Non-commutative moment problems

Abstract

We are concerned with two problems, called the generalized moment problem (GMP) and the quantum moment problem (QMP). Both problems can be considered as non-commutative versions of the classical moment problem et G be a connected Lie group and #7B-E(g) the universal enveloping algebra of the Lie algebra of G. Suppose f is a given linear functional on #7B-E(g) and ν is a fixed unitary representation of G. We then ask: (GMP) When do there exist a unitary representation U of G and a cyclic C ∞ -vector φ for U such that f(x)= , x∈#7B-E(g)? (QMP) When is f of the form f(x)=trdV(x)ρ, x∈#7B-E(g), where ρ is an operator such that dV(x)ρ is of trace class for every x∈#7B-E(g)?