Simultaneous stabilization in $A_{\mathbb R}(\mathbb D)$

Raymond Mortini¹ ; Brett D. Wick²

¹Département de Mathématiques LMAM, UMR 7122 Université Paul Verlaine Ile du Saulcy, F-57045 Metz, France
²Department of Mathematics University of South Carolina LeConte College 1523 Greene Street Columbia, SC 29208, U.S.A. and The Fields Institute 222 College Street, 2nd Floor Toronto, Ontario, M5T 3J1 Canada

Studia Mathematica, Tome 191 (2009) no. 3, pp. 223-235

Résumé

We study the problem of simultaneous stabilization for the algebra $A_\mathbb R(\mathbb D)$. Invertible pairs $(f_j,g_j)$, $j=1,\ldots, n$, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair $(\alpha,\beta)$ of elements such that $\alpha f_j+\beta g_j$ is invertible in this algebra for $j=1,\ldots, n$. For $n=2$, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_\mathbb R(\mathbb D)$ has stable rank two, we are faced here with a different situation. When $n=2$, necessary and sufficient conditions are given so that we have simultaneous stability in $A_\mathbb R(\mathbb D)$.For $n\geq 3$ we show that under these conditions simultaneous stabilization is not possible and further connect this result to the question of which pairs $(f,g)$ in $A_\mathbb R(\mathbb D)^2$ are totally reducible, that is, for which pairs there exist two units $u$ and $v$ in $A_\mathbb R(\mathbb D)$ such that $uf+vg=1$.

Détail

DOI : 10.4064/sm191-3-4

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