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

### Tom 191 / 2009

Studia Mathematica 191 (2009), 223-235 MSC: Primary 46E25; Secondary 46J15, 93D99. DOI: 10.4064/sm191-3-4

#### Streszczenie

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 $uf+vg=1$.

#### Autorzy

• Raymond MortiniDépartement de Mathématiques
LMAM, UMR 7122
Université Paul Verlaine
Ile du Saulcy, F-57045 Metz, France
e-mail
• Brett D. WickDepartment 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