# Sh:987

- Farah, I., & Shelah, S. (2014).
*Trivial automorphisms*. Israel J. Math.,**201**(2), 701–728. arXiv: 1112.3571 DOI: 10.1007/s11856-014-1048-5 MR: 3265300 -
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.

- 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} }