The truncated tracial moment problem

TL;DR

This paper extends classical moment matrix results to the non-commutative, tracial setting, providing conditions for representing sequences as matrix traces and exploring trace-positive polynomials.

Contribution

It introduces tracial analogs of moment matrix theorems, characterizes when sequences have matrix trace representations, and applies the theory to trace-positive polynomials.

Findings

Tracial sequences can be represented by matrix traces if their moment matrix is positive semidefinite and finite rank.

A truncated tracial sequence admits a matrix trace representation if it has a flat extension.

The theory helps analyze trace-positive polynomials in non-commuting variables.

Abstract

We present tracial analogs of the classical results of Curto and Fialkow on moment matrices. A sequence of real numbers indexed by words in non-commuting variables with values invariant under cyclic permutations of the indexes, is called a tracial sequence. We prove that such a sequence can be represented with tracial moments of matrices if its corresponding moment matrix is positive semidefinite and of finite rank. A truncated tracial sequence allows for such a representation if and only if one of its extensions admits a flat extension. Finally, we apply the theory via duality to investigate trace-positive polynomials in non-commuting variables.