We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (\emph{i.e.,} the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimpri\v c regarding $*$-orderability of quantum groups.

This paper develops a refinement of Lasserre's algorithm for optimizing a polynomial on a basic closed semialgebraic set via semidefinite programming and addresses an open question concerning the duality gap. It is shown that, under certain natural stability assumptions, the problem of optimization on a basic closed set reduces to the compact case.

The paper considers the relationship between positive polynomials, sums of squares and the multi-dimensional moment problem in the general context of basic closed semi-algebraic sets in real $n$-space. The emphasis is on the non-compact case and on quadratic module representations as opposed to quadratic preordering presentations. The paper clarifies the relationship between known results on the algebraic side and on the functional-analytic side and extends these results in a variety of ways.

A generalization of Schm\"udgen's Positivstellensatz is given which holds for any basic closed semialgebraic set in $\mathbb{R}^n$ (compact or not). The proof is an extension of W\"ormann's proof.

A preordering $T$ is constructed in the polynomial ring $A = \R [t_1,t_2, \dots]$ (countably many variables) with the following two properties: (1)~~For each $f \in A$ there exists an integer $N$ such that $-N \le f(P) \le N$ holds for all $P \in \Sper_T(A)$. (2)~~For all $f \in A$, if $N+f, N-f \in T$ for some integer $N$, then $f \in \R$. This is in sharp contrast with the Schm\"udgen-W\"ormann result that for any preordering $T$ in a finitely generated $\R$-algebra $A$, if property~(1) holds, then for any $f \in A$, $f > 0$ on $\Sper_T(A) \Rightarrow f \in T$. Also, adjoining to $A$ the square roots of the generators of $T$ yields a larger ring $C$ with these same two properties but with $\Sigma C^2$ (the set of sums of squares) as the preordering.