Analysis Seminar: "Matrix convexity, Choquet boundaries and Tsirelson problems"

Abstract: Following work of Evert, Helton, Klep and McCullough on free LMI 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, and some matrix convex sets can fail to have any finite-dimensional Choquet points. The general problem of determining whether a given matrix convex set has this property turns out to be difficult because for certain correlation sets studied by Tsirelson we show that a positive answer is equivalent to 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

Host: John McCarthy