  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 by Curto and Fialkow for
path *-algebras is presented and non-commutative positivstellensatz is proved.
We aslo present an analog of the flat extension theorem of Curto and Fialkow
for this class of algebras.
