Moment property and positivity for some algebras of fractions

TL;DR

This paper extends the moment property to certain algebras of fractions in multiple variables, identifying conditions under which positive linear functionals are moments and when positive elements are sums of squares.

Contribution

It generalizes Bisgaard's result to multivariable polynomial algebras with specific denominators and characterizes positivity and sum-of-squares properties in related real and semigroup algebras.

Findings

Existence of $3d-2$ denominators for the moment property in $d$ variables.

Failure of the moment property with 3 linear denominators in 2 variables.

All positive semidefinite elements are sums of squares in certain real and semigroup algebras.

Abstract

T. M. Bisgaard proved that the $*$-algebra ${\bf C}[z,\overline{z},1/z\overline{z}]$ has the moment property, that is, each positive linear functional on this $*$-algebra is a moment functional. We generalize this result to polynomials in $d$ variables $z_1,...,z_d$. We prove that there exist $3d-2$ linear polynomials as denominators such that the corresponding $*$-algebra has the moment property, while for 3 linear polynomials in case $d=2$ the moment property always fails. Further, it is shown that for the real algebras ${\bf R}[x,y,1/(x^2+y^2)]$ (the hermitean part of ${\bf C}[z,\overline{z},1/z\overline{z}]$) and ${\bf R}[x,y,x^2/(x^2+y^2),xy/(x^2+y^2)]$, all positive semidefinite elements are sums of squares. These results are used to prove that for the semigroup $*$-algebras of ${\bf Z}^2$, ${\bf N}_0\times{\bf Z}$ and ${\mathsf N}_+:=\{(k,n)\in{\bf Z}^2:k+n\geq 0\}$, all positive semidefinite elements are sums of hermitean squares.