Moment property and positivity for some algebras of fractions

T. M. Bisgaard proved that the $*$-algebra C[z,z̄,1/zz̄] 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 z₁,...,zd. 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 R[x,y,1/(x²+y²)] (the hermitean part of C[z,z̄,1/zz̄]) and R[x,y,x²/(x²+y²),xy/(x²+y²)], all positive semidefinite elements are sums of squares. These results are used to prove that for the semigroup $*$-algebras of Z², N₀× Z and N₊:=\(k,n)∈ Z²:k+n≥ 0\, all positive semidefinite elements are sums of hermitean squares.