Bass and topological stable ranks of complex and real algebras of measures, functions and sequences
In this important paper, the authors determine the Bass and topological stable ranks of a great variety of real and complex algebras of measures, functions and sequences. A question of the reviewer [\textit{R. Mortini}, ``An example of a subalgebra of \(H^ \infty\) on the unit disk whose stable rank is not finite'', Stud.\ Math.\ 103, No.\,3, 275--281 (1992; Zbl 0813.46041)], whether there exists a closed subalgebra of \(H^\infty\) with the corona property and having infinite stable rank, is answered positively. The results are summarized in a table right at the beginning of the text. Most symbols and algebras appearing in this work are presented in Section 2; some of them are missing though; for example, the set \(\ell^1(\mathbb N^n)\). Unfortunately, the symbol \(A_\mathbb R\) has two different meanings (on p.\,404 this stands for the algebra \(\{f\in A: f(\overline z)=\overline {f(\overline z)}\},\) \(z\in \overline {\mathbb D}^n\), or \(z\in \overline {\mathbb T}^n\), whereas on p.\,422 \(C(K)_\mathbb R=\{f\in C(K): f(x)\in \mathbb R\; \forall x\in K\cap \mathbb R^n\}\). The assertion on p.\,422 that \(A(\mathbb D^n)_{\mathbb R}= C( \overline {\mathbb D^n})_{\mathbb R}\) does not seem to be correct whatever meaning the authors give to their symbol. On pp.\,416--417, Corollary 5.3 and Corollary 5.6, the formula involving the solution to the Bezout equation is erroneous. On p.\,443, the formulas for the Bass and topological stable ranks of \(A^{n\times n}\) are not correct. Despite these few minor inconsistencies, we highly recommend this work to be read by anyone working in function theory and harmonic analysis and by engineers interested in system theory. Editorial note: Spurred by the above review the authors have compiled an errata list that can be found at \url{http://www.math.hut.fi/~kmikkola/research/art/errataMS_CAOT.pdf}.
scientific article; zbMATH DE number 5795260
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