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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Al-shakarchi, S., Doust, I.: Isomorphisms of AC(σ) spaces for linear graphs. Adv. Oper. Theory. 5, 474–488 (2020)
Al-shakarchi, S., Doust, I.: Isomorphisms of BV(σ) spaces. arXiv: 2007.05701
Ashton, B., Doust, I.: Functions of bounded variation on compact subsets of the plane. Stud. Math. 169, 163–188 (2005)
Ashton, B., Doust, I.: A comparison of algebras of functions of bounded variation. Proc. Edin. Math. Soc. 49, 575–591 (2006)
Ashton, B., Doust, I.: AC(σ) operators. J. Oper. Theory. 65, 255–279 (2011)
Berkson, E., Gillespie, T. A.: AC functions on the circle and spectral families. J. Oper. Theory. 13, 33–47 (1985)
Doust, I., Al-shakarchi, S.: Isomorphisms of AC(σ) spaces for countable sets. In: Böttcher A., Potts D., Stollmann P., Wenzel D. (eds.), The diversity and beauty of applied operator theory, Oper. Theory Adv. Appl., vol. 268. Birkhäuser, Cham (2018)
Doust, I., Leinert, M.: Isomorphisms of AC(σ) spaces. Stud. Math. 228, 7–31 (2015)
Doust, I., Leinert, M.: Approximation in AC(σ). arXiv: 1312.1806
Garrido, M. I., Jaramillo, J. A.: Variations on the Banach-Stone theorem, IV Curso Espacios de Banach y Operadores (Laredo, 2001). Extracta Math. 17, 351–383 (2002)
Gelfand, I., Kolmogoroff, A.: On rings of continuous functions on topological spaces. Dokl. Akad. Nauk. SSSR. 22, 11–15 (1939)
Gross, L., Tucker, W.: Topological Graph Theory. Wiley, New York (1987)
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The work of the first author was financially supported by the Ministry of Higher Education and Scientific Research of Iraq.
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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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DOI: https://doi.org/10.1007/s43037-020-00110-w