The Truncated Moment Problem for Unital Commutative R-Algebras

We investigate when a linear functional L defined on a linear subspace B of a unital commutative real algebra A admits an integral representation w.r.t. a positive Radon measure supported on a closed subset K of the character space of A. We provide a criterion for the existence of such a representation for L when A is equipped with a submultiplicative seminorm. We then build on this result to prove our main theorem for A not necessarily equipped with a topology. This allows us to extend well-known results on truncated moment problems.