Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommutative matrix polynomials with constrained terms. Given a tuple of symmetric polynomials $Γ$, a free set is called $Γ$-convex if it closed under isometric conjugation by isometries intertwining $Γ$. We establish an Effros-Winkler Hahn-Banach separation theorem for $Γ$-convex sets; they are delineated by linear pencils in the coordinates of $Γ$ and the variables $x$.
Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. The notion of xy-convexity, a strengthening of partial convexity arising in connection with BMIs, is also considered. A characterization of xy-convex polynomials is given.
This paper concerns free analytic maps on noncommutative domains. These maps are free analogs of classical holomorphic functions in several complex variables, and are defined in terms of noncommuting variables amongst which there are no relations - they are free variables. Free analytic maps include vector-valued polynomials in free (noncommuting) variables and form a canonical class of mappings from one noncommutative domain D in say g variables to another noncommutative domain D' in g' variables. Motivated by determining the possibilities for mapping a nonconvex noncommutative domain to a co…
It was shown classically that matrix monotone and matrix convex functions must be real analytic by Löwner and Kraus respectively. Recently, various analogues have been found in several noncommuting variables. We develop a general framework for lifting automatic analyticity theorems in matrix analysis from one variable to several variables, the so-called "royal road theorem." That is, we establish the principle that the hard part of proving any automatic analyticity theorem lies in proving the one variable theorem. We use our main result to prove the noncommutative Löwner and Kraus theorems ove…
We investigate the norm of sums of independent vector-valued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class Sq^n in Sp(lq^m) with optimal order m = n^2. Using these embeddings we show the surprising fact that the sharp type (cotype) index in the sense of operator spaces for Lp[0,1] is min(p,p') (max(p,p')). Similar techniques are used to show that the operator space notions of B-convexity and K-convexity are equivalent.
Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful semifinite normal weight $ϕ$ and $\mathcal{N}$ be a von Neumann subalgebra of $\mathcal{M}$ such that the restriction of $ϕ$ to $\mathcal{N}$ is semifinite and such that $\mathcal{N}$ is invariant by the modular group of $ϕ$. Let $\mathcal{E}$ be the weight preserving conditional expectation from $\mathcal{M}$ onto $\mathcal{N}$. We prove the following inequality: \[\|x\|_p^2\ge\bigl \|\mathcal{E}(x)\bigr\|_p^2+(p-1)\bigl\|x-\mathcal{E}(x)\bigr\|_p^2, \qquad x\in L_p(\mathcal{M}),1<p\le2,\] which extends the celebrated Ball-Ca…
We establish the dual equivalence of the category of (potentially nonunital) operator systems and the category of pointed compact nc (noncommutative) convex sets, extending a result of Davidson and the first author. We then apply this dual equivalence to establish a number of results about operator systems, some of which are new even in the unital setting. For example, we show that the maximal and minimal C*-covers of an operator system can be realized in terms of the C*-algebra of continuous nc functions on its nc quasistate space, clarifying recent results of Connes and van Suijlekom. We als…
This is a survey article of geometric properties of noncommutative symmetric spaces of measurable operators $E(\mathcal{M},τ)$, where $\mathcal{M}$ is a semifinite von Neumann algebra with a faithful, normal, semifinite trace $τ$, and $E$ is a symmetric function space. If $E\subset c_0$ is a symmetric sequence space then the analogous properties in the unitary matrix ideals $C_E$ are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value functio…
Solutions to Laplace's equation are called harmonic functions. Harmonic functions arise in many applications, such as physics and the theory of stochastic processes. Of interest classically are harmonic polynomials, which have a simple classification. Further, the work of Reznick, building on the work of others, namely Sylvester, Clifford, Rosanes, Gundelfinger, Cartan, Maass and Helgason, has led to a classification of all polynomial solutions to a differential equation of arising from a homogeneous polynomial over an algebraically closed field. The definition of harmonicity can be extended t…
Noncommutative Euclidean spaces -- otherwise known as Moyal spaces or quantum Euclidean spaces -- are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the opportunity to highlight some of their peculiar features. For example, the theory of nonlinear partial differential equations has unexpected properties in this noncommutative setting. We develop elementary aspects of paradifferential calculus for noncommutative Euclidean spaces and give some applications to nonlinear evolution equations. We demonstrate how the anal…
In classical complex analysis analyticity of a complex function $f$ is equivalent to differentiability of its real and imaginary parts $u$ and $v$, respectively, together with the Cauchy-Riemann equations for the partial derivatives of $u$ and $v$. We extend this result to the context of free noncommutative functions on tuples of matrices of arbitrary size. In this context, the real and imaginary parts become so called real noncommutative functions, as appeared recently in the context of Löwner's theorem in several noncommutative variables. Additionally, as part of our investigation of real no…
We propose to view hermitian metrics on trivial holomorphic vector bundles $E\toΩ$ as noncommutative analogs of functions defined on the base $Ω$, and curvature as the notion corresponding to the Laplace operator or $\partial\overline\partial$. We discuss noncommutative generalizations of basic results of ordinary potential theory, mean value properties, maximum principle, Harnack inequality, and the solvability of Dirichlet problems.
Let $p$ be a polynomial in the non-commuting variables $(a,x)=(a_1,...,a_{g_a},x_1,...,x_{g_x})$. If $p$ is convex in the variables $x$, then $p$ has degree two in $x$ and moreover, $p$ has the form $p = L + Λ^T Λ,$ where $L$ has degree at most one in $x$ and $Λ$ is a (column) vector which is linear in $x,$ so that $Λ^TΛ$ is a both sum of squares and homogeneous of degree two. Of course the converse is true also. Further results involving various convexity hypotheses on the $x$ and $a$ variables separately are presented.
We investigate the convexity problem for the Parisi functional defined on the space of the so-called functional ordered parameters in the Sherrington-Kirkpatrick model. In the recent work of Panchenko [3], he proved that this functional is convex along one-sided directions with a probabilistic method. In this paper, we will study this problem with a PDE approach that simplifies his original proof and presents more general results.
Trigonometric bases in noncommutative $L_p(\mathbb{T}^d_θ)$ spaces and associated partial sum operators
Al-Farabi Kazakh National University（阿尔法拉比哈萨克国立大学）；Institute of Mathematics and Mathematical Modeling（数学与数学建模研究所）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）
Let a and x denote tuples of (jointly) freely noncommuting variables. A square matrix valued polynomial p in these variables is naturally evaluated at a tuple (A,X) of symmetric matrices with the result p(A,X) a square matrix. The polynomial p is symmetric if it takes symmetric values. Under natural irreducibility assumptions and other mild hypothesis, the article gives an algebraic certificate for symmetric polynomials p with the property that for sufficiently many tuples A, the set of those tuples X such that p(A,X) is positive definite is convex. In particular, p has degree at most two in x…
As a first step at developing a theory of noncommutative nonlinear elliptic partial differential equations, we analyze noncommutative analogues of Laplace's equation and its variants (some of the them nonlinear) over noncommutative tori. Along the way we prove noncommutative analogues of many results in classical analysis, such as Wiener's Theorem on functions with absolutely convergent Fourier series, and standard existence and non-existence theorems on elliptic functions. We show that many many classical methods, including the Maximum Principle, the direct method of the calculus of variation…
Let $Ω$ be a bounded ${\mathcal{C}}^{\infty}$-smoothly bounded domain in ${\mathbb{C}}^{n}.$ For such a domain we define a new notion between strict pseudo-convexity and pseudo-convexity: the size of the set $W$ of weakly pseudo-convex points on $\partial Ω$ is small with respect to Minkowski dimension: near each point in the boundary $\partial Ω,$ there is at least one complex tangent direction in which the slices of $W$ has a upper Minkowski dimension strictly smaller than $2.$ We propose to call this notion "strong pseudo-convexity"; this word is free since "strict pseudo-convexity" gets th…
Let S be an operator system -- a self-adjoint linear subspace of a unital C*-algebra A such that contains 1 and A=C*(S) is generated by S. A boundary representation for S is an irreducible representation πof C*(S) on a Hilbert space with the property that $π\restriction_S$ has a unique completely positive extension to C*(S). The set $\partial_S$ of all (unitary equivalence classes of) boundary representations is the noncommutative counterpart of the Choquet boundary of a function system $S\subseteq C(X)$ that separates points of X. It is known that the closure of the Choquet boundary of a func…
In the first part of the paper the natural scheme for proving noncommutative individual ergodic theorems for multiple sequences is described and applied to obtain results on unrestricted convergence of multiaverages. In the second part convergence of ergodic averages induced by several maps satisfying specific recurrence relations, including so-called Multi Free Group Partial Sums, is proved. This is the multiindexed version of results obtained earlier jointly with V.I.Chilin and S.Litvinov.
We study positivity in C*-modules and operator spaces using open tripotents, and an ordered version of the `noncommutative Shilov boundary'. Because of their independent interest, we also systematically study open tripotents and their properties.
This chapter is a tutorial on techniques and results in free convex algebraic geometry and free real algebraic geometry (RAG). The term free refers to the central role played by algebras of noncommutative polynomials R<x> in free (freely noncommuting) variables x=(x_1,...,x_g). The subject pertains to problems where the unknowns are matrices or Hilbert space operators as arise in linear systems engineering and quantum information theory. The subject of free RAG flows in two branches. One, free positivity and inequalities is an analog of classical real algebraic geometry, a theory of polynomial…
