Positivstellensatz and flat functionals on path ∗-algebras

We consider the class of non-commutative ∗-algebras which are path algebras of doubles of quivers with the natural involutions. We study the problem of extending positive truncated functionals on such ∗-algebras. An analog of the solution of the truncated Hamburger moment problem (Curto and Fialkow, 1991 [Fia91]) for path ∗-algebras is presented and non-commutative positivstellensatz is proved. We also present an analog of the flat extension theorem of Curto and Fialkow for this class of algebras.