Isomorphisms of spaces of affine continuous complex functions
DOI:
https://doi.org/10.7146/math.scand.a-114989Abstract
Let $X$ and $Y$ be compact convex sets such that their each extreme point is a weak peak point. We show that $\operatorname{ext} X$ is homeomorphic to $\operatorname{ext} Y$ provided there exists a small-bound isomorphism of the space $\mathfrak{A}(X,\mathbb{C} )$ of continuous affine complex functions on $X$ onto $\mathfrak{A}(Y,\mathbb{C} )$. Further, we generalize a result of Cengiz to the context of compact convex sets.
References
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Rao, T. S. S. R. K., Isometries of $A_\bf C(K)$, Proc. Amer. Math. Soc. 85 (1982), no. 4, 544–546. https://doi.org/10.2307/2044062
Spurný, J., Representation of abstract affine functions, Real Anal. Exchange 28 (2002/03), no. 2, 337–354. https://doi.org/10.14321/realanalexch.28.2.0337
Spurný, J., Borel sets and functions in topological spaces, Acta Math. Hungar. 129 (2010), no. 1-2, 47–69. https://doi.org/10.1007/s10474-010-9223-6
Alfsen, E. M., Compact convex sets and boundary integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 57, Springer-Verlag, New York-Heidelberg, 1971.
Amir, D., On isomorphisms of continuous function spaces, Israel J. Math. 3 (1965), 205–210. https://doi.org/10.1007/BF03008398
Behrends, E., $M$-structure and the Banach-Stone theorem, Lecture Notes in Mathematics, vol. 736, Springer, Berlin, 1979.
Browder, A., Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969.
Cambern, M., A generalized Banach-Stone theorem, Proc. Amer. Math. Soc. 17 (1966), 396–400. https://doi.org/10.2307/2035175
Cengiz, B., On topological isomorphisms of $C_0(X)$ and the cardinal number of $X$, Proc. Amer. Math. Soc. 72 (1978), no. 1, 105–108. https://doi.org/10.2307/2042544
Chu, C.-H. and Cohen, H. B., Isomorphisms of spaces of continuous affine functions, Pacific J. Math. 155 (1992), no. 1, 71–85.
Cohen, H. B., A bound-two isomorphism between $C(X)$ Banach spaces, Proc. Amer. Math. Soc. 50 (1975), 215–217. https://doi.org/10.2307/2040542
Cohen, H. B., A second-dual method for $C(X)$ isomorphisms, J. Functional Analysis 23 (1976), no. 2, 107–118. https://doi.org/10.1016/0022-1236(76)90069-0
Dostál, P. and Spurný, J., The minimum principle for affine functions with the point of continuity property and isomorphisms of spaces of continuous affine function, preprint arXiv:1801.07940 [math.FA], 2018.
Drewnowski, L., A remark on the Amir-Cambern theorem, Funct. Approx. Comment. Math. 16 (1988), 181–190.
Fabian, M., Habala, P., Hájek, P., Montesinos, V., and 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. https://doi.org/10.1007/978-1-4419-7515-7
Fleming, R. J. and Jamison, J. E., Isometries on Banach spaces. Vol. 2: Vector-valued function spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 138, Chapman & Hall/CRC, Boca Raton, FL, 2008.
Fremlin, D. H., Measure theory. Vol. 4: Topological measure spaces. Part I, II, Torres Fremlin, Colchester, 2006, corrected second printing.
Fuhr, R. and Phelps, R. R., Uniqueness of complex representing measures on the Choquet boundary, J. Functional Analysis 14 (1973), 1–27. https://doi.org/10.1016/0022-1236(73)90027-x
Hess, H.-U., On a theorem of Cambern, Proc. Amer. Math. Soc. 71 (1978), no. 2, 204–206. https://doi.org/10.2307/2042833
Hirsberg, B., Représentations intégrales des formes linéaires complexes, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1222–A1224.
Hustad, O., A norm preserving complex Choquet theorem, Math. Scand. 29 (1971), 272–278. https://doi.org/10.7146/math.scand.a-11053
Jarosz, K., Perturbations of Banach algebras, Lecture Notes in Mathematics, vol. 1120, Springer-Verlag, Berlin, 1985. https://doi.org/10.1007/BFb0076885
Jarosz, K. and Pathak, V. D., Isometries and small bound isomorphisms of function spaces, in “Function spaces (Edwardsville, IL, 1990)”, Lecture Notes in Pure and Appl. Math., vol. 136, Dekker, New York, 1992, pp. 241–271.
Koumoullis, G., A generalization of functions of the first class, Topology Appl. 50 (1993), no. 3, 217–239. https://doi.org/10.1016/0166-8641(93)90022-6
Lazar, A. J., Affine products of simplexes, Math. Scand. 22 (1968), 165–175. https://doi.org/10.7146/math.scand.a-10880
Ludv\'ık, P. and Spurný, J., Isomorphisms of spaces of continuous affine functions on compact convex sets with Lindelöf boundaries, Proc. Amer. Math. Soc. 139 (2011), no. 3, 1099–1104. https://doi.org/10.1090/S0002-9939-2010-10534-8
Ludv\'ık, P. and Spurný, J., Descriptive properties of elements of biduals of Banach spaces, Studia Math. 209 (2012), no. 1, 71–99. https://doi.org/10.4064/sm209-1-6
Lukeš, J., Malý, J., Netuka, I., and Spurný, J., Integral representation theory: Applications to convexity, banach spaces and potential theory, De Gruyter Studies in Mathematics, vol. 35, Walter de Gruyter & Co., Berlin, 2010.
Rao, T. S. S. R. K., Isometries of $A_\bf C(K)$, Proc. Amer. Math. Soc. 85 (1982), no. 4, 544–546. https://doi.org/10.2307/2044062
Spurný, J., Representation of abstract affine functions, Real Anal. Exchange 28 (2002/03), no. 2, 337–354. https://doi.org/10.14321/realanalexch.28.2.0337
Spurný, J., Borel sets and functions in topological spaces, Acta Math. Hungar. 129 (2010), no. 1-2, 47–69. https://doi.org/10.1007/s10474-010-9223-6
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Published
2019-10-19
How to Cite
Rondoš, J., & Spurný, J. (2019). Isomorphisms of spaces of affine continuous complex functions. MATHEMATICA SCANDINAVICA, 125(2), 270–290. https://doi.org/10.7146/math.scand.a-114989
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