**Uniqueness of a positive extension in the complex moment problem and a representing measure**

A well known characterization of complex moment sequences on Sz.-Nagy semigroup N✕N (i.e. sequences admitting a representing measure) consist in requiring existence of a positive definite extensions on a larger semigroup of pairs of integers (m,n) with m+n nonnegative. In this talk we consider the question of determinacy for complex moment sequences with the connection to uniqueness of its positive definite extensions. This uniqueness is tightly related to the condition on the representing measures of the complex moment sequence having no atom at the point 0, which is in contrast with indeterminate Hamburger moment sequences (as they always admit a representing measure with atom at $0$). Considering examples of indeterminate sequences we employ representing measures supported on algebraic sets, which among others will be covered in the talk. The content of the talk is based on the joint work with J. Stochel and F.H. Szafraniec.