Bass and topological stable ranks of complex and real algebras of measures, functions and sequences

We compute the Bass stable rank and the topological stable rank of several convolution Banach algebras of complex measures on (-∞,∞) or on [0,∞) consisting of a discrete measure and/or of an absolutely continuous measure. We also compute the stable ranks of the convolution algebras , , ℓ1(S) and , where S is an arbitrary subgroup of , of the almost periodic algebra AP and of , etc. We answer affirmatively the question posed by Mortini (Studia Mathematica 103(3):275–281, 1992). For the above algebras, the polydisc algebra , the algebra of continuous functions, and others, we also study their subsets (real Banach algebras) of real-valued measures, real-valued sequences or real-symmetric functions, and of corresponding exponentially stable algebras (for example, the Callier–Desoer algebra of causal exponentially decaying measures and L1 functions), and we compute their stable ranks. Finally, we show that in some of these real algebras a variant of the parity interlacing property is equivalent to reducibility of a unimodular (or coprime) pair. Also corona theorems are presented and the existence of coprime fractions is studied; in particular, we list which of these algebras are Bézout domains.