A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

Abstract. We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford algebras for a new approach to the problem. Our main result is that if −1 is not a sum of hermitian squares in the Clifford algebra of a hyperbolic polynomial, then its hyperbolicity cone is spectrahedral. Our result also has computational applications, since this sufficient condition can be checked with a single semidefinite program.

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“…by the second statement. This is well known to imply that 1 is an interior point, see for example [4].…”

Section: Generalized Clifford Algebrasmentioning

confidence: 95%

Hyperbolic Polynomials and Generalized Clifford Algebras

Netzer

Thom

2014

Discrete Comput Geom

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“…The quadratic module (A, Q) is archimedean if − a * a ∈ Q for any a ∈ A and large enough , or equivalently − a ∈ Q for any a ∈ A h and large enough . It is also enough to require this for generators of A only (see [7] for technical details).…”

Section: Universal C * -Algebras Of Quadratic Modulesmentioning

confidence: 99%

“…which follows by separation from the archimedean cone Q, and the GNS construction. As in [7], the (separated) completion of A with respect to · Q is denoted by C * (A, Q) and called the universal C * -algebra of (A, Q). We denote by ι : A → C * (A, Q) the canonical map with dense image.…”

Section: Universal C * -Algebras Of Quadratic Modulesmentioning

confidence: 99%

Quadratic modules, 𝐶*-algebras, and free convexity

Alekseev

Netzer

Thom

2019

Trans. Amer. Math. Soc.

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Given a quadratic module, we construct its universal C * -algebra, and then use methods and notions from the theory of C * -algebras to study the quadratic module. We define residually finite-dimensional quadratic modules, and characterize them in various ways, in particular via a Positivstellensatz. We give unified proofs for several existing strong Positivstellensätze, and prove some new ones. Our approach also leads naturally to interesting new examples in free convexity. We show that the usual notion of a free convex hull is not able to detect residual finite-dimensionality. We thus propose a new notion of free convexity, which is coordinate-free. We characterize semialgebraicity of free convex hulls of semialgebraic sets, and show that they are not always semialgebraic, even at scalar level. This also shows that the membership problem for quadratic modules has a negative answer in the non-commutative setup.

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“…Let Σ 2 C[Γ] denote the set of sums of hermitian squares in C[Γ]. The following appears for example as Example 3 in [5]: .…”

Section: Sums Of Squares In the Group Algebramentioning

confidence: 99%

Real closed separation theorems and applications to group algebras

Netzer¹,

Thom²

2013

Pacific J. Math.

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In this paper we prove a strong Hahn-Banach theorem: separation of disjoint convex sets by linear forms is possible without any further conditions, if the target field R is replaced by a more general real closed extension field. From this we deduce a general Positivstellensatz for * -algebras, involving representations over real closed fields. We investigate the class of group algebras in more detail. We show that the cone of sums of squares in the augmentation ideal has an interior point if and only if the first cohomology vanishes. For groups with Kazhdan's property (T) the result can be strengthened to interior points in the ℓ 1 -metric. We finally reprove some strong Positivstellensätze by Helton and Schmüdgen, using our separation method.

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A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

Cited by 24 publications

References 22 publications

Hyperbolic Polynomials and Generalized Clifford Algebras

Hyperbolic Polynomials and Generalized Clifford Algebras

Quadratic modules, 𝐶*-algebras, and free convexity

Real closed separation theorems and applications to group algebras

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