Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A Remark in Automatic Continuity Theory

A Remark in Automatic Continuity Theory G. R. ALLAN The most striking recent results in that branch of functional analysis known as automatic continuity theory depend on making certain set-theoretic assumptions beyond ZFC (the usual Zermelo-Fraenkel axioms together with the axiom of choice). Notably, the work of Dales and Esterle [5], [6], showing the existence of discontinuous homomorphisms from C(X) into certain Banach algebras, makes use of the continuum hypothesis CH. (Here, C{X) is the uniform algebra of all continuous complex-valued functions on the infinite compact Hausdorff space X.) It is known that these results require some set-theoretic assumptions beyond ZFC. These matters are discussed in Dales's survey article [4]. Without wishing, in this note, to enter into the question of the significance of results that involve CH, there seems, at least, considerable interest in proving automatic continuity results on the basis of ZF C alone, in so far as that may be possible. For example, the proof in [1], of the existence of a discontinuous homomorphism from the disc algebra A(D), uses no special set-theoretic assumptions beyond ZFC. The purpose of this note is to point out that the recent example of O'Farrell [7], of a regular uniform algebra with a continuous point derivation http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Remark in Automatic Continuity Theory

Download PDF
3 pages

Loading next page...
 
/lp/wiley/a-remark-in-automatic-continuity-theory-tjBrLV76oJ

References (0)

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/12.6.452
Publisher site
See Article on Publisher Site

Abstract

G. R. ALLAN The most striking recent results in that branch of functional analysis known as automatic continuity theory depend on making certain set-theoretic assumptions beyond ZFC (the usual Zermelo-Fraenkel axioms together with the axiom of choice). Notably, the work of Dales and Esterle [5], [6], showing the existence of discontinuous homomorphisms from C(X) into certain Banach algebras, makes use of the continuum hypothesis CH. (Here, C{X) is the uniform algebra of all continuous complex-valued functions on the infinite compact Hausdorff space X.) It is known that these results require some set-theoretic assumptions beyond ZFC. These matters are discussed in Dales's survey article [4]. Without wishing, in this note, to enter into the question of the significance of results that involve CH, there seems, at least, considerable interest in proving automatic continuity results on the basis of ZF C alone, in so far as that may be possible. For example, the proof in [1], of the existence of a discontinuous homomorphism from the disc algebra A(D), uses no special set-theoretic assumptions beyond ZFC. The purpose of this note is to point out that the recent example of O'Farrell [7], of a regular uniform algebra with a continuous point derivation

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Nov 1, 1980

There are no references for this article.