Abstract. We consider the problem of determining the closure M of a quadratic module M in a commutative R-algebra with respect to the finest locally convex topology. This is of interest in deciding when the moment problem is solvable [26] [27] and in analyzing algorithms for polynomial optimization involving semidefinite programming [12]. The closure of a semiordering is also considered, and it is shown that the space Y M consisting of all semiorderings lying over M plays an important role in understanding the closure of M . The result of Schmüdgen for preorderings in [27] is strengthened and extended to quadratic modules. The extended result is used to construct an example of a non-archimedean quadratic module describing a compact semialgebraic set that has the strong moment property. The same result is used to obtain a recursive description of M which is valid in many cases.In Section 1 we consider the general relationship between the closure C and the sequential closure C ‡ of a subset C of a real vector space V in the finest locally convex topology. We are mainly interested in the case where C is a cone in V . We consider cones with non-empty interior and cones satisfying C ∪ −C = V .In Section 2 we begin our investigation of the closure M of a quadratic module M of a commutative R-algebra A; the focus is on finitely generated quadratic modules in finitely generated algebras. The closure of a semiordering Q of A is also considered, and it is shown that the space Y M consisting of all semiorderings of A lying over M plays an important role in understanding the closure of M; see Propositions 2.2, 2.3 and 2.4. The result of Schmüdgen for preorderings in [27] is strengthened and extended to quadratic modules; see Theorem 2.8.In Section 3 we consider the case of quadratic modules that describe compact semialgebraic sets. We use Theorem 2.8 to deduce various results; see Theorems 3.1 and 3.4; and also to construct an example where K M is compact, M satisfies the strong moment property (SMP), but M is not archimedean; see Example 3.7.
This article extends the classical Real Nullstellensatz of Dubois and Risler to left ideals in free *‐algebras ℝ 〈 x, x* 〉 with x=(x1, …, xn). First, we introduce notions of the (noncommutative) zero set of a left ideal and of a real left ideal. We prove that every element from ℝ 〈 x, x* 〉 whose zero set contains the intersection of zero sets of elements from a finite subset S of ℝ 〈 x, x* 〉 belongs to the smallest real left ideal containing S. Next, we give an implementable algorithm, which for every finite S⊂ℝ 〈 x, x* 〉, computes the smallest real left ideal containing S, and prove that the algorithm succeeds in a finite number of steps. Our definitions and some of our results also work for other *‐algebras. As an example, we treat real left ideals in Mn(ℝ[x1]).
Abstract. We present a new approach to noncommutative real algebraic geometry based on the representation theory of C * -algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand-Naimark representation theorem for commutative C * -algebras. A noncommutative version of Gelfand-Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.
Abstract. An involution # on an associative ring R is formally real if a sum of nonzero elements of the form r # r where r ∈ R is nonzero. Suppose that R is a central simple algebra (i.e. R = M n (D) for some integer n and central division algebra D) and # is an involution on R of the form r # = a −1 r * a, where * is some transpose involution on R and a is an invertible matrix such that a * = ±a. In section 1 we characterize formal reality of # in terms of a and * | D . In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on D = (K/F, Φ) that extend to a formally real involution on the split algebraEvery such involution is formally real but we show that there exist formally real involutions on D which are not of this form. In particular, there exists a formally real involution # for which the hermitian trace form x → tr(x # x) is not positive semidefinite. ǫ-hermitian cones on central simple algebrasWe say that an involution * on a central simple algebra R is formally real if any finite sum of nonzero elements of the form rr * where r ∈ R is nonzero. In this section we introduce our main technical tool for the study of formally real involutions -the notion of an ǫ-hermitian cone. The precise relationship between ǫ-hermitian cones and formally real involutions is explained by Corollary 4.Recall that a central simple algebra is a full matrix ring over a central division algebra. Let R be a central simple F -algebra with involution * and ǫ ∈ F such that ǫǫ * = 1. An element a ∈ R is ǫ-hermitian if ǫa * = a. The set of all ǫ-hermitian elements in R will be denoted by S ǫ (R). A subset M of S ǫ (R) such that M + M ⊆ M, aMa * ⊆ M for every a ∈ R and M ∩ −M = {0} will be called an ǫ-hermitian cone on R.
We define and study preorderings and orderings on rings of the form $M_n(R)$ where $R$ is a commutative unital ring. We extend the Artin-Lang theorem and Krivine-Stengle Stellens\"atze (both abstract and geometric) from $R$ to $M_n(R)$. While the orderings of $M_n(R)$ are in one-to-one correspondence with the orderings of $R$, this is not true for preorderings. Therefore, our theory is not Morita equivalent to the classical real algebraic geometry
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