The Arveson boundary of a Free Quadrilateral is given by a noncommutative variety

Abstract

Let $S{M}_n(\mathbb{R}{)}^g$ denote $g$-tuples of $n\times n$ real symmetric matrices and set $SM(\mathbb{R}{)}^g={\cup }_{n}S{M}_n(\mathbb{R}{)}^g$. A free quadrilateral is the collection of tuples $X\in SM(\mathbb{R}{)}^2$ which have positive semidefinite evaluation on the linear equations defining a classical quadrilateral. Such a set is closed under a rich class of convex combinations called matrix convex combination. That is, given elements $X=(X_1,\dots ,X_g)\in S{M}_{n_1}(\mathbb{R}{)}^g$ and $Y=(Y_1,\dots ,Y_g)\in S{M}_{n_2}(\mathbb{R}{)}^g$ of a free quadrilateral $\mathcal{Q}$, one has \[ V_1^T X V_1+V_2^T Y V_2 \in \mathcal{Q} \] for any contractions $V_1:{\mathbb{R}}^n\to {\mathbb{R}}^{n_1}$ and $V_2:{\mathbb{R}}^n\to {\mathbb{R}}^{n_2}$ satisfying $V_1^T{V}_1+V_2^T{V}_2=I_n$. These matrix convex combinations are a natural analogue of convex combinations in the dimension free setting. A natural class of extreme point for free quadrilaterals is free extreme points: elements of a free quadrilateral which cannot be expressed as a nontrivial matrix convex combination of elements of the free quadrilateral. These free extreme points serve as the minimal set which recovers a free quadrilateral through matrix convex combinations. In this article we show that the set of free extreme points of a free quadrilateral is determined by the zero set of a collection of noncommutative polynomials. More precisely, given a free quadrilateral $\mathcal{Q}$, we construct noncommutative polynomials $p_1,p_2,p_3,p_4$ such that a tuple $X\in SM(\mathbb{R}{)}^2$ is a free extreme point of a $\mathcal{Q}$ if and only if $X\in \mathcal{Q}$ and $p_i(X)=0$ for $i=1,2,3,4$ and $X$ is irreducible. In addition we establish several basic results for projective maps of free spectrahedra and for homogeneous free spectrahedra.

31 pages. Includes table of contents. An online appendix containing Mathematica notebooks which may be used to perform various computations described in the article is available at https://github.com/NCAlgebra/UserNCNotebooks/tree/master/Evert/FreeQuadrilaterals