Proper Analytic Free Maps

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain D' in g' variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of D'. Assuming that both domains contain 0, we show that if f:D->D' is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also g=g', then f is invertible and f^(-1) is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and D'.Comment: 17 pages, final version. To appear in the Journal of Functional Analysi

Matrix Convex Hulls of Free Semialgebraic Sets

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix convex sets $C$ and their projections $\hat C$. A free semialgebraic set which is convex as well as bounded and open can be represented as the solution set of a Linear Matrix Inequality (LMI), a result which suggests that convex free semialgebraic sets are rare. Further, Tarski's transfer principle fails in the free setting: The projection of a free convex semialgebraic set need not be free semialgebraic. Both of these results, and the importance of convex approximations in the optimization community, provide impetus and motivation for the study of the free (matrix) convex hull of free semialgebraic sets. This article presents the construction of a sequence $C^{(d)}$ of LMI domains in increasingly many variables whose projections $\hat C^{(d)}$ are successively finer outer approximations of the matrix convex hull of a free semialgebraic set $D_p=\{X: p(X)\succeq0\}$. It is based on free analogs of moments and Hankel matrices. Such an approximation scheme is possibly the best that can be done in general. Indeed, natural noncommutative transcriptions of formulas for certain well known classical (commutative) convex hulls does not produce the convex hulls in the free case. This failure is illustrated on one of the simplest free nonconvex $D_p$. A basic question is which free sets $\hat S$ are the projection of a free semialgebraic set $S$? Techniques and results of this paper bear upon this question which is open even for convex sets.Comment: 41 pages; includes table of contents; supplementary material (a Mathematica notebook) can be found at http://www.math.auckland.ac.nz/~igorklep/publ.htm

The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra

This article investigates matrix convex sets and introduces their tracial analogs which we call contractively tracial convex sets. In both contexts completely positive (cp) maps play a central role: unital cp maps in the case of matrix convex sets and trace preserving cp (CPTP) maps in the case of contractively tracial convex sets. CPTP maps, also known as quantum channels, are fundamental objects in quantum information theory. Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) L(x) = A_0 + A_1 x_1 + ... + A_g x_g > 0 and their matrix convex solution sets { X : L(X) is positive semidefinite }, called free spectrahedra. The Effros-Winkler Hahn-Banach Separation Theorem for matrix convex sets states that matrix convex sets are solution sets of LMIs with operator coefficients. Motivated in part by cp interpolation problems, we develop the foundations of convex analysis and duality in the tracial setting, including tracial analogs of the Effros-Winkler Theorem. The projection of a free spectrahedron in g+h variables to g variables is a matrix convex set called a free spectrahedrop. As a class, free spectrahedrops are more general than free spectrahedra, but at the same time more tractable than general matrix convex sets. Moreover, many matrix convex sets can be approximated from above by free spectrahedrops. Here a number of fundamental results for spectrahedrops and their polar duals are established. For example, the free polar dual of a free spectrahedrop is again a free spectrahedrop. We also give a Positivstellensatz for free polynomials that are positive on a free spectrahedrop.Comment: v2: 56 pages, reworked abstract and intro to emphasize the convex duality aspects; v1: 60 pages; includes an index and table of content

The convex Positivstellensatz in a free algebra

Given a monic linear pencil L in g variables let D_L be its positivity domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes making L(X) positive semidefinite. Because L is a monic linear pencil, D_L is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form D_L. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative polynomial p is positive semidefinite on D_L if and only if it has a weighted sum of squares representation with optimal degree bounds: p = s^* s + \sum_j f_j^* L f_j, where s, f_j are vectors of noncommutative polynomials of degree no greater than 1/2 deg(p). This noncommutative result contrasts sharply with the commutative setting, where there is no control on the degrees of s, f_j and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.Comment: 22 page

A Non-commutative Real Nullstellensatz Corresponds to a Non-commutative Real Ideal; Algorithms

This article takes up the challenge of extending the classical Real Nullstellensatz of Dubois and Risler to left ideals in a *-algebra A. After introducing the notions of non-commutative zero sets and real ideals, we develop three themes related to our basic question: does an element p of A having zero set containing the intersection of zero sets of elements from a finite set S of A belong to the smallest real ideal containing S? Firstly, we construct some general theory which shows that if a canonical topological closure of certain objects are permitted, then the answer is yes, while at the purely algebraic level it is no. Secondly for every finite subset S of the free *-algebra R of polynomials in g indeterminates and their formal adjoints, we give an implementable algorithm which computes the smallest real ideal containing S and prove that the algorithm succeeds in a finite number of steps. Lastly we provide examples of noncommutative real ideals for which a purely algebraic non-commutative real Nullstellensatz holds. For instance, this includes the real (left) ideals generated by a finite sets S in the *-algebra of n by n matrices whose entries are polynomials in one-variable. Further, explicit sufficient conditions on a left ideal in R are given which cover all the examples of such ideals of which we are aware and significantly more.Comment: Improved results compared to earlier version

Free bianalytic maps between spectrahedra and spectraballs in a generic setting

Given a tuple $E=(E_1,\dots,E_g)$ of $d\times d$ matrices, the collection of those tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $\| \sum E_j\otimes X_j\|\le 1$ is called a spectraball $\mathcal B_E$. Likewise, given a tuple $B=(B_1,\dots,B_g)$ of $e\times e$ matrices the collection of tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $I + \sum B_j\otimes X_j +\sum B_j^* \otimes X_j^*\succeq 0$ is a free spectrahedron $\mathcal D_B$. Assuming $E$ and $B$ are irreducible, plus an additional mild hypothesis, there is a free bianalytic map $p:\mathcal B_E\to \mathcal D_B$ normalized by $p(0)=0$ and $p'(0)=I$ if and only if $\mathcal B_E=\mathcal B_B$ and $B$ spans an algebra. Moreover $p$ is unique, rational and has an elegant algebraic representation.Comment: 19 page