Three spectral notions for representations of commutative Banach algebras

Domar, Yngve; Lindahl, Lars-Ake

Annales de l'Institut Fourier, Tome 25 (1975), p. 1-32 / Harvested from Numdam

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Résumé
Abstract

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 $∥ T_{b} x - \hat{b} (ϕ) x ∥ < ε ∥ x ∥$ , $b \in F$ , admettent une solution commune $x \neq 0$ , quels que soient $ε > 0$ et $F$ , sous-ensemble fini de $B$ . Un théorème de Beurling sur le spectre des fonctions de $L^{\infty}$ 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.

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 $∥ T_{b} x - \hat{b} (ϕ) x ∥ < ε ∥ x ∥$ , $b \in F$ , have a common solution $x \neq 0$ , for any $ε > 0$ and any finite subset $F$ of $B$ . A theorem of Beurling on the spectrum of $L^{\infty}$ -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.

Publié le : 1975-01-01

MR 53 #3714 | Zbl 0301.46045

DOI : https://doi.org/10.5802/aif.553

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     author = {Domar, Yngve and Lindahl, Lars-Ake},
     title = {Three spectral notions for representations of commutative Banach algebras},
     journal = {Annales de l'Institut Fourier},
     volume = {25},
     year = {1975},
     pages = {1-32},
     doi = {10.5802/aif.553},
     mrnumber = {53 \#3714},
     zbl = {0301.46045},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1975__25_2_1_0}
}

Domar, Yngve; Lindahl, Lars-Ake. Three spectral notions for representations of commutative Banach algebras. Annales de l'Institut Fourier, Tome 25 (1975) pp. 1-32. doi : 10.5802/aif.553. http://gdmltest.u-ga.fr/item/AIF_1975__25_2_1_0/

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