Discontinuous homomorphisms, selectors, and automorphisms of the complex field
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- by Paul B. Larson and Jindřich Zapletal
- Proc. Amer. Math. Soc. 147 (2019), 1733-1737
- DOI: https://doi.org/10.1090/proc/14338
- Published electronically: December 6, 2018
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Abstract:
We show, assuming a weak form of the Axiom of Choice, that the existence of a discontinuous homomorphism between separable Banach spaces induces a selector for the Vitali equivalence relation $\mathbb {R}/\mathbb {Q}$. In conjunction with a result of Di Prisco and Todorcevic, this shows that a nonprincipal ultrafilter on the integers is not sufficient to construct a discontinuous automorphism of the complex field, confirming a conjecture of Simon Thomas.References
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Bibliographic Information
- Paul B. Larson
- Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
- MR Author ID: 646854
- Email: larsonpb@miamioh.edu
- Jindřich Zapletal
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32601
- Email: zapletal@math.ufl.edu
- Received by editor(s): February 19, 2018
- Received by editor(s) in revised form: July 24, 2018, and July 26, 2018
- Published electronically: December 6, 2018
- Additional Notes: The research of the first author was partially supported by NSF grant DMS-1201494.
The research of the second author was partially supported by NSF grant DMS-1161078. - Communicated by: Heike Mildenberger
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1733-1737
- MSC (2010): Primary 03E25; Secondary 12D99, 54H11
- DOI: https://doi.org/10.1090/proc/14338
- MathSciNet review: 3910437