Moment property and positivity for some algebras of fractions

T. M. Bisgaard proved that the *-algebra Cz, z, 1/zz 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 Rx, y, 1/ (x²+y²) (the hermitean part of Cz, z, 1/zz) and Rx, 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.