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1. CJM 2009 (vol 62 pp. 646)

Rupp, R.; Sasane, A.
 Reducibility in AR(K), CR(K), and A(K) 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, reducibilityCategories:46J15, 19B10, 30H05, 93D15

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