Extending Positive Definiteness

The main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of ∗-semigroups to positive definite ones. By specifying the mappings in question we obtain new solutions of relevant issues in harmonic analysis concerning truncations of some important multivariate moment problems, like complex, two-sided complex and multidimensional trigonometric moment problems. In addition, unbounded subnormality and existence of unitary power dilation of several contractions is treated as an application of our general scheme. Introduction In [61] a fairly general concept of ∗-semigroups, which includes groups, ∗-algebras and quite a number of instances in between, as well as positive definite functions on them, has been originated by Sz.-Nagy on occasion of his general dilation theorem. On the other hand, the parallel notion of complete positivity, which is involved in the so-called Stinespring dilation theorem, turns out to be equivalent to the former; cf. [59]. This is so as long as bounded operators are on the dilation level. However, Sz.-Nagy’s approach offers a much more useful environment to work in, which is apparent when dealing with moment problems. If one goes beyond bounded operator dilations, the two notions, positive definiteness and complete positivity, still make sense but are no longer equivalent. This happens especially when one deals with moment problems on unbounded sets. In this case positivity understood as in the sense of Marcel Riesz and Haviland [43, 28, 29] plays a role, too. Therefore, there is a need for common treatment of these by means of forms over ∗-semigroups, as in [58]. The aforesaid cases are represented in our paper by unbounded subnormal operators and the complex Received by the editors December 22, 2008 and, in revised form, December 9, 2009. 2010 Mathematics Subject Classification. Primary 43A35, 44A60; Secondary 47A20, 47B20.