Abstract
We prove a more general vector-valued variant of the following statement: If \(X_1\) and \(X_2\) are Choquet simplices such that the spaces of affine continuous functions \({\mathfrak {A}}(X_1, {\mathbb {R}})\) and \({\mathfrak {A}}(X_2, {\mathbb {R}})\) are isomorphic, then the cardinality of \({\text {ext}}X_1\) is equal to the cardinality of \({\text {ext}}X_2\).
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References
Alfsen, E.: Compact Convex Sets and Boundary Integrals. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57. Springer-Verlag, New York, Heidelberg (1971)
Bessaga, C., Pełczyński, A.: On bases and unconditional convergence of series in Banach spaces. Studia Math. 17(2), 151–164 (1958)
Cengiz, B.: On topological isomorphisms of \(C_{0}(X)\) and the cardinal number of \(X\). Proc. Amer. Math. Soc. 72(1), 105–108 (1978)
Chu, C.H., Cohen, H.B.: Isomorphisms of spaces of continuous affine functions. Pacific J. Math. 155(1), 71–85 (1992)
Dostál, P., Spurný, J.: The minimum principle for affine functions and isomorphisms of continuous affine function spaces. Arch. Math. (Basel) 114(1), 61–70 (2020)
Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. The Basis for Linear and Nonlinear Analysis. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011)
Galego, E.M., Rincón-Villamizar, M.A.: Weak forms of Banach-Stone theorem for \(C_0(K, X)\) spaces via the \(\alpha \)th derivatives of \(K\). Bull. Sci. Math. 139(8), 880–891 (2015)
Hess, H.U.: On a theorem of Cambern. Proc. Amer. Math. Soc. 71(2), 204–206 (1978)
Koumoullis, G.: A generalization of functions of the first class. Topol. Appl. 50(3), 217–239 (1993)
Lazar, A.J.: Affine products of simplexes. Math. Scand. 22(165–175), 1968 (1969)
Lukeš, J., Malý, J., Netuka, I., Spurný, J.: Integral Representation Theory. Applications to Convexity, Banach Spaces and Potential Theory. De Gruyter Studies in Mathematics, 35. Walter de Gruyter & Co. Berlin (2010)
Morrison, T.: Functional Analysis: An Introduction to Banach Space Theory. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley (2001)
Rondoš, J., Spurný, J.: Isomorphisms of subspaces of vector-valued continuous functions. Acta Math. Hungar., to appear
Rondoš, J., Spurný, J.: Small-bound isomorphisms of function spaces. J. Aust. Math. Soc., to appear
Rondoš, J., Spurný, J.: Isomorphisms of spaces of affine continuous complex functions. Math. Scand. 125(2), 270–290 (2019)
Samuel, C.: Sur la reproductibilite des espaces \(l_p\). Math. Scand. 45, 103–117 (1979)
Spurný, J.: Borel sets and functions in topological spaces. Acta Math. Hungar. 129(1–2), 47–69 (2010)
Acknowledgements
We thank the anonymous referee for a careful reading of the paper, which led to a better presentation of the paper. Funding was provided by České Vysoké Učení Technické v Praze (Grant No. OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778).
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The first author was supported by the Project OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778.
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Rondoš, J., Spurný, J. A weak vector-valued Banach–Stone theorem for Choquet simplices. Arch. Math. 117, 529–536 (2021). https://doi.org/10.1007/s00013-021-01629-6
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DOI: https://doi.org/10.1007/s00013-021-01629-6