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Abstract. Let A R (D) denote the set of functions belonging to the disc algebra having real Fourier coefficients. We show that A R (D) has Bass and topological stable ranks equal to 2, which settles the conjecture made by Brett Wick in [18]. We also give a necessary and sufficient condition for reducibility in some real algebras of functions on symmetric domains with holes, which is a generalization of the main theorem in [18]. A sufficient topological condition on the symmetric open set D is given for the corresponding real algebra A R (D) to have Bass stable rank equal to 1.

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 that 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.

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