It is proved that there is a unique metrizable simplex whose extreme points are dense. This simplex is homogeneous in the sense that for every 2 affinely homeomorphic faces and there is an automorphism of which maps onto . Every metrizable simplex is affinely homeomorphic to a face of . The set of extreme points of is homeomorphic to the Hilbert space . The matrices which represent are characterized.
On démontre ici qu’il existe un seul simplexe métrisable dont les points extrémaux sont denses. Ce simplexe est homogène au sens que pour tout couple de face , affinement homéomorphes, il existe un automorphisme de qui transforme en . Tout simplexe métrisable est affinement homéomorphe à une face de . L’ensemble des points extrémaux de est homéomorphe à l’espace de Hilbert . On caractérise les matrices qui représentent .
@article{AIF_1978__28_1_91_0, author = {Lindenstrauss, Joram and Olsen, Gunnar and Sternfeld, Y.}, title = {The {Poulsen} simplex}, journal = {Annales de l'Institut Fourier}, pages = {91--114}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {28}, number = {1}, year = {1978}, doi = {10.5802/aif.682}, mrnumber = {80b:46019a}, zbl = {0363.46006}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.682/} }
TY - JOUR AU - Lindenstrauss, Joram AU - Olsen, Gunnar AU - Sternfeld, Y. TI - The Poulsen simplex JO - Annales de l'Institut Fourier PY - 1978 SP - 91 EP - 114 VL - 28 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.682/ DO - 10.5802/aif.682 LA - en ID - AIF_1978__28_1_91_0 ER -
Lindenstrauss, Joram; Olsen, Gunnar; Sternfeld, Y. The Poulsen simplex. Annales de l'Institut Fourier, Volume 28 (1978) no. 1, pp. 91-114. doi : 10.5802/aif.682. http://archive.numdam.org/articles/10.5802/aif.682/
[1] Compact convex sets and boundary integrals, Springer-Verlag, 1971. | MR | Zbl
,[2] Selected topics from infinite dimensional topology, Warsaw, 1975. | Zbl
and ,[3] On the characterization of the dimension of a compact metric space K by the representing matrices of C(K), Israel. J. of Math., 22 (1975), 148-167. | MR | Zbl
and ,[4] A new proof that every polish space is the extreme boundary of a simplex, Bull. London Math, Soc., 7 (1975), 97-100. | MR | Zbl
,[5] Spaces of affine continuous functions on simplexes, A.M.S. Trans., 134 (1968), 503-525. | MR | Zbl
,[6] Affine product of simplexes, Math. Scand., 22 (1968), 165-175. | MR | Zbl
,[7] Banach spaces whose duals are L1 spaces and their representing matrices. Acta Math., 120 (1971), 165-193. | MR | Zbl
and ,[8] The Gurari space is unique, Arch. Math., 27 (1976), 627-635. | MR | Zbl
,[9] On separable Lindenstrauss spaces, J. Funct. Anal., 26 (1977), 103-120. | MR | Zbl
,[10] A simplex with dense extreme points, Ann. Inst. Fourier, Grenoble, 11 (1961), 83-87. | Numdam | MR | Zbl
,[11]
, Characterization of Bauer simplices and some other classes of Choquet simplices by their representing matrices, to appear.[12] Some remarks on the Gurari space, Studia Math., XLI (1972), 207-210. | MR | Zbl
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