The Poulsen simplex
Annales de l'institut Fourier, Tome 28 (1978) no. 1 , p. 91-114
doi : 10.5802/aif.682
URL stable : http://www.numdam.org/item?id=AIF_1978__28_1_91_0

On démontre ici qu’il existe un seul simplexe métrisable $S$ dont les points extrémaux sont denses. Ce simplexe est homogène au sens que pour tout couple de face ${F}_{1}$, ${F}_{2}$ affinement homéomorphes, il existe un automorphisme de $S$ qui transforme ${F}_{1}$ en ${F}_{2}$. Tout simplexe métrisable est affinement homéomorphe à une face de $S$. L’ensemble des points extrémaux de $S$ est homéomorphe à l’espace de Hilbert ${\ell }_{2}$. On caractérise les matrices qui représentent $A\left(S\right)$.
It is proved that there is a unique metrizable simplex $S$ whose extreme points are dense. This simplex is homogeneous in the sense that for every 2 affinely homeomorphic faces ${F}_{1}$ and ${F}_{2}$ there is an automorphism of $S$ which maps ${F}_{1}$ onto ${F}_{2}$. Every metrizable simplex is affinely homeomorphic to a face of $S$. The set of extreme points of $S$ is homeomorphic to the Hilbert space ${\ell }_{2}$. The matrices which represent $A\left(S\right)$ are characterized.

### Bibliographie

[1] E.M. Alfsen, Compact convex sets and boundary integrals, Springer-Verlag, 1971. MR 56 #3615 | Zbl 0209.42601

[2] C. Bessaga and A. Pelczynski, Selected topics from infinite dimensional topology, Warsaw, 1975. Zbl 0304.57001

[3] A. B. Hansen and Y. Sternfeld, 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 53 #11351 | Zbl 0318.46036

[4] R. Haydon, A new proof that every polish space is the extreme boundary of a simplex, Bull. London Math, Soc., 7 (1975), 97-100. MR 50 #10778 | Zbl 0302.46003

[5] A. Lazar, Spaces of affine continuous functions on simplexes, A.M.S. Trans., 134 (1968), 503-525. MR 38 #1511 | Zbl 0174.17102

[6] A. Lazar, Affine product of simplexes, Math. Scand., 22 (1968), 165-175. MR 40 #4727 | Zbl 0176.42803

[7] A. Lazar and J. Lindenstrauss, Banach spaces whose duals are L1 spaces and their representing matrices. Acta Math., 120 (1971), 165-193. MR 45 #862 | Zbl 0209.43201

[8] W. Lusky, The Gurari space is unique, Arch. Math., 27 (1976), 627-635. MR 55 #6177 | Zbl 0338.46023

[9] W. Lusky, On separable Lindenstrauss spaces, J. Funct. Anal., 26 (1977), 103-120. MR 58 #12303 | Zbl 0358.46016

[10] E.T. Poulsen, A simplex with dense extreme points, Ann. Inst. Fourier, Grenoble, 11 (1961), 83-87. Numdam | MR 23 #A1224 | Zbl 0104.08402

[11] Y. Sternfeld, Characterization of Bauer simplices and some other classes of Choquet simplices by their representing matrices, to appear.

[12] P. Wojtaszczyk, Some remarks on the Gurari space, Studia Math., XLI (1972), 207-210. MR 46 #7860 | Zbl 0233.46024