A strict Positivstellensatz for the Weyl algebra

Schmüdgen, Konrad

doi:10.1007/s00208-004-0604-4

Cited by 30 publications

(32 citation statements)

References 17 publications

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“…The aim of this section is to prove a variant of Schmüdgen's Strict Positivstellensatz for the Weyl algebra, see [16,Theorem 1.1]. We will refer to Schmüdgen's original proof several times.…”

Section: An Application Of the Intersection Theoremmentioning

confidence: 98%

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Maximal Quadratic Modules on ∗-rings

Cimpric

2007

Algebr Represent Theor

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We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to * -rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime * -ideal, that every maximal proper quadratic module in a Noetherian * -ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schmüdgen's Strict Positivstellensatz for the Weyl algebra: Let c be an element of the Weyl algebra W(d) which is not negative semidefinite in the Schrödinger representation. It is shown that under some conditions there exists an integer k and elements r 1 , . . . , r k ∈ W(d) such that k j=1 r j cr * j is a finite sum of hermitian squares. This result is not a proper generalization however because we don't have the bound k ≤ d.

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Section: An Application Of the Intersection Theoremmentioning

confidence: 98%

“…The study of quadratic modules in * -rings is suggested by the recent developments in noncommutative real algebraic geometry, see [1,5,8,16,17].…”

Section: Introductionmentioning

confidence: 99%

Maximal Quadratic Modules on ∗-rings

Cimpric

2007

Algebr Represent Theor

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“…The fraction algebras of the next two examples have been the main technical tools in the proofs of a strict Positivstellensatz in [25] and in [26].…”

Section: Example 14 Compact Quantum Spacesmentioning

confidence: 99%

Noncommutative Real Algebraic Geometry Some Basic Concepts and First Ideas

Schmüdgen

2008

Emerging Applications of Algebraic Geometry

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“…In this case, we would say p(X, X * ) ≻ 0, since there are no X satisfying q(X, X * ), and voila p ∈ QM(q) as the theorem says. A nonarchimedean Positivstellensatz for the Weyl algebra, which treats unbounded representations and eigenvalues of polynomial partial differential operators, is given in [Sch05].…”

Section: Theorem 52 ([Hm04]mentioning

confidence: 99%

Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

Helton

Klep

McCullough

2011

International Series in Operations Research &Amp; Management Science

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Abstract. One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the formulas naturally contain non-commutative polynomials in matrices. These polynomials depend only on the system layout and do not change with the size of the matrices involved, hence such problems are called "dimension-free". Analyzing dimension-free problems has led to the development recently of a non-commutative (nc) real algebraic geometry (RAG) which, when combined with convexity, produces dimension-free Semidefinite Programming. This article surveys what is known about convexity in the non-commutative setting and nc SDP and includes a brief survey of nc RAG. Typically, the qualitative properties of the non-commutative case are much cleaner than those of their scalar counterpartsvariables in R g . Indeed we describe how relaxation of scalar variables by matrix variables in several natural situations results in a beautiful structure.

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A strict Positivstellensatz for the Weyl algebra

Abstract: Let c be an element of the Weyl algebra W(d) which is given by a strictly positive operator in the Schrödinger representation. It is shown that, under some conditions, there exist elements b 1 , . . ., b d ∈ W(d) such that d j=1 b j cb * j is a finite sum of squares.

Cited by 30 publications

References 17 publications

Maximal Quadratic Modules on ∗-rings

Maximal Quadratic Modules on ∗-rings

Noncommutative Real Algebraic Geometry Some Basic Concepts and First Ideas

Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

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