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

Published online by Cambridge University Press:  16 January 2023

Alexander Brudnyi*
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada


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.

© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Research is supported in part by NSERC.


Alexander, H., Polynomial approximation and hulls in sets of finite linear measure in ${\mathbb{C}}^n$ . Amer. J. Math. 93(1971), 6574.CrossRefGoogle Scholar
Bass, H., K-theory and stable algebra . Publ. Math. Inst. Hautes Études Sci. 22(1964), 560.CrossRefGoogle Scholar
Brudnyi, A., Oka principle on the maximal ideal space of ${H}^{\infty }$ . St. Petersburg Math. J. 31(2020), 769817.CrossRefGoogle Scholar
Brudnyi, A., On homotopy invariants of tensor products of Banach algebras . Integral Equations Operator Theory 92(2020), 19.CrossRefGoogle Scholar
Brudnyi, A., Dense stable rank and Runge type approximation theorems for ${H}^{\infty }$ maps. J. Aust. Math. Soc. 113(2022), 289317. CrossRefGoogle Scholar
Brudnyi, A. and Brudnyi, Y., Multivariate bounded variation functions of Jordan–Wiener type . J. Approx. Theory 251(2020), 105346, 70 pp.CrossRefGoogle Scholar
Brudnyi, A. and Kinzebulatov, D., On uniform subalgebras of ${L}^{\infty }$ on the unit circle generated by almost periodic functions. St. Petersburg Math. J. 19(2008), no. 4, 495518.CrossRefGoogle Scholar
Brudnyi, A. and Sasane, A., Projective freeness and Hermiteness of complex function algebras. Preprint, 2022. arXiv:2208.04901 Google Scholar
Cohn, P., Free rings and their relations. 2nd ed., Academic Press, London, 1985.Google Scholar
Dineen, S., Harte, R., and Taylor, C., Spectra of tensor product elements. I. Basic theory . Math. Proc. R. Ir. Acad. 101A(2001), no. 2, 177196.Google Scholar
Eilenberg, S. and Steenrod, N., Foundations of algebraic topology, Princeton University Press, Princeton, NJ, 1952.CrossRefGoogle Scholar
Engelking, R., Dimension theory, North–Holland, Amsterdam, Oxford, New York, 1978.Google Scholar
Falconer, K. J., The geometry of fractal sets, Cambridge University Press, Cambridge, 1985.CrossRefGoogle Scholar
Gamelin, T., Uniform algebras, Prentice-Hall, New Jersey, 1969.Google Scholar
Heinonen, J., Lectures on analysis of metric spaces, Springer, New York, 2001.CrossRefGoogle Scholar
Hu, S.-T., Mappings of a normal space into an absolute neighborhood retract . Trans. Amer. Math. Soc. 64(1948), 336358.CrossRefGoogle Scholar
Lam, T., Serre’s conjecture, Lecture Notes in Mathematics, 635, Springer, Berlin, Heidelberg, New York, 1978.CrossRefGoogle Scholar
Nagami, K., Dimension theory, Academic Press, New York, London, 1970.Google Scholar
Royden, H., Function algebras . Bull. Amer. Math. Soc. 69(1963), 281298.CrossRefGoogle Scholar
Spanier, E., Algebraic topology, McGraw-Hill, New York, 1966.Google Scholar
Stout, E., Polynomial convexity, Progress in Mathematics, 261, Birkhäuser, Boston, Basel, Berlin, 2007.Google Scholar
Suárez, D., Čech cohomology and covering dimension for the ${H}^{\infty }$ maximal ideal space. J. Funct. Anal. 123(1994), 233263.CrossRefGoogle Scholar
Taylor, J., Topological invariants of the maximal ideal space of a Banach algebra . Adv. Math. 19(1976), no. 2, 149206.CrossRefGoogle Scholar
Vaserstein, L. N., Bass’s first stable range condition . J. Pure Appl. Algebra 34(1984), 319330.CrossRefGoogle Scholar
Vidyasagar, M., Control system synthesis: A factorization approach, Series in Signal Processing, Optimization, and Control, 7, MIT Press, Cambridge, 1985.Google Scholar