\(AC(\sigma)\) spaces for polygonally inscribed curves

Abstract

For certain families of compact subsets of the plane, the isomorphism class of the algebra of absolutely continuous functions on a set is completely determined by the homeomorphism class of the set. This is analogous to the Gelfand–Kolmogorov theorem for C(K) spaces. In this paper, we define a family of compact sets comprising finite unions of convex curves and show that this family has the ‘Gelfand–Kolmogorov’ property.

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Acknowledgements

The work of the first author was financially supported by the Ministry of Higher Education and Scientific Research of Iraq.

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Correspondence to Ian Doust.

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Communicated by Jörg Eschmeier.

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Al-shakarchi, S., Doust, I. \(AC(\sigma)\) spaces for polygonally inscribed curves. Banach J. Math. Anal. 15, 31 (2021). https://doi.org/10.1007/s43037-020-00110-w

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Keywords

  • Functions of bounded variation
  • \(AC(\sigma)\)
  • Isomorphisms of function spaces

Mathematics Subject Classification

  • 46J10
  • 05C10
  • 46J45
  • 47B40
  • 26B30