# Analysis Seminar Friday, October 19, 2018 — 3:00 PM EDT

Ben Passer, Department of Pure Mathematics, University of Waterloo

"Containment of Matrix Convex Sets"

This talk contains joint work with Orr Shalit and Baruch Solel.

A set $S \subseteq \bigcup_{n=1}^\infty M_n(\mathbb{C})^d$ is called matrix convex if it is closed under direct sums and the application of unital completely positive maps. We may view $S$ as a disjoint union of its levels $S_n \subseteq M_n(\mathbb{C})^d$, each of which is convex and is closed under simultaneous unitary conjugations. Following work of Davidson, Dor-On, Shalit, and Solel, we seek to determine how much information about $S$ is contained in its first level, $S_1 = K$.

If $K$ is a compact convex set in $\mathbb{C}^d$, then there could be many matrix convex sets which have $K$ as the first level. However, there are always two extremal options $\mathcal{W}^\text{min}(K)$ and $\mathcal{W}^{\text{max}}(K)$. We study the disparity between these two matrix convex sets by searching for an optimal constant $C$ with
\begin{equation*}
\mathcal{W}^{\text{max}}(K) \subseteq C \cdot \mathcal{W}^{\text{min}}(K).
\end{equation*}

Among our results, we find an optimal scaling constant when $K$ is an $\ell^p$ ball, and we completely characterize when the constant $C = 1$ may be achieved. In doing so, we make use of theorems in convex geometry, and we find that it suffices to check a single matrix level $n = 2^{d-1}$.

Time permitting (which occurs with probability zero), I will discuss related/generalized problems concerning products of matrix convex sets, dilation theory, and extreme points.

MC 5417

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