Reducibility in A R(K), C R(K), and A(K)

Rupp, R. ; Sasane, A.

Canadian journal of mathematics, Tome 62 (2010) no. 3, pp. 646-667

Résumé

Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let ${{A}_{\mathbb{R}}}\left( K \right)$ 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 $\left( f,\,g \right)$ in ${{A}_{\mathbb{R}}}{{\left( K \right)}^{2}}$ 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 $A\left( K \right)$ is 1. Finally, we also characterize all compact real symmetric sets $K$ such that ${{A}_{\mathbb{R}}}\left( K \right)$ , respectively ${{C}_{\mathbb{R}}}\left( K \right)$ , has Bass stable rank 1.

Détail

DOI : 10.4153/CJM-2010-025-9

Mots-clés : 46J15, 19B10, 30H05, 93D15, real Banach algebras, Bass stable rank, topological stable rank, reducibility