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By A. Hartkämper, H. Neumann

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E. e. semi-exposed faces implies either F = K A(K) K = co(F~ or if G) G = K. In connection with Theorem 6 we recall that a unital B -algebra is commutative if and only if Let let K A is a lattice. be a function algebra on a compact Hausdorff space-(~, be the state space of split faces of of A(K) Z A and let Z = co(K ~ -iK) . The are also connected with the algebraic structure A , as the following result shows (cf. [14] and [15]) • Theorem 7- Let F be a closed face of is a split face of set for Z A ; (ii) F if and only if K .

We shall say that X is a simplex if every x in X is the barycenter of a unique maximal measure. 1. EXAMPLE. Let ~ b e any open set in R 2 with compact closure. Let A ( ~ ) continuous functions on ~ , harmonic in ~ , then A ( ~ ) be the space of is a simplex space. 2. THEOREM. The following statements are equivalent : (a) X is a simplex ; (b) E is a simplex space ; (c) E' is lattice (when ordered by the positive cone ~) ; (d) Edward's separation property : For any f, -g in Qs(X) with f sg, there exists h~A c (X) with f ~ h 4 g .

And sources, see the Chapter 7 of [6]. This book of Luxemburg and Zaanen is so complete that it becomes confusing. 2.. 3. is due to Kakutani, see [5]. 4. B. Davies. See theorem IO of [3]. For another proof, see [7~, theorem 1. BIBLIOGRAPHIE [U A. GOULLET de RUGY. La th~orie des cSnes bir~ticulgs. Ann. Inst. Fourier (Grenoble), 2 1 (4), 1-64, 1971 [2j A. GOULLET de RUGY. La structure id~ale des M-espaces. J. Math. Pures et Appl. B. DAVIES. The Choquet theory and representation of ordered Banach spaces.

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