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The Truncated Moment Problem for Unital Commutative R-Algebras, Part II

Professor Raúl Curto; Department of Mathematics, University of Iowa

Let A be a unital commutative R-algebra, K a closed subset of the character space of A, and B a linear subspace 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 A, K, B and L which allow us to construct a seminormed structure on A, so as to exploit our previous result to get an integral representation for L.

Our main theorems allow us to extend some well-known results on the Classical Truncated Moment Problem, the Truncated Moment Problem for point processes, and the Subnormal Completion Problem for 2-variable weighted shifts.  We also analyze the relation between the Full and the Truncated Moment Problem in our general setting; we obtain a suitable generalization of Stochel's Theorem, which readily applies to Full Moment Problems for localized algebras.

The talk is based on joint work with Mehdi Ghasemi, Maria Infusino, and Salma Kuhlmann.