Bass and topological stable ranks of complex and real algebras of measures, functions and sequences
type: Item
instance of: scholarly article
title: Bass and topological stable ranks of complex and real algebras of measures, functions and sequences
zbMATH Open document ID: 1211.46056
published in: Complex Analysis and Operator Theory
publication date: 2010-10-04T00:00:00Z
review text: 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}.
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 43A10
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 93D25
Mathematics Subject Classification ID: 19B10
zbMATH DE Number: 5795260
zbMATH Keywords: Bass stable range
zbMATH Keywords: topological stable rank
zbMATH Keywords: parity interlacing property
zbMATH Keywords: Callier-Desoer algebra
zbMATH Keywords: polydisc algebra
zbMATH Keywords: real-symmetric functions
zbMATH Keywords: almost periodic functions
zbMATH Keywords: absolutely summable sequences
zbMATH Keywords: Bézout domains
zbMATH Keywords: bounded analytic functions
author: Kalle M. Mikkola
author: Amol Sasane
MaRDI profile type: Publication
full work available at URL: s11785-009-0009-1
OpenAlex ID: W2089395863
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DOI: 10.1007/S11785-009-0009-1
description: scientific article; zbMATH DE number 5795260