A Note On A Choquet-deny-type Theorem

We present a Choquet-Deny-type theorem for downward filtering convex sets of continuous functions and show that the Identity Korovkin cone of a downward filtering convex cone S is exactly the uniform closure of S.434861868Choquet, G., Deny, J., Ensembles semi-réticulés et ensembles réticulés de fonctions continues (1957) J. Math. Pures Appl., 36 (9), pp. 179-189Flösser, H.O., Irmisch, R., Roth, W., Infimum-stable convex cones and approximation (1981) Proc. London Math. Soc., 42 (1-3), pp. 104-120Nachbin, L., Topology and Order (1976), Van Nostrand, Princeton, NJ, 1965reprinted by Keirgar, Huntington, NYNachbin, L., Elements of Approximation Theory (1976), Van Nostrand, Princeton, NJ, 1967reprinted by Krieger, Huntington, N. YPriestley, H.A., A Choquet-Deny theorem for affine functions on a Choquet simplex (1968) Proc. Edinburgh Math. Soc., 16 (2), pp. 325-327Prolla, J.B., The uniform closure of convex semi-lattices (1994) Approximation, Probability and Related Fields, pp. 413-421. , Plenum, New Yor