Closure of the cone of sums of 2d-powers in commutative real topological algebras

Let R be a unitary commutative R-algebra and K ⊆ X(R) = Hom(R,R), closed with respect to the product topology. We consider R endowed with the topology T (K), induced by the family of seminorms ρα(a) := |α(a)|, for α ∈ K and a ∈ R. In case K is compact, we also consider the topology induced by a K := supα∈K |α(a)| for a ∈ R. If K is Zariski dense, then those topologies are Hausdorff. In this paper we prove that the closure of the cone of sums of 2d-powers, R2d , with respect to those two topologies is equal to Psd(K) := {a ∈ R: α(a) 0, for all α ∈ K}. In particular, any continuous linear functional L on the polynomial ring R = R[X] = R[X1, . . . , Xn] with L(h2d ) 0 for each h ∈ R[X] is integration with respect to a positive Borel measure supported on K. Finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies. © 2012 Elsevier Inc. All rights reserved.