Hostname: page-component-669899f699-7xsfk Total loading time: 0 Render date: 2025-04-24T13:03:09.991Z Has data issue: false hasContentIssue false
  • English
  • Français

DISCONTINUOUS HOMOMORPHISMS OF $C(X)$ WITH $2^{\aleph _0}>\aleph _2$

Part of: Topological algebras, normed rings and algebras, Banach algebras Model theory Set theory

Published online by Cambridge University Press:  15 April 2024

BOB A. DUMAS Open the ORCID record for BOB A. DUMAS [Opens in a new window]
BOB A. DUMAS*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF WASHINGTON SEATTLE, WA 98195, USA
*
E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Assume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$-morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P-generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$-linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $, into the Esterle algebra of formal power series. Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X, there exists a discontinuous homomorphism of $C(X)$, the algebra of continuous real-valued functions on X.

Keywords

discontinuous homomorphisms of C(X)morassesEsterle Algebrasaturated modelsultrapowers of the real numbers

MSC classification

Primary: 03C25: Model-theoretic forcing 03E17: Cardinal characteristics of the continuum 03E35: Consistency and independence results 46H40: Automatic continuity
Type
Article
Information
The Journal of Symbolic Logic , Volume 89 , Issue 2 , June 2024 , pp. 665 - 696
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Dales, H. G., Automatic continuity: A survey. Bulletin of the London Mathematical Society, vol. 10 (1978), pp. 129–183.CrossRefGoogle Scholar
Dales, H. G., A discontinuous homomorphism of C(X). American Journal of Mathematics, vol. 101 (1979), no. 3, pp. 647–734.CrossRefGoogle Scholar
Dumas, B. A., Order-isomorphic ${\eta}_1$ -orderings in Cohen extensions. Annals of Pure and Applied Logic, vol. 158 (2009), pp. 1–22.CrossRefGoogle Scholar
Dumas, B. A., Gap-2 Morass-definable ${\eta}_1$ -orderings. Mathematical Logic Quarterly, vol. 68 (2022), no. 2, pp. 227–242.CrossRefGoogle Scholar
Erdös, P., Gillman, L., and Henriksen, M., An isomorphism theorem for real closed fields. Annals of Mathematics, vol. 61 (1955), pp. 552–554.CrossRefGoogle Scholar
Esterle, J., Sur l’existence d’un homomorphisme discontinu do C(K). Acta Mathematica Academiae Scientiarum Hungarica, vol. 30 (1977), pp. 113–127.CrossRefGoogle Scholar
Esterle, J., Semi-normes on $C(K)$ . Proceedings of the London Mathematical Society, vol. 36 (1978), no. 3, pp. 27–45.CrossRefGoogle Scholar
Esterle, J., Injections de semi-groupes divisible dans des algébras de convolution et construction d’homomorphismes discontinus des C(K), Proceedings of the London Mathematical Society, vol. 36 (1978), no. 3, pp. 59–85.CrossRefGoogle Scholar
Hahn, H., Ober die nichtarchimedischen Grössensysteme. Sitzungsberichte Kaiserlichen der Akademie Wissenschaften, Wien, Mathematisch–Naturwissenschaftliche Klasse 116 (Abteilung IIa, 1907), pp. 601–655.Google Scholar
Johnson, B. E., Norming $C(\varOmega)$ and related algebras. Transactions of the American Mathematical Society, vol. 220 (1976), pp. 37–58.Google Scholar
Maclane, S., The universality of power series fields. Bulletin of the American Mathematical Society, vol. 45 (1939), pp. 888–890.CrossRefGoogle Scholar
Szalkai, I., An inductive definition of higher gap simplified morasses. Publicationes Mathematicae Debrecen, vol. 58 (2001), no. 4, pp. 605–634.CrossRefGoogle Scholar
Velleman, D., Simplified morasses, Journal of Symbolic Logic, vol. 49 (1984), pp. 257–271.Google Scholar
Velleman, D., Simplified gap-2 morasses. Annals of Pure and Applied Logic, vol. 34 (1987), pp. 171–208.CrossRefGoogle Scholar
Woodin, H., A discontinuous homomorphism from C(X) without CH. Journal of the London Mathematical Society, vol. s2-48 (1993), no. 2, pp. 299–315.CrossRefGoogle Scholar