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Discontinuous homomorphisms, selectors, and automorphisms of the complex field


Authors: Paul B. Larson and Jindřich Zapletal
Journal: Proc. Amer. Math. Soc. 147 (2019), 1733-1737
MSC (2010): Primary 03E25; Secondary 12D99, 54H11
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.


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Additional Information

Paul B. Larson
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
Email: larsonpb@miamioh.edu

Jindřich Zapletal
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32601
Email: zapletal@math.ufl.edu

DOI: https://doi.org/10.1090/proc/14338
Keywords: Axiom of Choice, discontinuous homomorphisms, equivalence relations, automorphisms of the complex field
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
Article copyright: © Copyright 2018 American Mathematical Society