T. M. Bisgaard [1] proved that the \(*\) -algebra \(\mathbb {C}[z,\overline{z},z^{-1},\overline{z}^{-1}]\) 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 \(z_1,\dots ,z_d.\) 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 \(\mathbb {R}[x,y,\frac{1}{x^2+y^2}]\) (the hermitean part of \(\mathbb {C}[z,\overline{z},z^{-1},\overline{z}^{-1}]\) ) and \(\mathbb {R}[x,y,\frac{x^2}{x^2+y^2},\frac{xy}{x^2+y^2}]\) , all positive semidefinite elements are sums of squares. These results are used to prove that for the semigroup \(*\) -algebras of \(\mathbb {Z}^2\) , \(\mathbb {N}_0\times \mathbb {Z}\) and \({\textsf{N}}_+:=\{(k,n)\in \mathbb {Z}^2:k+n\ge 0\}\) , all positive semidefinite elements are sums of hermitean squares.
