Abstract view
Reducibility in A_{R}(K), C_{R}(K), and A(K)


Published:20091204
Printed: Jun 2010
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 $1$.
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
