Noncommutative Real Algebraic Geometry Some Basic Concepts and First Ideas

Schmüdgen, Konrad

doi:10.1007/978-0-387-09686-5_9

Cited by 63 publications

(68 citation statements)

References 30 publications

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“…The referee has kindly pointed out that subsequent to [42], alternate proofs of Schmüdgen's Positivstellensatz were developed, based on properties of Archimedean quadratic modules. The approach of T. Wörmann appears in M. Marshall's monograph [29], and a unified exposition of this approach appears in K. Schmüdgen's recent expository paper [44]. In response to the referee's question, we believe that this alternate approach to Schmüdgen's Positivstellensatz cannot be adapted to give a simplified proof of Theorem 3.1.…”

Section: Remark 37mentioning

confidence: 91%

An analogue of the Riesz–Haviland theorem for the truncated moment problem

Curto

Fialkow

2008

Journal of Functional Analysis

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Let β ≡ β (2n) = {β i } |i| 2n denote a d-dimensional real multisequence, let K denote a closed subset of R d , and let P 2n := {p ∈ R[x 1 , . . . , x d ]: degp 2n}. Corresponding to β, the Riesz functional L ≡ L β : P 2n → R is defined by L( a i x i ) := a i β i . We say that L is K-positive if whenever p ∈ P 2n and p| K 0, then L(p) 0. We prove that β admits a K-representing measure if and only if L β admits a K-positive linear extensionL : P 2n+2 → R. This provides a generalization (from the full moment problem to the truncated moment problem) of the Riesz-Haviland theorem. We also show that a semialgebraic set solves the truncated moment problem in terms of natural "degree-bounded" positivity conditions if and only if each polynomial strictly positive on that set admits a degree-bounded weighted sum-of-squares representation.

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Section: Remark 37mentioning

confidence: 91%

An analogue of the Riesz–Haviland theorem for the truncated moment problem

Curto

Fialkow

2008

Journal of Functional Analysis

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“…The following is similar to Proposition 4 in [19]. For every element of F d we can define its degree as the total degree in X i and X * j .…”

Section: Mutually Inverse Homomorphisms Of Left M N (A)-modules Whichmentioning

confidence: 93%

Noncommutative Positivstellensätze for pairs representation-vector

Cimpric

2010

Positivity

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We study non-commutative real algebraic geometry for a unital associative * -algebra A viewing the points as pairs (π, v) where π is an unbounded * -representation of A on an inner product space which contains the vector v. We first consider the * -algebras of matrices of usual and free real multivariate polynomials with their natural subsets of points. If all points are allowed then we can obtain results for general A. Finally, we compare our results with their analogues in the usual (i.e. Schmüd-gen's) non-commutative real algebraic geometry where the points are unbounded * -representation of A.

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“…Some results on the decompositions of non-commutative polynomials which are positive in every representation into sums of hermitian squares in the free * -algebras were obtained in [Hel02], and, for some other classes of algebras, in [Put04,Sch08]. In the present paper we study the non-commutative moment problem for the path * -algebras A Γ associated with the finite graphs Γ .…”

Section: Theorem (See Curto and Fialkowmentioning

confidence: 95%

Positivstellensatz and flat functionals on path ∗-algebras

Popovych

2010

Journal of Algebra

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We consider the class of non-commutative * -algebras which are path algebras of doubles of quivers with the natural involutions. We study the problem of extending positive truncated functionals on such * -algebras. An analog of the solution of the truncated Hamburger moment problem (Curto and Fialkow, 1991 [Fia91]) for path * -algebras is presented and non-commutative positivstellensatz is proved. We also present an analog of the flat extension theorem of Curto and Fialkow for this class of algebras.

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Noncommutative Real Algebraic Geometry Some Basic Concepts and First Ideas

Cited by 63 publications

References 30 publications

An analogue of the Riesz–Haviland theorem for the truncated moment problem

An analogue of the Riesz–Haviland theorem for the truncated moment problem

Noncommutative Positivstellensätze for pairs representation-vector

Positivstellensatz and flat functionals on path ∗-algebras

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