2017/06/30 by Benjamin Passer, Orr Shalit, Baruch Solel · 1 citation
Mathematics · #math.OA #msc:47A20 #msc:47A13 #msc:46L07 #msc:47L25
paper · pdf · doi:10.1016/j.jfa.2017.11.011
published as J. Funct. Anal. 274:11 (2018), 3197-3253 · 47 pages, 5 figures. Version 2 corrects minor errors and clarifies some of the theorem statements. Remarks have been added throughout the text, and section 4 has been expanded a bit. To appear in Journal of Functional Analysis
arxiv created 2017/11/27 · arxiv updated 2019/07/04
To every convex body K ⊆ ℝd, one may associate a minimal matrix convex set W^\textrmmin(K), and a maximal matrix convex set W^\textrmmax(K), which have K as their ground level. The main question treated in this paper is: under what conditions on a given pair of convex bodies K,L ⊆ ℝd does W^\textrmmax(K) ⊆ W^\textrmmin(L) hold? For a convex body K, we aim to find the optimal constant θ(K) such that W^\textrmmax(K) ⊆ θ(K) ⋅ W^\textrmmin(K); we achieve this goal for all the ℓp unit balls, as well as for other sets. For example, if \mathbbBp,d is the closed unit ball in ℝd with the ℓp norm, then θ(\mathbbBp,d) = d1-|1/p - 1/2|. This constant is sharp, and it is new for all p ≠ 2. Moreover, for some sets K we find a minimal set L for which W^\textrmmax(K) ⊆ W^\textrmmin(L). In particular, we obtain that a convex body K satisfies W^\textrmmax(K) = W^\textrmmin(K) if and only if K is a simplex. These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. We discuss and exploit these connections as well. For example, our results show that every d-tuple of self-adjoint operators of norm less than or equal to 1, can be dilated to a commuting family of self-adjoints, each of norm at most √(d). We also introduce new explicit constructions of these (and other) dilations.