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2009 | 191 | 3 | 223-235
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Simultaneous stabilization in $A_{ℝ}(𝔻)$

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We study the problem of simultaneous stabilization for the algebra $A_{ℝ}(𝔻)$. Invertible pairs $(f_{j},g_{j})$, j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $αf_{j} + βg_{j}$ is invertible in this algebra for j = 1,..., 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 $A_{ℝ}(𝔻)$ 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 $A_{ℝ}(𝔻)$.
For n ≥ 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 $A_{ℝ}(𝔻)²$ are totally reducible, that is, for which pairs there exist two units u and v in $A_{ℝ}(𝔻)$ such that uf + vg = 1.
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  • Département de Mathématiques, LMAM, UMR 7122, Université Paul Verlaine, Ile du Saulcy, F-57045 Metz, France
  • Department of Mathematics, University of South Carolina, LeConte College, 1523 Greene Street, Columbia, SC 29208, U.S.A.
  • The Fields Institute, 222 College Street, 2nd Floor, Toronto, Ontario, M5T 3J1 Canada
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-3-4
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