Minimal and maximal matrix convex sets
by Orr Shalit
The final version of the paper Minimal and maximal matrix convex sets, written by Ben Passer, Baruch Solel and myself, has recently appeared online. The publisher (Elsevier) sent us a link through which the official final version is downloadable, for anyone who clicks on the following link before May 26, 2018. Here is the link for the use of the public:
Of course, if you don’t click by May 26 – don’t panic! We always put our papers on the arXiv, and here is the link to that. Here is the abstract:
Abstract. For every convex body , there is a minimal matrix convex set , and a maximal matrix convex set , which have as their ground level. We aim to find the optimal constant such that . For example, if is the unit ball in with the -norm, then we find that
This constant is sharp, and it is new for all . Moreover, for some sets we find a minimal set for which . In particular, we obtain that a convex body satisfies only if is a simplex.
These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. For example, our results show that every -tuple of self-adjoint contractions, can be dilated to a commuting family of self-adjoints, each of norm at most . We also introduce new explicit constructions of these (and other) dilations.