Noncommutative Real Algebraic Geometry Research Articles

The Connes embedding problem: A guided tour

The Connes embedding problem (CEP) is a problem in the theory of tracial von Neumann algebras and asks whether or not every tracial von Neumann algebra embeds into an ultrapower of the hyperfinite II 1 _1 factor. The CEP has had interactions with a wide variety of areas of mathematics, including C ∗ \mathrm {C}^* -algebra theory, geometric group theory, free probability, and noncommutative real algebraic geometry, to name a few. After remaining open for over 40 years, a negative solution was recently obtained as a corollary of a landmark result in quantum complexity theory known as MIP ∗ = RE \operatorname {MIP}^*=\operatorname {RE} . In these notes, we introduce all of the background material necessary to understand the proof of the negative solution of the CEP from MIP ∗ = RE \operatorname {MIP}^*=\operatorname {RE} . In fact, we outline two such proofs, one following the “traditional” route that goes via Kirchberg’s QWEP problem in C ∗ \mathrm {C}^* -algebra theory and Tsirelson’s problem in quantum information theory and a second that uses basic ideas from logic.

Finsler's Lemma for matrix polynomials

Finsler's Lemma characterizes all pairs of symmetric n×n real matrices A and B which satisfy the property that vTAv>0 for every nonzero v∈Rn such that vTBv=0. We extend this characterization to all symmetric matrices of real multivariate polynomials, but we need an additional assumption that B is negative semidefinite outside some ball. We also give two applications of this result to Noncommutative Real Algebraic Geometry which for n=1 reduce to the usual characterizations of positive polynomials on varieties and on compact sets.

About the Connes embedding conjecture

In his celebrated paper in 1976, A. Connes casually remarked that any finite von Neumann algebra ought to be embedded into an ultraproduct of matrix algebras, which is now known as the Connes embedding conjecture or problem. This conjecture became one of the central open problems in the field of operator algebras since E. Kirchberg’s seminal work in 1993 that proves it is equivalent to a variety of other seemingly totally unrelated but important conjectures in the field. Since then, many more equivalents of the conjecture have been found, also in some other branches of mathematics such as noncommutative real algebraic geometry and quantum information theory. In this note, we present a survey of this conjecture with a focus on the algebraic aspects of it.

Noncommutative Positivstellensätze for pairs representation-vector

We study non-commutative real algebraic geometry for a unital associative *-algebra \({\mathcal {A}}\) viewing the points as pairs (π, v) where π is an unbounded *-representation of \({\mathcal A}\) 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 \({\mathcal {A}}\). Finally, we compare our results with their analogues in the usual (i.e. Schmüdgen’s) non-commutative real algebraic geometry where the points are unbounded *-representation of \({\mathcal {A}}\).

A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming

A method for computing global minima of real multivariate polynomials based on semidefinite programming was developed by N.Z. Shor, J.B. Lasserre and P.A. Parrilo. The aim of this article is to extend a variant of their method to noncommutative symmetric polynomials in variables X and Y satisfying Y X − X Y = 1 and X * = X , Y * = − Y . Global minima of such polynomials are defined and showed to be equal to minima of the spectra of the corresponding differential operators. We also discuss how to exploit sparsity and symmetry. Several numerical experiments are included. The last section explains how our theory fits into the framework of noncommutative real algebraic geometry.

A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

AbstractWe 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.

Maximal Quadratic Modules on ∗-rings

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 $\mathcal{W}(d)$ which is not negative semidefinite in the Schrödinger representation. It is shown that under some conditions there exists an integer k and elements $r_1,\ldots,r_k \in \mathcal{W}(d)$ such that ∑ j=1 k r j c r 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.

Orderings of higher level on noetherian rings

The noncommutative algebraic geometry has found fruitful applications in quantum geometry. Similar applications are expected to be found for its younger sister the noncommutative real algebraic geometry One of the basic results in real algebraic geometry is the Positivestellensatz. The original results of Dubois and Risler (see section 3.3 of [13]) have been extended in many directions. We refer to [14], [1], [2], [3] for commutative rings and [9], [4] for associative rings. The aim of this paper is to prove the higher level Posit ivstellensatz for noncommutative Noetherian rings. Our proof depends on the intersection theorem for orderings of higher level on skew fields ([11], Theorem 3.13). The general case of orderings of higher level on associative rings remains open.