On a fundamental theorem of Choquet theory (Q1064515)

In 1977 \textit{M. M. Neumann} [Arch. Math. 28, 182-192 (1977; Zbl 0366.46015)] after \textit{B. Fuchssteiner} [J. Funct. Analysis 17, 377-387 (1974; Zbl 0292.46026)] obtained a general theorem on boundaries for cones of functions which constitutes a wide extension of the general Choquet-Bishop-de Leeuw theorem. But his proof was technically complicated because of the extensive use of the state space embedding and Gelfand transform method. In our paper we present a short and transparent proof of Neumann's result and even of a much more precise new result. Our essential contribution is a new method, which consists of the construction of appropriate extensions for cones of functions.