Affine Products of Simplexes.

Tensor products of compact convex sets

“…The following proposition was proved in [1,3,4] under the additional hypothesis that each S x was a simplex. PROPOSITION 1.3.…”

“…We first investigate further the extreme points of all three tensor products. This has been done before [1,3,4], but except for [3] it was assumed that both sets were simplexes. The proof of this proceeds by a series of simple lemmas.…”

“…Indeed, an arbitrary element of P x * 0 P* may be written (not uniquely) in the form g x 0 ^ + g 2 0 c? 2 + g 3 0 (di + cZ 8 ) + g 4 0 (d 2 + rf 3 ), where ^eP 2 (i = 1, 2, 3,4). Suppose that there is a net {g a } = {gΐ ®d 1 + g%…”

“…Those results which can be formulated for finite tensor products can, however, be readily proved by appropriate induction arguments and can then be extended by standard methods to infinite tensor products. The techniques for doing this are well known [1,3,4] so we have restricted ourselves to the simplest case in order to exhibit the essential ideas of the proofs. Similarly, these results can be extended fairly easily to sets of states which are invariant under appropriate actions of semigroups, as was done in [3].…”

Approximation par des opérateurs compacts ou faiblement compacts à valeurs dans $C(X)$

“…Dans le cas où V est un espace simplicial (resp. avec unité) la démons-tration peut être aménagée pour obtenir le résultat suivant, qui se trouve aussi dans [1] et [8]. …”