Minimal and maximal matrix convex sets

Benjamin Passer, Orr Moshe Shalit, Baruch Solel

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

Abstract

For every convex body K⊆Rd, there is a minimal matrix convex set Wmin(K), and a maximal matrix convex set Wmax(K), which have K as their ground level. We aim to find the optimal constant θ(K) such that Wmax(K)⊆θ(K)⋅Wmin(K). For example, if B‾p,d is the unit ball in Rd with the ℓp norm, then we find that θ(B‾p,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 Wmax(K)⊆Wmin(L). In particular, we obtain that a convex body K satisfies Wmax(K)=Wmin(K) only if K is a simplex. These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. For example, our results show that every d-tuple of self-adjoint contractions, 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.

Original languageEnglish
Pages (from-to)3197-3253
Number of pages57
JournalJournal of Functional Analysis
Volume274
Issue number11
DOIs
StatePublished - 1 Jun 2018
Externally publishedYes

Keywords

  • Abstract operator system
  • Dilation
  • Matrix convex set
  • Matrix range

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