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Uniqueness of the norm topology for Banach algebras with finite-dimensional radical

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Uniqueness of the norm topology for Banach algebras with finite-dimensional radical. / Dales, H.G.; Loy, Richard J.

In: Proceedings of the London Mathematical Society, Vol. 74, No. 3, 05.1997, p. 633-661.

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Dales, HG & Loy, RJ 1997, 'Uniqueness of the norm topology for Banach algebras with finite-dimensional radical', Proceedings of the London Mathematical Society, vol. 74, no. 3, pp. 633-661. https://doi.org/10.1112/S002461159700021X

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Dales, H.G. ; Loy, Richard J. / Uniqueness of the norm topology for Banach algebras with finite-dimensional radical. In: Proceedings of the London Mathematical Society. 1997 ; Vol. 74, No. 3. pp. 633-661.

Bibtex

@article{0c775859f45646d790c5398e9dee4fcf,
title = "Uniqueness of the norm topology for Banach algebras with finite-dimensional radical",
abstract = "Semisimple Banach algebras are well-known to have a unique (complete) algebra norm topology, but such uniqueness may fail if the radical is even one-dimensional. We obtain a necessary condition for uniqueness of norm when the algebra has finite-dimensional radical. In the case where the Banach algebra is separable, the condition is shown to be also sufficient for a large class of algebras, and in particular under various hypotheses of commutativity. Examples are given to show the limitations of the various sufficiency results, and these also give a good indication of where the difficulties lie in general. We conjecture that, at least in the separable case, our condition is both necessary and sufficient. ",
keywords = "Banach algebras, uniqueness of norm, radical, analytic space",
author = "H.G. Dales and Loy, {Richard J.}",
year = "1997",
month = may,
doi = "10.1112/S002461159700021X",
language = "English",
volume = "74",
pages = "633--661",
journal = "Proceedings of the London Mathematical Society",
issn = "0024-6115",
publisher = "Oxford University Press",
number = "3",

}

RIS

TY - JOUR

T1 - Uniqueness of the norm topology for Banach algebras with finite-dimensional radical

AU - Dales, H.G.

AU - Loy, Richard J.

PY - 1997/5

Y1 - 1997/5

N2 - Semisimple Banach algebras are well-known to have a unique (complete) algebra norm topology, but such uniqueness may fail if the radical is even one-dimensional. We obtain a necessary condition for uniqueness of norm when the algebra has finite-dimensional radical. In the case where the Banach algebra is separable, the condition is shown to be also sufficient for a large class of algebras, and in particular under various hypotheses of commutativity. Examples are given to show the limitations of the various sufficiency results, and these also give a good indication of where the difficulties lie in general. We conjecture that, at least in the separable case, our condition is both necessary and sufficient.

AB - Semisimple Banach algebras are well-known to have a unique (complete) algebra norm topology, but such uniqueness may fail if the radical is even one-dimensional. We obtain a necessary condition for uniqueness of norm when the algebra has finite-dimensional radical. In the case where the Banach algebra is separable, the condition is shown to be also sufficient for a large class of algebras, and in particular under various hypotheses of commutativity. Examples are given to show the limitations of the various sufficiency results, and these also give a good indication of where the difficulties lie in general. We conjecture that, at least in the separable case, our condition is both necessary and sufficient.

KW - Banach algebras

KW - uniqueness of norm

KW - radical

KW - analytic space

U2 - 10.1112/S002461159700021X

DO - 10.1112/S002461159700021X

M3 - Journal article

VL - 74

SP - 633

EP - 661

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 3

ER -