An open subset$D$of$R$^{$d$},$d≧2$, is called Poissonian iff every bounded harmonic function on the set is a Poisson integral of a bounded function on its boundary. We show that the intersection of two Poissonian open sets is itself Poissonian and give a sufficient condition for the union of two Poissonian open sets to be Poissonian. Some necessary and sufficient conditions for an open set to be Poissonian are also given. In particular, we give a necessary and sufficient condition for a Greenian$D$to be Poissonian in terms of its Martin boundary.

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Let ℒ($H$) denote the space of operators on a Hilbert space$H$. We show that the extreme points of the unit ball of the space of continuous functions$C(K, ℒ(H))$($K$-compact Hausdorff) are precisely the functions with extremal values. We show also that these extreme points are (a) strongly exposed if and only if dim$H$<∞ and card$K$<∞, (b) exposed if and only if$H$is separable and$K$carries a strictly positive measure.

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