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

Let K denote a compact real symmetric subset of C and let AR(K) denote the real Banach algebra of all real symmetric continuous functions on K which are analytic in the interior K of K, endowed with the supremum norm. We characterize all unimodular pairs (f, g) in AR(K) 2 which are reducible. In addition, for an arbitrary compact K in 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 AR(K), respectively CR(K), has Bass stable rank 1.