Holomorphic Functional Calculus on Quotients of Fréchet Algebras and Michael's Problem
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | ESTERLE, Jean | |
| dc.date.accessioned | 2024-04-04T02:37:05Z | |
| dc.date.available | 2024-04-04T02:37:05Z | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/190775 | |
| dc.description.abstractEn | The purpose of the paper is to give the state of the art on Michael's problem, the long-standing open question odf continuity of characters on commutative Fréchet algebras. We first quote two well-known consequences of the "abstract Mittag-Leffler theorem", the theorem of Arens, which shows that characters on finitely rationally generated Fréchet algebras are continuous, and the fact that the existence of a nonincreasing sequence (Ωn) n≥1 of Fatou-Bieberbach domains in C p such that ∩ n≥1 Ωn = ∅ would imply that all characters on commutative Fréchet algebras are continuous. In the opposite direction the existence of a discontinuous character on some commutative unital Fréchet algebra is equivalent to the existence of a character on a quotient algebra of the form U/I where U is a 'test algebra' for Michael's problem and where I is a dense ideal of U which is a Picard-Borel ideal, which means that every family of pairwise linearly independent invertible elements of U/I is linearly independent. It was recently shown that all Picard-Borel ideals in commutative unital Fréchet algebras are prime, and Picard-Borel ideals of H(C) can be easily described. We raise a question concerning Picard-Borel ideals of H(C p), p ≥ 2 which could lead to important general information about the quotient of commutative unital Fréchet algebras by Picard-Borel ideals. The fact that entire functions of several variables oerate on quotients of Fréchet algebras by ideals which are not necessarily closed plays an essential role in the paper. | |
| dc.language.iso | en | |
| dc.subject.en | Borel theorem Picard theorem entire function Fréchet algebra Michael's problem prime ideal Picard-Borel ideal weak Picard-Borel ideal holomorphic functional calculus MSC: 30D20 30H50 32A15 46H10 46J05 N. Surname N. Surname: Short Title (pp. 1 -9) | |
| dc.subject.en | Borel theorem | |
| dc.subject.en | Picard theorem | |
| dc.subject.en | entire function | |
| dc.subject.en | Fréchet algebra | |
| dc.subject.en | Michael's problem | |
| dc.subject.en | prime ideal | |
| dc.subject.en | Picard-Borel ideal | |
| dc.subject.en | weak Picard-Borel ideal | |
| dc.subject.en | holomorphic functional calculus MSC: 30D20 | |
| dc.subject.en | 30H50 | |
| dc.subject.en | 32A15 | |
| dc.subject.en | 46H10 | |
| dc.subject.en | 46J05 N. Surname | |
| dc.subject.en | N. Surname: Short Title (pp. 1 -9) | |
| dc.title.en | Holomorphic Functional Calculus on Quotients of Fréchet Algebras and Michael's Problem | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Analyse fonctionnelle [math.FA] | |
| dc.subject.hal | Sciences de l'ingénieur [physics]/Autre | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | hal-03902192 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-03902192v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=ESTERLE,%20Jean&rft.genre=preprint |
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