Positivstellensatz and flat functionals on path *-algebras

TL;DR

This paper develops a non-commutative Positivstellensatz and flat extension theorem for path *-algebras, extending classical moment problem solutions to a non-commutative algebraic setting.

Contribution

It introduces a non-commutative Positivstellensatz and flat extension theorem for path *-algebras, generalizing classical results to non-commutative *-algebras.

Findings

Proved a non-commutative Positivstellensatz for path *-algebras.

Established an analog of the flat extension theorem for these algebras.

Presented an extension of the truncated Hamburger moment problem to non-commutative path *-algebras.

Abstract

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.