Intersection properties of balls in spaces of compact operators
Annales de l'Institut Fourier, Tome 28 (1978) no. 3, pp. 35-65.

Cet article est une étude des rapports entre les propriétés d’intersection et l’existence des grandes faces pour les boules d’un espace de Banach. D’après un résultat classique de Hanner un espace de dimension finie a la propriété d’intersection 3.2 (la “p.i. 3.2”) si et seulement si deux faces disjointes quelconques sont contenues dans deux hyperplans parallèles. Nous donnons ici une démonstration pour le cas général. Nous prouvons aussi que l’espace C(X,Y) des opérateurs compacts de X dans Y a la p.i. 3.2 si et seulement si X et Y ont la p.i. 3.2 et de plus ou bien X ou bien Y * est isométrique à un espace L 1 (μ).

We study the connection between intersection properties of balls and the existence of large faces of the unit ball in Banach spaces. Hanner’s result that a real space has the 3.2 intersection property if an only if disjoint faces of the unit ball are contained in parallel hyperplanes is extended to infinite dimensional spaces. It is shown that the space of compact operators from a space X to a space Y has the 3.2 intersection property if and only if X and Y have the 3.2 intersection property and either X or Y * is isometric to an L 1 (μ)-space.

@article{AIF_1978__28_3_35_0,
     author = {Lima, Asvald},
     title = {Intersection properties of balls in spaces of compact operators},
     journal = {Annales de l'Institut Fourier},
     pages = {35--65},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {28},
     number = {3},
     year = {1978},
     doi = {10.5802/aif.700},
     mrnumber = {80g:47048},
     zbl = {0347.46018},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.700/}
}
TY  - JOUR
AU  - Lima, Asvald
TI  - Intersection properties of balls in spaces of compact operators
JO  - Annales de l'Institut Fourier
PY  - 1978
SP  - 35
EP  - 65
VL  - 28
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.700/
DO  - 10.5802/aif.700
LA  - en
ID  - AIF_1978__28_3_35_0
ER  - 
%0 Journal Article
%A Lima, Asvald
%T Intersection properties of balls in spaces of compact operators
%J Annales de l'Institut Fourier
%D 1978
%P 35-65
%V 28
%N 3
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.700/
%R 10.5802/aif.700
%G en
%F AIF_1978__28_3_35_0
Lima, Asvald. Intersection properties of balls in spaces of compact operators. Annales de l'Institut Fourier, Tome 28 (1978) no. 3, pp. 35-65. doi : 10.5802/aif.700. http://archive.numdam.org/articles/10.5802/aif.700/

[1] E. Alfsen and E. Effros, Structure in real Banach spaces, Ann. of Math., 96 (1972), 98-173. | MR | Zbl

[2] S.J. Bernau and H.E. Lacey. Bicontractive projections and reordering of Lp-spaces. | Zbl

[3] J. Diestel, Geometry of Banach spaces-Selected Topics, Lecture notes in Mathematics, 485, Springer Verlag, 1975. | MR | Zbl

[4] A.J. Ellis, The duality of partially ordered normed linear spaces, J. London Math. Soc., 39 (1964), 713-744. | MR | Zbl

[5] O. Hanner, Intersection of translates of convex bodies, Math. Scand., 4 (1956), 65-87. | MR | Zbl

[6] J. Hennefeld, A decomposition for B(X)* and unique Hahn-Banach extensions, Pacific J. Math., 46 (1973), 197-199. | MR | Zbl

[7] B. Hirsberg and A.J. Lazar, Complex Lindenstrauss spaces with extreme points, Trans. Amer. Math. Soc., 186 (1973), 144-150. | MR | Zbl

[8] O. Hustad, Intersection properties of balls in complex Banach spaces whose duals are L1-spaces, Acta Math., 132 (1974), 283-313. | MR | Zbl

[9] E.H. Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der math. Wissenschaften, Band 208, Springer-Verlag, 1974. | MR | Zbl

[10] A.J. Lazar, Affine functions on simplexes and extreme operators, Israel J. Math., 5 (1967), 31-43. | MR | Zbl

[11] A.J. Lazar and J. Lindenstrauss, Banach spaces whose duals are L1-spaces and their representing matrices, Acta Math., 126 (1971), 165-193. | MR | Zbl

[12] A. Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc., 227 (1977), 1-62. | MR | Zbl

[13] A. Lima, Complex Banach spaces whose duals are L1-spaces, Israel J. Math., 24 (1976), 59-72. | MR | Zbl

[14] A. Lima, An application of a theorem of Hirsberg and Lazar, Math. Scand., 38 (1976), 325-340. | MR | Zbl

[15] J. Lindenstrauss, Extensions of compact operators, Memoirs Amer. Math. Soc., 48 (1964). | MR | Zbl

[16] J. Lindenstrauss and H.P. Rosenthal, The Lp-spaces, Israel J. Math., 7 (1969), 325-349. | MR | Zbl

[17] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Vol. 1, Ergebnisse der math., 92, Springer-Verlag, (1977). | MR | Zbl

[18] N.J. Nielsen and G.H. Olsen, Complex preduals of L1 and subspaces of 1n∞(C), Math. Scand., 40 (1977), 271-287. | MR | Zbl

[19] G.H. Olsen, Edwards separation theorem for complex Lindenstrauss spaces with applications to selection and embedding theorems, Math. Scand., 38 (1975), 97-105. | MR | Zbl

[20] M. Sharir, A note on extreme elements in A0(K,E), Proc. Amer. Math. Soc., 46 (1974), 244-246. | MR | Zbl

[21] M. Sharir, Extremal structure in operator spaces, Trans. Amer. Math. Soc., 186 (1973), 91-111. | MR | Zbl

[22] R.M. Blumenthal, J. Lindenstrauss and R.R. Phelps, Extreme operators into C(K), Pacific J. Math., 15 (1965), 747-756. | MR | Zbl

[23] H. Fakhoury, Préduaux de L-espaces et éléments extrémaux, C.R. Acad. Sci., Paris Sér A-B, 272 (1971), A1703-A1706. | MR | Zbl

[24] H. Fakhoury, Approximation par des opérateurs compacts ou faiblement compacts à valeurs dans C(X), C.R. Acad. Sci., Paris, t. 283 (1976) Série A, 615-618. | MR | Zbl

[25] G. Choquet and P.M. Meyer, Existence et unicité des représentations intégrales dans les convexes compacts quelconques, Ann. Inst. Fourier, Grenoble, 13 (1963), 133-154. | Numdam | MR | Zbl

[26] M. Hasumi, The extension property of complex Banach spaces, Tohoku Math. J. (sec. series), 10 (1958), 135-142. | MR | Zbl

[27] H.H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der math. Wissenschaften, Band 215, Springer-Verlag, 1974. | MR | Zbl

[28] V. Zizler, On some extremal problems in Banach spaces, Math. Scand., 32 (1973), 214-224. | MR | Zbl

[29] E. Alfsen, Compact convex sets and boundary integrals, Ergebnisse der math., 57, Springer Verlag, 1971. | MR | Zbl

Cité par Sources :