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Normed algebras of differentiable functions on compact plane sets

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Normed algebras of differentiable functions on compact plane sets. / Dales, H.G.; Feinstein, J. F.

In: Indian Journal of Pure and Applied Mathematics, Vol. 41, No. 1, 01.02.2010, p. 153-187.

Research output: Contribution to journalJournal articlepeer-review

Harvard

Dales, HG & Feinstein, JF 2010, 'Normed algebras of differentiable functions on compact plane sets', Indian Journal of Pure and Applied Mathematics, vol. 41, no. 1, pp. 153-187. https://doi.org/10.1007/s13226-010-0005-1

APA

Dales, H. G., & Feinstein, J. F. (2010). Normed algebras of differentiable functions on compact plane sets. Indian Journal of Pure and Applied Mathematics, 41(1), 153-187. https://doi.org/10.1007/s13226-010-0005-1

Vancouver

Dales HG, Feinstein JF. Normed algebras of differentiable functions on compact plane sets. Indian Journal of Pure and Applied Mathematics. 2010 Feb 1;41(1):153-187. https://doi.org/10.1007/s13226-010-0005-1

Author

Dales, H.G. ; Feinstein, J. F. / Normed algebras of differentiable functions on compact plane sets. In: Indian Journal of Pure and Applied Mathematics. 2010 ; Vol. 41, No. 1. pp. 153-187.

Bibtex

@article{167dfe8d2e874ce1ab452866342e0678,
title = "Normed algebras of differentiable functions on compact plane sets",
abstract = "We investigate the completeness and completions of the normed algebras (D (1)(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D (1)(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which the completeness of (D (1)(X), ‖ · ‖) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ.In an earlier paper of Bland and Feinstein, the notion of an F -derivative of a function was introduced, where F is a suitable set of rectifiable paths, and with it a new family of Banach algebras D(1)F(X) corresponding to the normed algebras D (1)(X). In the present paper, we obtain stronger results concerning the questions when D (1)(X) and D(1)F(X) are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever X is {\textquoteleft} F -regular{\textquoteright}.An example of Bishop shows that the completion of (D (1)(X), ‖ · ‖) need not be semisimple. We show that the completion of (D (1)(X), ‖ · ‖) is semisimple whenever the union of all the rectifiable Jordan arcs in X is dense in X.We prove that the character space of D (1)(X) is equal to X for all perfect, compact plane sets X, whether or not (D (1)(X), ‖ · ‖) is complete. In particular, characters on the normed algebras (D (1)(X), ‖ · ‖) are automatically continuous.",
keywords = "Normed algebra, differentiable functions, Banach function algebra, completions , pointwise regularity of compact plane sets",
author = "H.G. Dales and Feinstein, {J. F.}",
year = "2010",
month = feb,
day = "1",
doi = "10.1007/s13226-010-0005-1",
language = "English",
volume = "41",
pages = "153--187",
journal = "Indian Journal of Pure and Applied Mathematics",
issn = "0019-5588",
publisher = "Indian National Science Academy",
number = "1",

}

RIS

TY - JOUR

T1 - Normed algebras of differentiable functions on compact plane sets

AU - Dales, H.G.

AU - Feinstein, J. F.

PY - 2010/2/1

Y1 - 2010/2/1

N2 - We investigate the completeness and completions of the normed algebras (D (1)(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D (1)(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which the completeness of (D (1)(X), ‖ · ‖) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ.In an earlier paper of Bland and Feinstein, the notion of an F -derivative of a function was introduced, where F is a suitable set of rectifiable paths, and with it a new family of Banach algebras D(1)F(X) corresponding to the normed algebras D (1)(X). In the present paper, we obtain stronger results concerning the questions when D (1)(X) and D(1)F(X) are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever X is ‘ F -regular’.An example of Bishop shows that the completion of (D (1)(X), ‖ · ‖) need not be semisimple. We show that the completion of (D (1)(X), ‖ · ‖) is semisimple whenever the union of all the rectifiable Jordan arcs in X is dense in X.We prove that the character space of D (1)(X) is equal to X for all perfect, compact plane sets X, whether or not (D (1)(X), ‖ · ‖) is complete. In particular, characters on the normed algebras (D (1)(X), ‖ · ‖) are automatically continuous.

AB - We investigate the completeness and completions of the normed algebras (D (1)(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D (1)(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which the completeness of (D (1)(X), ‖ · ‖) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ.In an earlier paper of Bland and Feinstein, the notion of an F -derivative of a function was introduced, where F is a suitable set of rectifiable paths, and with it a new family of Banach algebras D(1)F(X) corresponding to the normed algebras D (1)(X). In the present paper, we obtain stronger results concerning the questions when D (1)(X) and D(1)F(X) are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever X is ‘ F -regular’.An example of Bishop shows that the completion of (D (1)(X), ‖ · ‖) need not be semisimple. We show that the completion of (D (1)(X), ‖ · ‖) is semisimple whenever the union of all the rectifiable Jordan arcs in X is dense in X.We prove that the character space of D (1)(X) is equal to X for all perfect, compact plane sets X, whether or not (D (1)(X), ‖ · ‖) is complete. In particular, characters on the normed algebras (D (1)(X), ‖ · ‖) are automatically continuous.

KW - Normed algebra

KW - differentiable functions

KW - Banach function algebra

KW - completions

KW - pointwise regularity of compact plane sets

U2 - 10.1007/s13226-010-0005-1

DO - 10.1007/s13226-010-0005-1

M3 - Journal article

VL - 41

SP - 153

EP - 187

JO - Indian Journal of Pure and Applied Mathematics

JF - Indian Journal of Pure and Applied Mathematics

SN - 0019-5588

IS - 1

ER -