A full description of extreme points in 𝐶(Ω,𝐿^{𝜙}(𝜇))

Wisła, Marek

doi:10.1090/S0002-9939-1991-1072351-6

A full description of extreme points in $C(\Omega ,L^ \phi (\mu ))$
HTML articles powered by AMS MathViewer

by Marek Wisła PDF

Proc. Amer. Math. Soc. 113 (1991), 193-200 Request permission

Abstract:

Let ${L^\varphi }(\mu )$ be an Orlicz space endowed with the Luxemburg norm. The main result of this paper reads as follows: $f$ is an extreme point of the unit ball of the space of continuous functions from a compact Hausdorff space $\Omega$ into ${L^\varphi }(\mu )$ with the supremum norm if and only if the inverse image of the set of all extreme points of the unit ball of ${L^\varphi }(\mu )$ under $f$ is dense in $\Omega$.

References

R. M. Blumenthal, Joram Lindenstrauss, and R. R. Phelps, Extreme operators into $C(K)$, Pacific J. Math. 15 (1965), 747–756. MR 209862, DOI 10.2140/pjm.1965.15.747
A. Clausing and S. Papadopoulou, Stable convex sets and extremal operators, Math. Ann. 231 (1977/78), no. 3, 193–203. MR 467249, DOI 10.1007/BF01420240
Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
Ryszard Grząślewicz, Extreme points in $C(K,L^\varphi (\mu ))$, Proc. Amer. Math. Soc. 98 (1986), no. 4, 611–614. MR 861761, DOI 10.1090/S0002-9939-1986-0861761-3
R. Grząślewicz, H. Hudzik, and W. Kurc, Extreme and exposed points in Orlicz spaces, Canad. J. Math. 44 (1992), no. 3, 505–515. MR 1176367, DOI 10.4153/CJM-1992-032-3
A. J. Lazar, Affine functions on simplexes and extreme operators, Israel J. Math. 5 (1967), 31–43. MR 211246, DOI 10.1007/BF02771594
Wilhelmus Anthonius Josephus Luxemburg, Banach function spaces, Technische Hogeschool te Delft, Delft, 1955. Thesis. MR 0072440
Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 77107, DOI 10.2307/1969615
Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. MR 724434, DOI 10.1007/BFb0072210

Über eine gewisse Klasse von Räumen vom Typus

Susanna Papadopoulou, On the geometry of stable compact convex sets, Math. Ann. 229 (1977), no. 3, 193–200. MR 450938, DOI 10.1007/BF01391464
M. Sharir, A note on extreme elements in $A_{0}(K,\,E)$, Proc. Amer. Math. Soc. 46 (1974), 244–246. MR 374879, DOI 10.1090/S0002-9939-1974-0374879-0
Antonio Suárez Granero, Stable unit balls in Orlicz spaces, Proc. Amer. Math. Soc. 109 (1990), no. 1, 97–104. MR 1000154, DOI 10.1090/S0002-9939-1990-1000154-6
Dirk Werner, Extreme points in function spaces, Proc. Amer. Math. Soc. 89 (1983), no. 4, 598–600. MR 718980, DOI 10.1090/S0002-9939-1983-0718980-3
Marek Wisła, Extreme points and stable unit balls in Orlicz sequence spaces, Arch. Math. (Basel) 56 (1991), no. 5, 482–490. MR 1100574, DOI 10.1007/BF01200093
Cong Xin Wu and Shu Tao Chen, Extreme points and rotundity of Orlicz-Musielak sequence spaces, J. Math. Res. Exposition 8 (1988), no. 2, 195–200. MR 959844