The Truncated Moment Problem for Unital Commutative R-Algebras

Let A be a unital commutative R-algebra, B a linear subspace of A and K a closed subset of the character space of A. For a linear functional L: B --> R, we investigate conditions under which L admits an integral representation with respect to a positive Radon measure supported in K. When A is equipped with a submultiplicative seminorm, we employ techniques from the theory of positive extensions of linear functionals to prove a criterion for the existence of such an integral representation for L. When no topology is prescribed on A, we identify suitable assumptions on B, A, L and K which allow us to construct a seminormed structure on A, so as to exploit our previous result to get an integral representation for L. We then use our main theorems to obtain, as applications, several well known results on the classical truncated moment problem, the moment problem for point processes, and the subnormal completion problem for 2-variable weighted shifts.