QLunch: Matrix convexity, Choquet boundaries and Tsirelson problems – University of Copenhagen

QMath > Events > Quantum Lunch > w44 Adam Dor-On

QLunch: Matrix convexity, Choquet boundaries and Tsirelson problems

Speaker: Adam Dor-On

Title: Matrix convexity, Choquet boundaries and Tsirelson problems

Abstract: Following work of Evert, Helton, Klep and McCullough on free linear matrix inequality domains, we ask when a matrix convex set is the closed convex hull of its (finite dimensional) Choquet points. This is a finite-dimensional version of Arveson's non-commutative Krein-Milman theorem, which may generally fail completely since some matrix convex sets fail to have any (finite dimensional) Choquet points. We show that the finite dimensional Arveson-Krein-Milman property for a given matrix convex set is difficult to determine. More precisely, based on works of Junge et. al., Fritz and Ozawa, we show that for commuting correlation sets studied by Tsirelson, the finite dimensional Arveson-Krein-Milman property is equivalent to Connes' embedding conjecture. We do more than just provide another equivalent formulation of Connes' embedding conjecture. Our approach provides new geometric variants of Tsirelson type problems for pairs of convex polytopes, which may be easier to rule out than the original Tsirelson problems.

Based on joint work with Roy Araiza and Thomas Sinclair.