Three spectral notions for representations of commutative Banach algebras

Domar, Yngve; Lindahl, Lars-Ake

Annales de l'Institut Fourier, Tome 25 (1975) no. 2, pp. 1-32.

Soit $T$ une représentation bornée d’une algèbre de Banach commutative $B$. Les ensembles spectraux suivants sont étudiés. ${\Lambda }_1\left(T\right)$ : le spectre de l’algèbre quotient $B/\mathrm{Ker}T$. ${\Lambda }_2\left(T\right)$ : le spectre de l’algèbre d’opérateurs $\mathrm{Im}T$. ${\Lambda }_3\left(T\right)$ : les caractères $\varphi$ de $B$, tels que les inégalités $\parallel T_{b}x-\widehat{b}\left(\varphi \right)x\parallel <\epsilon \parallel x\parallel$, $b\in F$, admettent une solution commune $x\ne 0$, quels que soient $\epsilon >0$ 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. ${\Lambda }_1\left(T\right)$: the Gelfand space of the quotient algebra $B/\mathrm{Ker}T$. ${\Lambda }_2\left(T\right)$: the Gelfand space of the operator algebra $\mathrm{Im}T$. ${\Lambda }_3\left(T\right)$: those characters $\varphi$ of $B$ for which the inequalities $\parallel T_{b}x-\widehat{b}\left(\varphi \right)x\parallel <\epsilon \parallel x\parallel$, $b\in F$, have a common solution $x\ne 0$, for any $\epsilon >0$ 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.

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