Closure of the cone of sums of 2d-powers in commutative 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 ρ α (a) := |α(a)|, for α ∈ K and a ∈ R. In case K is compact, we also consider the topology induced by a K := sup α∈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, R 2d , 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[X 1 , . . . , X n ] with L(h 2d ) 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.

Infinite dimensional moment problem: open questions and applications

“…This approach has been extended to more general settings (see e.g. [6], [7], [43]) and recently also to the one of Problem 1.1 in [16], [19], [20], where the authors analyze integral representations of linear functionals continuous w.r.t. locally multiplicatively convex (lmc) topologies.…”

“…In this section we shortly present some closure results of the type (1.2) and their corresponding solutions for Problem 1.1 which have been recently developed for locally multiplicatively convex R−algebras (see [16] and [19]). In particular we are going to highlight some consequences of these results which led us to interesting open questions.…”

Moment Problem for Symmetric Algebras of Locally Convex Spaces

“…For instance, [11] and [24] deal with linear functionals continuous with respect to weighted norm topologies, generalizing [6] and [33]. In [10], [12] and [14] the authors analyze integral representations of linear functionals that are continuous with respect to locally multiplicatively convex topologies. [4, Theorem 2.1], [5], [7], [16], [34,Section 12.5], [17], [20] consider linear functionals on the symmetric algebra of a nuclear space under certain quasi-analyticity assumptions which are less restrictive than continuity.…”

On the Determinacy of the Moment Problem for Symmetric Algebras of a Locally Convex Space

“…There is a huge literature about moment problems belonging to this general set up (see, e.g., [3], [6], [7], [8], [9], [12], [15], [18]) and in particular several works have been devoted to the study of the moment problem for linear functionals defined on the symmetric algebra of certain locally convex spaces (see, e.g., [1, Chapter 5, Section 2], [2], [4], [10], [11], [13], [19,Section 12.5]). In [5] we deal with the general case of linear functionals defined on the symmetric algebra S(V ) of any locally convex space (V, τ ).…”