Multivariable Moment Problems

In this paper we solve moment problems for Poisson transforms and, more generally, for completely positive linear maps on unital C∗-algebras generated by “universal” row contractions associated with F+n , the free semigroup with n generators. This class of C ∗-algebras includes the Cuntz-Toeplitz algebra C∗(S1, . . . , Sn) (resp. C ∗(B1, . . . , Bn)) generated by the creation operators on the full (resp. symmetric, or anti-symmetric)) Fock space with n generators. As consequences, we obtain characterizations for the orbits of contractive Hilbert modules over complex free semigroup algebras such as CF+n , C[z1, . . . , zn], and, more generally, the quotient algebra CF+n /J , where J is an arbitrary two-sided ideal of CF + n . All these results are extended to the generalized Cuntz algebra O(∗ni=1G + i ), where G + i are the positive cones of discrete subgroups G+i of the real line R. Moreover, we characterize the orbits of Hilbert modules over the quotient algebra C ∗ni=1 G + i /J , where J is an arbitrary two-sided ideal of the free semigroup algebra C ∗ni=1 G + i .