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Search: MSC category 46J20 ( Ideals, maximal ideals, boundaries )

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1. CJM 2006 (vol 58 pp. 859)

 Nonstandard Ideals from Nonstandard Dual Pairs for $L^1(\omega)$ and $l^1(\omega)$ The Banach convolution algebras $l^1(\omega)$ and their continuous counterparts $L^1(\bR^+,\omega)$ are much studied, because (when the submultiplicative weight function $\omega$ is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of nice'' weights $\omega$, the only closed ideals they have are the obvious, or standard'', ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in $l^1(\omega)$. His proof was successfully exported to the continuous case $L^1(\bR^+,\omega)$ by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in $l^1(\omega)$ and $L^1(\bR^+,\omega)$. The new proof is based on the idea of a nonstandard dual pair'' which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in $L^1(\bR^+,\omega)$ containing functions whose supports extend all the way down to zero in $\bR^+$, thereby solving what has become a notorious problem in the area. Keywords:Banach algebra, radical, ideal, standard ideal, semigroupCategories:46J45, 46J20, 47A15
 Division par un polynÃ´me hyperbolique On se donne un intervalle ouvert non vide $\omega$ de $\mathbb R$, un ouvert connexe non vide $\Omega$ de $\mathbb R_s$ et un polyn\^ome unitaire $P_m(z, \lambda) = z^m + a_1(\lambda)z^{m-1} = +\dots + a_{m-1}(\lambda) z + a_m(\lambda),$ de degr\'e $m>0$, d\'ependant du param\etre $\lambda \in \Omega$. Un tel polyn\^ome est dit $\omega$-hyperbolique si, pour tout $\lambda \in \Omega$, ses racines sont r\'eelles et appartiennent \a $\omega$. On suppose que les fonctions $a_k, \, k=1, \dots, m$, appartiennent \a une classe ultradiff\'erentiable $C_M(\Omega)$. On sint\'eresse au probl\eme suivant. Soit $f$ appartient \a $C_M(\Omega)$, existe-t-il des fonctions $Q_f$ et $R_{f,k},\, k=0, \dots, m-1$, appartenant respectivement \a $C_M(\omega \times \Omega)$ et \a $C_M(\Omega)$, telles que l'on ait, pour $(x,\lambda) \in \omega \times \Omega$, $f(x) = P_m(x,\lambda) Q_f (x,\lambda) + \sum^{m-1}_{k=0} x^k R_{f,k}(\lambda)~?$ On donne ici une r\'eponse positive d\es que le polyn\^ome est $\omega$-hyperbolique, que la class untradiff\'eren\-tiable soit quasi-analytique ou non ; on obtient alors, des exemples d'id\'eaux ferm\'es dans $C_M(\mathbb R^n)$. On compl\ete ce travail par une g\'en\'eralisation d'un r\'esultat de C.~L. Childress dans le cadre quasi-analytique et quelques remarques. Categories:26E10, 46E25, 46J20
 Homeomorphic Analytic Maps into the Maximal Ideal Space of $H^\infty$ Let $m$ be a point of the maximal ideal space of $\papa$ with nontrivial Gleason part $P(m)$. If $L_m \colon \disc \rr P(m)$ is the Hoffman map, we show that $\papa \circ L_m$ is a closed subalgebra of $\papa$. We characterize the points $m$ for which $L_m$ is a homeomorphism in terms of interpolating sequences, and we show that in this case $\papa \circ L_m$ coincides with $\papa$. Also, if $I_m$ is the ideal of functions in $\papa$ that identically vanish on $P(m)$, we estimate the distance of any $f\in \papa$ to $I_m$. Categories:30H05, 46J20