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Projective freeness and stable rank of algebras of complex-valued BV functions

Part of: Methods of category theory in functional analysis Theory of modules and ideals Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  16 January 2023

Alexander Brudnyi
Alexander Brudnyi*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada
*
e-mail: abrudnyi@ucalgary.ca
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Abstract

The paper investigates the algebraic properties of weakly inverse-closed complex Banach function algebras generated by functions of bounded variation on a finite interval. It is proved that such algebras have Bass stable rank 1 and are projective-free if they do not contain nontrivial idempotents. These properties are derived from a new result on the vanishing of the second Čech cohomology group of the polynomially convex hull of a continuum of a finite linear measure described by the classical H. Alexander theorem.

Keywords

Weakly inverse-closed Banach algebrafunction of bounded variationprojective moduleidempotentstable rankHausdorff measurecontinuumČech cohomologycovering dimensionpolynomially convex hull

MSC classification

Primary: 46J10: Banach algebras of continuous functions, function algebras
Secondary: 46M10: Projective and injective objects 13C10: Projective and free modules and ideals
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 10
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

Research is supported in part by NSERC.

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