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Trivial automorphisms |
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| Authors: | Ilijas Farah Saharon Shelah |
| Institution: | 1. Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3 2. Matematicki Institut, Kneza Mihaila 35, Belgrade, Serbia 3. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem, 91904, Israel 4. Department of Mathematics, Hill Center-Busch Campus, Rutgers The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA |
| Abstract: | We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω. We can also assure that the dominating number, σ, is equal to ?1 and that \({2^{{\aleph _1}}} > {2^{{\aleph _0}}}\) . Therefore, the Calkin algebra has outer automorphisms while all automorphisms of P(ω)/Fin are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of Suslin proper forcings. |
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