The classical theory of regularity of embeddings of compact convex sets was developed in the 1970s, exclusively in the real case, and even there it does not appear to have been stated in its simplest form. We begin by revisiting this setting, showing that under a reasonable condition, every locally convex topological vector space E that contains and is spanned by a compact convex set lying in a hyperplane not passing through the origin, is a (specific) dual Banach space equipped with the weak* topology. Second, we establish the corresponding regularity theory for convex sets in complex LCTVS's. Third, we develop a theory of regular embeddings for complex noncommutative convex sets, in the sense of Davidson and Kennedy. Finally, we use the complex theory to derive a theory of regular embeddings for real noncommutative convex sets. Interestingly, at present there appears to be no direct route to the latter.
Classically, Jensen's Inequality asserts that if $X$ is a compact convex set, and $f:K\to \mathbb{R}$ is a convex function, then for any probability measure $μ$ on $K$, that $f(\text{bar}(μ))\le \int f\;dμ$, where $\text{bar}(μ)$ is the barycenter of $μ$. Recently, Davidson and Kennedy proved a noncommutative ("nc") version of Jensen's inequality that applies to nc convex functions, which take matrix values, with probability measures replaced by ucp maps. In the classical case, if $f$ is only a separately convex function, then $f$ still satisfies the Jensen inequality for any probability measu…
Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity properties of usual analytic functions) and admit a good difference-differential calculus. Noncommutative functions appear naturally in a large variety of settings: noncommutative algebra, systems and control, spectral theory, and free probability. Starting with pioneering work of J.L. Taylor, the theory was further developed by D.-V. Voiculescu, and established …
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…
给定g元幺一线性束L，设D_L为其正性域，即使L（X）半正定的所有大小的对称矩阵的所有g元组X的集合。由于L是幺一线性束，所以DL是具有内部的凸集，反之，具有内部的凸有界非交换半代数集都是DL形式的。本文的主要结果建立了凸半代数集上的完备非交换Nichtnegativstellenetrix。也就是说，一个非交换多项式p在D_L上是半正定的当且仅当它有一个带最优次数界的加权平方和表示：p = s^* s + \sum_j f_j^* L f_j，其中s，f_j是次数不大于大于1/2 deg（p）的非交换多项式的向量。这个非对易的结果与对易的情况形成了鲜明的对比，在对易的情况下，对s，f_j的度数没有控制，并且假设只有p是非负的，而不是p严格为正，产生一个干净的Positivstellenance是如此罕见，以至于这种情况是值得注意的。
This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set $D_L$, called a free Hilbert spectrahedron, of the linear operator inequality (LOI) $L(X)=A_0\otimes I+\sum_{j=1}^g A_{j}\otimes X_j\succeq 0,$ where $A_j$ are self-adjoint linear operators on a separable Hilbert space, $X_j$ matrices and $I$ is an identity matrix. If $A_j$ are matrices, then $L(X)\succeq 0$ is called a linear matrix inequality (LMI) and $D_L$ a free spectrahedron. For monic LMIs, i.e., $A_0=I$, and nc matrix-valued polyn…
