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

Abstract

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 View the MathML source with L(h2d)⩾0 for each View the MathML source 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.