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

Research output: Contribution to journal › Journal article › peer-review

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

Dales, H. G., & Loy, R. J. (1997). Uniqueness of the norm topology for Banach algebras with finite-dimensional radical. *Proceedings of the London Mathematical Society*, *74*(3), 633-661. https://doi.org/10.1112/S002461159700021X

Dales HG, Loy RJ. Uniqueness of the norm topology for Banach algebras with finite-dimensional radical. Proceedings of the London Mathematical Society. 1997 May;74(3):633-661. https://doi.org/10.1112/S002461159700021X

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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",

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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 -