The moment problem asks which linear functionals on certain vector spaces of functions are integration with respect to a measure. The famous Riesz representation theorem implies that the moment problem is solvable for every positive linear functional on the Banach space \(C(X)\) of continuous real-valued functions on a compact space \(X\). Haviland’s theorem states that a linear functional \(L\) on the real polynomial algebra \(\mathbb{R}[x_1,\ldots,x_n]\) is integration with respect to a Borel measure on some closed set \(K\subset\mathbb{R}^n\) if and only if \(L\) maps polynomials that are nonnegative on \(K\) to nonnegative reals. To see whether the moment problem has a solution in this case one considers the following question: Is there a finitely generated preordering on \(\mathbb{R}[x_1,\ldots,x_n]\) whose double dual cone consists of all polynomials nonnegative on \(K\)? By Schmüdgen’s theorem [K. Schmüdgen, “The \(K\)-moment problem for compact semi-algebraic sets”, Math. Ann. 289, No. 2, 203–206 (1991; Zbl 0744.44008)] the answer is yes in the case that \(K\) is a compact basic closed semialgebraic set in \(\mathbb{R}^n\).
Let us explain the above terms in the general setting of an \(\mathbb{R}\)-algebra \(A\). A preordering of \(A\) is a subset \(P\) of \(A\) satisfying \(P+P\subset P\), \(P\cdot P\subset P\) and \(A^2\subset P\) where \(A^2\) denotes the set of squares of \(A\). A cone in \(A\) is a subset \(C\) of \(A\) satisfying \(C+C\subset C\) and \(\mathbb{R}\cdot C\subset C\). Clearly a preordering is a cone. The dual cone \(C^\vee\) of \(C\) is the set of all linear functionals \(L:A\to\mathbb{R}\) such that \(L(v)\geq 0\) for all \(v\in C\). The double dual cone \(C^{\vee\vee}\) of \(C\) is the set of all \(v\in V\) such that \(L(v)\geq 0\) for all \(L\in C^\vee\). The double dual cone of \(C\) equals the closure \(\overline{C}\) of \(C\) in the finest locally convex topology on \(A\). A neighbourhood base of zero in this topology is given by the set of all convex, absorbent and symmetric subsets of \(A\). So the question above includes the problem of describing the closure of a preordering with respect to the finest locally convex topology. Note that the closure of a preordering is again a preordering.
In the paper under review, the authors do this for the more general notion of a quadratic module. A quadratic module of \(A\) is a subset \(Q\) of \(A\) satisfying \(Q+Q\subset Q\), \(A^2\cdot Q\subset Q\) and \(1\in Q\). A quadratic module is clearly a cone and the closure of a quadratic module is again a quadratic module. One of the main theorems is the following.
Let \(A\) and \(Q\) be finitely generated. If \(Q\) is stable then \(\overline{Q}=Q^\ddagger\) and \(\overline{Q}\) is stable.
The finitely generated quadratic module \(Q\) of \(A\) is said to be stable if for each finite-dimensional subspace \(V\) of \(A\) there exists a finite-dimensional subspace \(W\) of \(A\) such that every \(f\in Q\cap V\) is expressible as \(f=\sigma_0+\sigma_1g_1+\ldots+\sigma_sg_s\), where \(g_1,\ldots,g_s\) are the fixed generators of \(Q\) and the \(\sigma_i\) are sums of elements of \(W\). The sequential closure \(C^\ddagger\) of a cone \(C\) of \(A\) is the set of all elements of \(A\) that are the limit of some sequence in \(C\).
The authors show also a strengthening and an extension to quadratic modules of K. Schmüdgen’s fibre theorem [“On the moment problem of closed semi-algebraic sets”, J. Reine Angew. Math. 558, 225–234 (2003; Zbl 1047.47012)].