  The main result of the paper gives criteria for extendibility of sesquilinear
form-valued mappings defined on symmetric subsets of *-semigroups to positive
definite ones. By specifying this we obtain new solutions of:
  * the truncated complex moment problem,
  * the truncated multidimensional trigonometric moment problem,
  * the truncated two-sided complex moment problem, as well as
characterizations of unbounded subnormality and criteria for the existence of
unitary power dilation.
