)/{\mathcal I} and {\mathcal P}(\omega)/{\mathcal 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 {\mathcal I} on \omega every isomorphism between {\mathcal P} (\omega)/{\mathcal I} and any other quotient {\mathcal P}(\omega)/{\mathcal J} over a Borel ideal is trivial.

We can also assure that in this model the dominating number, {\mathfrak 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 {\mathcal P}(\omega)/{\mathop{Fin}} are trivial.

• Current version: 2013-04-30_11 (22p) published version (28p)
Bib entry
@article{Sh:987,
author = {Farah, Ilijas and Shelah, Saharon},
title = {{Trivial automorphisms}},
journal = {Israel J. Math.},
fjournal = {Israel Journal of Mathematics},
volume = {201},
number = {2},
year = {2014},
pages = {701--728},
issn = {0021-2172},
mrnumber = {3265300},
mrclass = {03E35 (03E17 03E40 03E57 03G05 06E99)},
doi = {10.1007/s11856-014-1048-5},
note = {\href{https://arxiv.org/abs/1112.3571}{arXiv: 1112.3571}},
arxiv_number = {1112.3571}
}