Three spectral notions for representations of commutative Banach algebras
Annales de l'Institut Fourier, Volume 25 (1975) no. 2, pp. 1-32.

Let T be a bounded representation of a commutative Banach algebra B. The following spectral sets are studied. Λ 1 (T): the Gelfand space of the quotient algebra B/ Ker T. Λ 2 (T): the Gelfand space of the operator algebra Im T. Λ 3 (T): those characters ϕ of B for which the inequalities T b x-b ^(ϕ)x<εx, bF, have a common solution x0, for any ε>0 and any finite subset F of B. A theorem of Beurling on the spectrum of L -functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of our theory by taking for T the regular representation and its adjoint.

Soit T une représentation bornée d’une algèbre de Banach commutative B. Les ensembles spectraux suivants sont étudiés. Λ 1 (T) : le spectre de l’algèbre quotient B/ Ker T. Λ 2 (T) : le spectre de l’algèbre d’opérateurs Im T. Λ 3 (T) : les caractères ϕ de B, tels que les inégalités T b x-b ^(ϕ)x<εx, bF, admettent une solution commune x0, quels que soient ε>0 et F, sous-ensemble fini de B. Un théorème de Beurling sur le spectre des fonctions de L et des résultats de Slodkowski et Zelazko sur les diviseurs de zéro topologiques simultanés sont obtenus comme des cas particuliers de notre théorie en prenant pour T la représentation régulière et son adjoint.

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     title = {Three spectral notions for representations of commutative {Banach} algebras},
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Domar, Yngve; Lindahl, Lars-Ake. Three spectral notions for representations of commutative Banach algebras. Annales de l'Institut Fourier, Volume 25 (1975) no. 2, pp. 1-32. doi : 10.5802/aif.553. http://www.numdam.org/articles/10.5802/aif.553/

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