Closed convex hulls,contents, and K‐spectral states

Our general result says that the closed convex hull of a set K consists of barycentres of probability contents (i.e., finitely additive set functions) on K. (Here K can be any nonempty subset of any nonempty compact convex set in any real or complex locally convex Hausdorff vector space.) In the equivalent setting of dual spaces, we give a very handy analytic criterion for a linear functional to be in the closed convex hull of a given nonempty point‐wise bounded set K of linear functionals (under some mild additional assumption). This is the notion of a K‐spectral state. Our criterion enhances the Abstract Bochner Theorem for unital commutative Banach *‐algebras (which easily follows from our result), in that it allows us to prescribe the set K on which a representing content should live. The content can be chosen to be a Radon measure if K is weak* compact. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)