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Reducibility in AR(K), CR(K), and A(K)

  Published:2009-12-04
 Printed: Jun 2010
  • R. Rupp
  • A. Sasane
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Abstract

Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let $A_{\mathbb R}(K)$ denote the real Banach algebra of all real symmetric continuous functions on $K$ that are analytic in the interior $K^\circ$ of $K$, endowed with the supremum norm. We characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)^2$ which are reducible. In addition, for an arbitrary compact $K$ in $\mathbb C$, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of $A(K)$ is $1$. Finally, we also characterize all compact real symmetric sets $K$ such that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass stable rank $1$.
Keywords: real Banach algebras, Bass stable rank, topological stable rank, reducibility real Banach algebras, Bass stable rank, topological stable rank, reducibility
MSC Classifications: 46J15, 19B10, 30H05, 93D15 show english descriptions Banach algebras of differentiable or analytic functions, $H^p$-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30]
Stable range conditions
Bounded analytic functions
Stabilization of systems by feedback
46J15 - Banach algebras of differentiable or analytic functions, $H^p$-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30]
19B10 - Stable range conditions
30H05 - Bounded analytic functions
93D15 - Stabilization of systems by feedback
 

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