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Extreme points in spaces of continuous functions. (English) Zbl 0832.46030
Summary: We study the $$\lambda$$-property for the space $${\mathcal C}(T, X)$$ of continuous and bounded functions from a topological space $$T$$ into a strictly convex Banach space $$X$$. We prove that the $$\lambda$$-property for $${\mathcal C}(T, X)$$ is equivalent to an extension property for continuous functions of the pair $$(T, X)$$. We show also that, when $$X$$ has even dimension, the $$\lambda$$-property is equivalent to the fact that the unit ball of $${\mathcal C}(T, X)$$ is the convex hull of its extreme points and that this last property is true if $$X$$ is infinite- dimensional. As a result we get that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space can be expressed as the average of four retractions of the unit ball onto the unit sphere.

MSC:
 46E40 Spaces of vector- and operator-valued functions 46B20 Geometry and structure of normed linear spaces 46E15 Banach spaces of continuous, differentiable or analytic functions
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References:
 [1] Richard M. Aron and Robert H. Lohman, A geometric function determined by extreme points of the unit ball of a normed space, Pacific J. Math. 127 (1987), no. 2, 209 – 231. · Zbl 0662.46020 [2] Richard M. Aron, Robert H. Lohman, and Antonio Suárez, Rotundity, the C.S.R.P., and the \?-property in Banach spaces, Proc. Amer. Math. Soc. 111 (1991), no. 1, 151 – 155. · Zbl 0739.46003 [3] John Cantwell, A topological approach to extreme points in function spaces, Proc. Amer. Math. Soc. 19 (1968), 821 – 825. · Zbl 0175.13403 [4] J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353 – 367. · Zbl 0043.38105 [5] Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. · Zbl 0684.54001 [6] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. · Zbl 0362.46013 [7] David Oates, A sequentially convex hull, Bull. London Math. Soc. 22 (1990), no. 5, 467 – 468. · Zbl 0745.46031 [8] N. T. Peck, Extreme points and dimension theory, Pacific J. Math. 25 (1968), 341 – 351. · Zbl 0157.29603 [9] Yu. M. Smirnov, On the dimension of proximity spaces, Mat. Sb. N.S. 38(80) (1956), 283 – 302 (Russian). · Zbl 0070.18002 [10] Stanisław J. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (1986), no. 3, 437 – 444. · Zbl 0604.46019
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