Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2007-07

February 2007


Bass and Topological Stable Ranks of Complex and Real Algebras of Measures, Functions and Sequences

Kalle Mikkola and Amol Sasane

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 (modelling delays, possibly commensurate or having n generators) and/or of an absolutely continuous measure (an L¹ function).

We also compute the stable ranks of the convolution algebras l¹(Nn), l¹(Zn), l¹(S) and l¹(S ∩R+), where S is an arbitrary subgroup of R, of the almost periodic algebra AP and its causal (holomorphic) subalgebra AP+⊂ H, etc. As a by-product, we answer armatively the question posed by R. Mortini in [20].

In addition to the above algebras, we also consider the polydisc algebra A(Dn), the algebra C(Tn) 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 (or to strong stabilizability) 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.


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